1,2

The the Reingold-Tarjan sequence is based on the following function defined on even positive integers and range of the rational numbers:

f(2*n) = if n is even then 2*f(n)/3 else (f(n+1)+f(n-1))/3 for n>1, f(2*1)=1.

f(2*n) = a(n)/A072404(n) for n>1, a(1)=1 and A072404(1)=1.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences (Example 33), Theoretical Computer Science, 98 (1992), 163-197.

E. M. Reingold and R. E. Tarjan, On a greedy heuristic for complete matching, SIAM J. Computing 10 (1981), 676-681.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

Sequence in context: A114393 A100710 A069913 this_sequence A010078 A074639 A002314

Adjacent sequences: A072400 A072401 A072402 this_sequence A072404 A072405 A072406

nonn,frac

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 16 2002

1,2

The the Reingold-Tarjan sequence is based on the following function defined on even positive integers and range of the rational numbers:

f(2*n) = if n is even then 2*f(n)/3 else (f(n+1)+f(n-1))/3 for n>1, f(2*1)=1.

f(2*n) = A072403(n)/a(n) for n>1, A072403(1)=1 and a(1)=1.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences (Example 33), Theoretical Computer Science, 98 (1992), 163-197.

E. M. Reingold and R. E. Tarjan, On a greedy heuristic for complete matching, SIAM J. Computing 10 (1981), 676-681.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

Sequence in context: A143225 A099720 A162349 this_sequence A125824 A038227 A080292

Adjacent sequences: A072401 A072402 A072403 this_sequence A072405 A072406 A072407

nonn,frac

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 16 2002

0,3

Also the power of 2 in (2n)!.

Write n in binary: 1ab..yz, then a(n) = 1ab..yz + ... + 1ab + 1a + 1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 27 2003

Also numbers having a partition into distinct Mersenne numbers >0; A079559(a(n))=1; complement of A055938. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 18 2009]

Laurent Alonso; Edward M. Reingold; Ren\`e Schott, Determining the majority, Inform. Process. Lett. 47 (1993), no. 5, 253-255.

Laurent Alonso; Edward M. Reingold; Ren\`e Schott, The average-case complexity of determining the majority, SIAM J. Comput. 26 (1997), no. 1, 1-14.

Michael E. Saks; Michael Werman, On computing majority by comparisons. Combinatorica 11 (1991), no. 4, 383-387.

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..1000

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

For m>0, let q=[ log_2(m) ]; a(2m+1)=2^q+3m+sum([ (m-2^q)/2^k ], k=1..infinity); a(2m)=a(2m+1)-1 - Len Smiley (smiley(AT)math.uaa.alaska.edu)

a(n) = Sum{k >= 0}[ n/2^k ] = n+A011371(n). - Henry Bottomley (se16(AT)btinternet.com), Jul 03 2001

G.f.: A(x) = Sum(k=0, infinity, x^(2^k)/((1-x)*(1-x^(2^k)))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Dec 24 2002

a(n) = Sum(k = 1 through n) A001511(k) - number of 1's in binary representation of n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003

Conjecture : a(n)=2n+O(log(n)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 07 2003

Sum_{n, 2^k<=n<2^(k+1)} a(n) = 3*4^k - (k+4)*2^(k-1) = A085354(k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 19 2004

a(n)= A011371(n)+n, n>=0.

Recurrence: a(n)=n+a(floor(n/2)); a(2n)=2n+a(n); a(n*2^m)=2*n*(2^m-1)+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

a(2^m)=2^(m+1)-1, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

Asymptotic behavior: a(n)=2n+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

a(n)<=2n-1; equality holds for powers of 2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

a(n)>=2n-1-floor(log_2(n)); equality holds for n=2^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

lim inf (2n-a(n))=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

lim sup (2n-log_2(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

lim sup (a(n+1)-a(n)-log_2(n))=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007

a(n)=2n-A000120(n); - Paul Barry (pbarry(AT)wit.ie), Oct 26 2007

PURRS demo results: Bounds for a(n) = n+a(1/2*n) with initial conditions a(1) = 1: a(n) >= -2+2*n-log(n)*log(2)^(-1), a(n) <= 1+2*n for each n >= 1 - Alexander R. Povolotsky (pevnev(AT)juno.com), Apr 06 2008

If n = 2^a_1 + 2^a_2 + ... + 2^a_k, then a(n) = n-k. This can be used to prove that 2^n maximally divides (2n!)/n!. [From Jon Perry (johnandruth(AT)jrperry.orangehome.co.uk), Jul 16 2009]

a[0] = 0; a[n_] := a[n] = a[Floor[n/2]] + n; Table[ a[n], {n, 0, 50}] (* or *)

Table[IntegerExponent[(2n)!, 2], {n, 0, 65}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Apr 19 2006).

