0,5

Also 2n-bead balanced binary necklaces of fundamental period 2n that are equivalent to their complements; binary Lyndon words of length n with an odd number of 1's; number of binary irreducible polynomials of degree n having trace 1.

Also number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 1 (mod n+1) = size of Varshamov-Tenengolts code VT_1(n).

The trace of a polynomial of degree n is the coefficient of x^(n-1); the subtrace is the coefficient of x^(n-2).

Also number of binary Lyndon words with trace 1 over GF(2).

Number of self-reciprocal polynomials of degree 2n over GF(2).

Contribution from Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009: (Start)

Also the number of dynamical cycles of period 2n of a threshold Boolean automata

network which is a quasi-minimal negative circuit of size nq where q is odd and

which is updated in parallel. (End)

Also the number of 3-elements orbits of the symmetric group S3 action on irreducible polynomials of degree 2n, n>1, over GF(2). [From Jean-Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Oct 04 2009]

L. Carlitz, A theorem of Dickson on irreducible polynomials. Proc. Amer. Math. Soc. 3, (1952). 693-700.

J. Demongeot, M. Noual and S. Sene, On the number of attractors of positive and negative threshold Boolean automata circuits", preprint (2009) [From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009]

N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.

R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.

H. Kawakami, Table of rotation sequences of x_{n+1} = x_n^2 - lambda, pp. 73-92 of G. Ikegami, Editor, Dynamical Systems and Nonlinear Oscillations, Vol. 1, World Scientific, 1986.

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.

N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44; reprinted in P. Cvitanovic, ed., Universality in Chaos, Hilger, Bristol, 1986, pp. 187-206.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200

Joerg Arndt, Fxtbook

H. Meyn and W. G\"otz, Self-reciprocal polynomials over finite fields

F. Ruskey, Number of q-ary Lyndon words with given trace mod q

F. Ruskey, Number of Lyndon words of given trace

N. J. A. Sloane, On single-deletion-correcting codes

J.-Y. Thibon, The cycle enumerator of unimodal permutations.

Index entries for "core" sequences

Index entries for sequences related to Lyndon words

Index entries for sequences related to subset sums modulo m

(Sum_{odd d divides n } mu(d)*2^(n/d)) / (2n).

a(5) = 3 corresponding to the necklaces 00001, 00111, 01011; a(6) = 5 from 000001, 000011, 000101, 000111, 001011.

with(numtheory); A000048 := proc(n) local d, t1; if n = 0 then RETURN(1) else t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+mobius(d)*2^(n/d)/(2*n); fi; od; RETURN(t1); fi; end;

(PARI) A000048(n) = sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n) [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 09 2009]

Like A000013, but primitive necklaces. Half of A064355.

Equals A042981 + A042982. Cf. A002823, A000016, A053633, A051841, A001037, A002075, A002076.

Cf. A001037 [From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009]

Sequence in context: A143961 A128023 A056303 this_sequence A074099 A006788 A054650

Adjacent sequences: A000045 A000046 A000047 this_sequence A000049 A000050 A000051

nonn,core,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Frank Ruskey (fruskey(AT)cs.uvic.ca), Dec 13 1999

0,5

Trees are unrooted, nodes are unlabeled. Every node has degree <= 4.

Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000628 for the analogous sequence with stereoisomers counted.

In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The degree of each node is then <= 4.

K. Adam, Title?, MNU 1983, 36, 29 (in German).

M. van Almsick, H. Dolhaine and H. Honig, Efficient algorithms to enumerate isomers and diamutamers with more than one type of substituent, J. Chem. Info. and Computer Science, 40 (2000), 956-966.

A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (No. 4, 1988), 304-313.

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.

L. Bytautas and D. J. Klein, Chemical combinatorics for alkane-isomer enumeration and more, J. Chem. Inf. Comput. Sci., 38 (1998), 1063-1078.

A. Cayley, Ueber die analytischen Figuren, welche in der Mathematik Baeume genannt werden..., Chem. Ber. 8 (1875), 1056-1059.

R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278-281.

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.

Handbook of Combinatorics, North-Holland '95, p. 1963.

F. Harary and R. Z. Norman, Dissimilarity characteristic theorems for graphs, Proc. Amer. Math. Soc. 11 (1960), 332-334.

J. B. Hendrickson and C. A. Parks, "Generation and Enumeration of Carbon skeletons", J. Chem. Inf. Comput. Sci, vol. 31 (1991) pp. 101-107. See Table 2, column 2 on page 103.

H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.

H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-686.

M. D. Jackson and T. I. Bieber, Applications of degree distribution, 2: construction and enumeration of isomers in the alkane series, J. Chem. Info. and Computer Science, 33 (1993), 701-708.

E. V. Konstantinova and M. V. Vidyuk, Discriminating tests of information and toplogical indices; animals and trees; J. Chem. Inf. Comput. Sci., 43 (2003), 1860-1871.

J. Lederberg et al., Applications of artificial intelligence for chemical systems, I: The number of possible organic compounds. Acyclic structures containing C, H, O and N, J. Amer. Chem. Soc., 91 (1969), 2973-2097.

P. Leroux and B. Miloudi, ``G\'{e}n\'{e}ralisations de la formule d'Otter,'' Ann. Sci. Math. Qu\'{e}bec, Vol. 16, No. 1, pp. 53-80, 1992.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

L. M. Masinter, Applications of artificial intelligence for chemical systems, XX, Exhaustive generation of cyclic and acyclic isomers, J. Amer. Chem. Soc., 96 (1974), 7702-7714.

W. R. Mueller et al., Molecular topological index, J. Chem. Inf. Comput. Sci., 30 (1990),160-163.

D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly - compare first link below]

M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443.

R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 28.

R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes..., Tetrahedron 32 (1976), 355-361.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marten J. ten Hoor, Formula for Success?, Education in Chemistry, 2005, 42(1), 10.

N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, COMPUTER GENERATION OF ISOMERIC STRUCTURES, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-39O, 1983.

S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb., 2006.

N. J. A. Sloane, Table of n, a(n) for n = 0..60

R. Aringhieri, P. Hansen and F. Malucelli, Chemical Tree Enumeration Algorithms, Report TR-99-09, Dept. Informatica, Univ. Pisa, 1999.

H. Bottomley, Illustration of initial terms of A000022, A000200, A000602

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478

Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. Daylight CIS, Inc., EuroMUG '04;4-5 Nov, 2004.

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees)., J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.

N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678

Index entries for sequences related to trees

Index entries for "core" sequences

a(n) = A010372(n) + A010373(n/2) for n even, a(n) = A010372(n) for n odd.

Also equals A000022 + A000200 (n>0), both of which have known generating functions. Also g.f. = A000678(x)-A000599(x)+A000598(x^2) = (x+x^2+2x^3...)-(x^2+x^3+3x^4...)+(1+x^2+x^4+...) = 1+x+x^2+x^3+2x^4+3x^5...

a(6)=5 because hexane has five isomers: n-hexane, 2-methylhexane, 3-methylhexane, 2,2-dimethylhexane, 2,3-dimethylhexane. - Michael Lugo (mtlugo(AT)mit.edu), Mar 15 2003

A000602 := proc(n) if n=0 then RETURN(1) else A000022(n)+A000200(n); fi; end;

Cf. A000598, A000625, A000628, A067608-A067610.

Sequence in context: A047031 A056766 A080091 this_sequence A034790 A047121 A096753

Adjacent sequences: A000599 A000600 A000601 this_sequence A000603 A000604 A000605

nonn,easy,core,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20, 2003.

Kappler reference from Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 15 2005

0,4

The entries are restricted to 0 and 1; the determinant is computed in the field of real numbers.

Suppose M = (m(i,j)) is an n X n matrix of real numbers. Let

a(n) = max det M subject to m(i,j) = 0 or 1 [this sequence],

g(n) = max det M subject to m(i,j) = -1 or 1 [A003433],

h(n) = max det M subject to m(i,j) = -1, 0 or 1 [A003433],

F(n) = max det M subject to 0 <= m(i,j) <= 1 [this sequence],

G(n) = max det M subject to -1 <= m(i,j) <= 1 [A003433].

Then a(n) = F(n), g(n) = h(n) = G(n), g(n) = 2^(n-1)*a(n-1). Thus all five problems are equivalent.

Hadamard proved that a(n) <= 2^(-n)*(n+1)^((n+1)/2), with equality if and only if a Hadamard matrix of order n+1 exists. Equivalently, g(n) <= n^(n/2), with equality if and only if a Hadamard matrix of order n exists. It is believed that a Hadamard matrix of order n exists if and only if n = 1, 2 or a multiple of 4 (see A036297).

J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626-630.

H. Ehlich, Determinantenabschaetzungen fuer binaere Matrizen, Math. Z. 83 (1964), 123-132.

H. Ehlich and K. Zeller, Binaere Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 20-21.

J. Hadamard, R\'{e}solution d'une question relative aux d\'{e}terminants, Bull. des Sciences Math. 2 (1893), 240-246.

Hudelson, Matthew; Klee, Victor and Larman, David, Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem. Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 519-598.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 54.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.

C. Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.

W. P. Orrick, The maximal {-1, 1}-determinant of order 15.

W. P. Orrick and B. Solomon, Large determinant sign matrices of order 4k+1

W. P. Orrick and B. Solomon, The Hadamard Maximal Determinant Problem (website)

W. P. Orrick, B. Solomon, R. Dowdeswell and W. D. Smith, New lower bounds for the maximal determinant problem

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, (0, 1)-Matrix

Eric Weisstein's World of Mathematics, (-1, 0, 1)-Matrix

Index entries for sequences related to binary matrices

One of 2 ways to get determinant 9 with a 6 X 6 matrix, found by Williamson:

100110

001111

111001

010101

010011

011110

A003433(n) = 2^(n-1)*a(n-1). Cf. A013588, A036297, A051752.

Sequence in context: A105180 A094206 A118998 this_sequence A081938 A129500 A109204

Adjacent sequences: A003429 A003430 A003431 this_sequence A003433 A003434 A003435

nonn,hard,nice

N. J. A. Sloane (njas(AT)research.att.com).

For a(18) through a(22) we have a(18) = 3411968?, a(19) = 19531250, a(20) = 56640625, a(21) = 195312500?, a(22) = 662671875?. See the Orrick-Solomon web site for further information.

Entry revised Jan 18 2004.

0,3

f[n_]:=First[IntegerDigits[n!, 13]]; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 2*5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 14 2009]

Cf. A000142, A136699, A136754, A136755, A136756, A136757, A136758, A136759, A136760, A008905, A136761, A136762, A136764, A136765, A136766.

Sequence in context: A136765 A011041 A100831 this_sequence A109530 A111519 A008855

Adjacent sequences: A136760 A136761 A136762 this_sequence A136764 A136765 A136766

base,easy,nonn

Carl R. White (oeisfan(AT)phodd.net), Jan 21 2008

1,4

S. Even and A. Lempel, Generation and enumeration of all solutions of the characteristic sum condition. Information and Control 21 (1972), 476-482.

P. Flajolet and H. Prodinger, Level number sequences for trees, Discrete Math., 65 (1987), 149-156.

E. N. Gilbert, Codes based on inaccurate source probabilities, IEEE Trans. Inform. Theory, 17 (1971), 304-315.

H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid. Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

E. Norwood, The Number of Different Possible Compact Codes, IEEE Transactions on Information Theory, Vol. 13, P. 614, 1967.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. R. Stein, personal communication.

T. D. Noe, Table of n, a(n) for n = 1..201

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 200

Index entries for "core" sequences

Index entries for sequences related to trees

Index entries for sequences related to rooted trees

Math. Rev. 22 #11020, Minc, H. A problem in partitions ... 1959: v(c, d) is the number of partitions of d into positive integers of the form d = c + c_1 + c_2 + ... + c_n, where c_1 <= 2*c, c_{i+1} <= 2*c_i.

{1}; {1/2 + 1/2}; { 1/2 + 1/4 + 1/4 }; { 1/2 + 1/4 + 1/8 + 1/8, 1/4 + 1/4 + 1/4 + 1/4 }; ...

v := proc(c, d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i, d-c), i=1..2*c); fi; end; [ seq(v(1, n), n=1..50) ];

(PARI) v(c, d) = if ( d<0 | c<0, 0, if ( d==c, 1, sum(i=1, 2*c, v(i, d-c) ) ) )

Cf. A002573, A047913, A002574, A049284, A049285, A007178.

Sequence in context: A005314 A099529 A088352 this_sequence A114834 A143961 A128023

Adjacent sequences: A002569 A002570 A002571 this_sequence A002573 A002574 A002575

core,nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

0,4

A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row.

a(n)=number of Dyck paths for which the sum of the heights of the vertices that terminate an upstep (i.e. peaks and doublerises) is n. Example: a(4)=3 beacuse we have UDUUDD, UUDDUD and UDUDUDUD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

a(n)=number of ordered trees with path length n (follows from previous comment via a standard bijection). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

Row sums of A047998. Column sums of A138158. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

Probably first studied by Jim Propp (unpublished).

Glasser, M. L.; Privman, V.; Svrakic, N. M.; Temperley's triangular lattice compact cluster model: exact solution in terms of the $q$ series. J. Phys. A 20 (1987), no. 18, L1275-L1280.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..500

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.