(PARI) a(n)=if(n<0, 0, valuation((2*n)!, 2))

(PARI) a(n)=if(n<0, 0, sum(k=1, n, (2*n)\2^k))

(PARI) {a(n) = if( n < 0, 0, 2*n - subst( Pol( binary( n ) ), x, 1) ) } /* Michael Somos Aug 28 2007 */

Cf. A001790, A011371, A001511(n) = a(n+1)-a(n), A046161(n)=2^a(n).

a(n)=A011371(2n+1)

Partial sums of A001511.

Cf. A067080, A098844, A132027, A004128, A054899.

Sequence in context: A075752 A046541 A047344 this_sequence A091934 A024515 A138971

Adjacent sequences: A005184 A005185 A005186 this_sequence A005188 A005189 A005190

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Allan Wilks (allan(AT)research.att.com)

More terms from Jud McCranie (j.mccranie(AT)comcast.net), Jul 04 2000

0,5

a(n) = A007814(A000142(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 09 2004

Entries of A005187 appearing twice. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 06 2004

This sequence shows why in binary 0 and 1 are the only two numbers n such that n equals the sum of its digits raised to the consecutive powers (equivalent to the base 10 sequence A032799). 1 raised to any consecutive power is still 1 and thus any the sum of digits raised to consecutive powers for any n > 1 falls short of equaling the value of n by the n-th number of this sequence. - Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 27 2004

Also the number of trailing zeros in the base-2 representation of n!. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007

Laurent Alonso, Edward M. Reingold and Ren\`e Schott, Determining the majority, Inform. Process. Lett. 47 (1993), no. 5, 253-255.

Laurent Alonso, Edward M. Reingold and Ren\`e Schott, The average-case complexity of determining the majority, SIAM J. Comput. 26 (1997), no. 1, 1-14.

K. Atanassov, On Some of Smarandache's Problems, section 7, on the 61-st problem, page 42, American Research Press, 1999, 16-21.

G. Bachman, Introduction to p-Adic Numbers and Valuation Theory, Academic Press, 1964; see Lemma 3.1.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 305.

Michael E. Saks and Michael Werman, On computing majority by comparisons. Combinatorica 11 (1991), no. 4, 383-387.

T. D. Noe, Table of n, a(n) for n=0..1000

K. Atanassov, On Some of Smarandache's Problems

K. Matthews, Computing the prime-power factorization of n!

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

a(n) =a([n/2])+[n/2] =[n/2]+[n/4]+[n/8]+[n/16]+... - Henry Bottomley (se16(AT)btinternet.com), Apr 24 2001

G.f.: A(x) = 1/(1-x)*Sum(k=1, infinity, x^(2^k)/(1-x^(2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 11 2002

a(n) = n - A000120(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 01 2003

a(n)=A005187(n)-n, n>=0.

a(n)=sum{2<=k<=n, sum{j|k,j>=2, floor(log_2(j))-floor(log_2(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007

The g.f. can be expressed in terms of a Lambert series, in that g(x)=L[b(k)](x)/(1-x) where L[b(k)](x)=sum{k>=0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k)=1, if k is a power of 2, else b(k)=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007

G.f.: g(x)=sum{k>0, c(k)*x^k}/(1-x), where c(k)=sum{j>1,j|k, floor(log_2(j))-floor(log_2(j-1))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007

Recurrence: a(n)=floor(n/2)+a(floor(n/2)); a(2*n)=n+a(n); a(n*2^m)=n*(2^m-1)+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(2^m)=2^m-1, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

Asymptotic behavior: a(n)=n+O(log(n)), a(n+1)-a(n)=O(log(n)), which follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(n)<=n-1; equality holds for powers of 2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(n)>=n-1-floor(log_2(n)); equality holds for n=2^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

lim inf (n-a(n))=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

lim sup (n-log_2(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

lim sup (a(n+1)-a(n)-log_2(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007

a(n) = RETURN(((2^(l))-1)+sum('(j*floor((n-(2^l)+2^j)/(2^(j+1))))', 'j'=1..l)); # after K. Atanassov. Here l is [ log2(n) ].

a := n -> n - add(i, i=convert(n, base, 2)) [From Peter Luschny (peter(AT)luschny.de), May 02 2009]