A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 10.7 (pdf, ps)

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 331

A005169(n) = f(n, 1), where f(n, p) = 0 if p > n, 1 if p = n, Sum(1 <= q <= p+1; f(n-p, q)) if p < n.

G.f.=F(t)=Sum(P[k],k=0..infinity), where P[0]=1, P[n]=t*sum(P[j]*P[n-j-1]*t^(n-j-1),j= 0..n-1) for n>=1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

An example of a fountain with 19 coins:

... O . O O

.. O O O O O O . O

. O O O O O O O O O

P[0]:=1: for n to 40 do P[n]:=sort(expand(t*(sum(P[j]*P[n-j-1]*t^(n-j-1), j= 0..n-1)))) end do: F:=sort(sum(P[k], k=0..40)): seq(coeff(F, t, j), j=0..36); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2008

Cf. A047998, A138158.

Sequence in context: A096816 A018157 A003065 this_sequence A129852 A065954 A067847

Adjacent sequences: A005166 A005167 A005168 this_sequence A005170 A005171 A005172

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net), Apr 30 2001

0,3

M. Elia and F. Neri, A note on addition chains ..., pp. 166-181 of R. M. Capocelli, ed., Sequences, Springer-Verlag, NY 1990.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 459.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

See A003313 for a much more extensive list of references and links.

Daniel Bleichenbacher, Efficiency and Security of Cryptosystems based on Number Theory. PhD Thesis, Diss. ETH No. 11404, Zuerich 1996. See p. 61.

Achim Flammenkamp, Shortest addition chains

a(6) = 15 because 15 numbers have shortest addition chains involving 6 additions. These numbers are 19,21,22,23,25,26,27,28,30,33,34,36,40,48,64.

Cf. A003064, A003313.

Cf. A114623 [Number of integers for which Knuth's power tree method produces an addition chain of length n].

Sequence in context: A114140 A096816 A018157 this_sequence A005169 A129852 A065954

Adjacent sequences: A003062 A003063 A003064 this_sequence A003066 A003067 A003068

nonn,nice,hard

N. J. A. Sloane (njas(AT)research.att.com), D. E. Knuth

Updated through a(28) from the Flammenkamp web site Feb 01 2005.

a(28) corrected from 6465773 to 6465774, based on information received from N. Clift (neillclift(AT)msn.com). - Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 29 2006

a(29)=11947258 and a(30)=22087489 computed by N. Clift (neillclift(AT)msn.com), Jun 15 2007

40886910, 75763102, 140588339 from N. Clift (neillclift(AT)msn.com), May 21 2008

1,4

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure. Identical to A000048 for n>1.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.

Sum mu(d)*A056295(n/d) where d divides n . Alternatively, A000048(n)-A000007(n-1)..

Cf. A001037.

Sequence in context: A114834 A143961 A128023 this_sequence A000048 A074099 A006788

Adjacent sequences: A056300 A056301 A056302 this_sequence A056304 A056305 A056306

nonn

Marks R. Nester (nesterm(AT)dpi.qld.gov.au)

0,3

Peter Boros (borospet(AT)freemail.hu): Lectures on Fibonacci's World at the SOTERIA Foundation, 1999.

P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 580.)

a(n)=a(n-1)+a(n-2)+a(n-5)+a(n-10), a(0)=1.

G.f. 1/(1-x-x^2-x^5-x^10). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 24 2006

a(4)=5 because 4=1+1+1+1=1+1+2=1+2+1=2+1+1=2+2: five possible exchange. a(15)=a(14)+a(13)+a(10)+a(5)=1883=1101+644+129+9.

a:= n-> (Matrix(10, (i, j)-> if i+1=j or j=1 and member (i, [1, 2, 5, 10]) then 1 else 0 fi)^n)[1, 1]: seq (a(n), n=0..35); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 07 2008]

Cf. A079971.

Sequence in context: A034073 A114623 A079971 this_sequence A114138 A114140 A096816

Adjacent sequences: A073028 A073029 A073030 this_sequence A073032 A073033 A073034

easy,nonn

Miklos Kristof (kristmikl(AT)freemail.hu), Aug 22 2002

1,3

Terms computed by Hugo Pfoertner; see entry A114622 for definition, references and links.

Cf. A114622 (power tree), A114624 (row sums), A114625 (left-most nodes in rows).

Sequence in context: A005517 A034063 A034073 this_sequence A079971 A073031 A114138

Adjacent sequences: A114620 A114621 A114622 this_sequence A114624 A114625 A114626

nonn

Hugo Pfoertner (hugo(AT)pfoertner.org) and Paul D. Hanna (pauldhanna(AT)juno.com), Dec 20 2005