-1+Length[ Last[ Split[ IntegerDigits[ 2(n!), 2 ] ] ] ], FoldList[ Plus, 0, Fold[ Flatten[ {#1, #2, #1} ]&, 0, Range[ 6 ] ] ]

Table[IntegerExponent[n!, 2], {n, 0, 127}]; Table[n-DigitCount[n, 2, 1], {n, 0, 127}]

Table[t = 0; p = 2; While[s = Floor[n/p]; t = t + s; s > 0, p *= 2]; t, {n, 0, 100} ]

(PARI) a(n)=if(n<0, 0, valuation(n!, 2))

(PARI) a(n)=if(n<0, 0, sum(k=1, n, n\2^k))

(PARI) {a(n) = if( n < 0, 0, n - subst( Pol( binary( n ) ), x, 1) ) } /* Michael Somos Aug 28 2007 */

Cf. A000120, A005187, A054861.

a(n)=sum(A007814(k), k=1..n), n >= 1, a(0)=0.

Cf. A032799.

Cf. A067080, A098844, A132027.

Sequence in context: A025561 A145814 A007768 this_sequence A097355 A003860 A108216

Adjacent sequences: A011368 A011369 A011370 this_sequence A011372 A011373 A011374

nonn,nice,easy

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

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 02 2001

0,3

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

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

J.-P. Allouche & J. Shallit, Automatic sequences, Cambrige University Press, 2003, p. 94

E. N. Gilbert, Games of identification or convergence, SIAM Review, 4 (1962), 16-24.

R. L. Graham, On primitive graphs and optimal vertex assignments, pp. 170-186 of Internat. Conf. Combin. Math. (New York, 1970), Annals of the NY Academy of Sciences, Vol. 175, 1970.

Z. Li and E. M. Reingold, Solution of a divide-and-conquer maximin recurrence, SIAM J. Comput., 18 (1989), 1188-1200.

B. Lindstrom, On a combinatorial problem in number theory, Canad. Math. Bull., 8 (1965), 477-490.

R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.9. [From N. J. A. Sloane, Mar 12 2009]

T. D. Noe, Table of n, a(n) for n=0..1000

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

P. J. Grabner, H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Eric Weisstein's World of Mathematics, Binary

Index entries for sequences related to binary expansion of n

a(n)=sum(k=1, n, A000120(k)). - Benoit Cloitre, Dec 19, 2002

a(0) = 0, a(2n) = a(n)+a(n-1)+n, a(2n+1) = 2a(n)+n+1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 13 2003

a(n)=(1/2)*log2(n)*n + O(n); a(2^n)=n*2^(n-1)+1 - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 25 2003

a(n)=(1/2)*n*log(n)/log(2)+n*F(log(n)/log(2)) where F is a nowhere differentiable continuous function of period 1 (see Allouche & Shallit) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 08 2004

G.f.: 1/(1-x)^2 * Sum(k>=0, x^2^k/(1+x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 19 2003

The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.

Cf. A005183.

Sequence in context: A140206 A007818 A158618 this_sequence A053039 A027861 A062428

Adjacent sequences: A000785 A000786 A000787 this_sequence A000789 A000790 A000791

nonn,nice,easy

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

More terms from Larry Reeves (larryr(AT)acm.org), Jan 15 2001

0,4

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

Index entries for sequences related to Beatty sequences

G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3

a(n)=-1+Sum{k=0..n}{(1/8)*((k mod 4)+((k+1) mod 4)-((k+2) mod 4)+3*((k+3) mod 4)} [From Paolo P. Lava (ppl(AT)spl.at), Nov 17 2008]

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A086525 A120503 A083544 this_sequence A076539 A074184 A093700

Adjacent sequences: A057350 A057351 A057352 this_sequence A057354 A057355 A057356

nonn,easy

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

1,4

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

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

J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.

J. Freeman, A neural network solution to the n-queens problem, The Mathematica J., 3 (No. 3, 1993), 52-56.

M. Gardner, The Unexpected Hanging, pp. 190-2, Simon & Shuster NY 1969

Jieh Hsiang, Yuh-Pyng Shieh and Yao-Chiang Chen, The cyclic complete mappings counting problems, in Problems and Problem Sets for ATP, volume 02-10 of DIKU technical reports, G. Sutcliffe, J. Pelletier and C. Suttner, eds., 2002.

Kenji Kise, Takahiro Katagiri, Hiroki Honda and Toshitsugu Yuba: Solving the 24-queens Problem using MPI on a PC Cluster, Technical Report UEC-IS-2004-6, Graduate School of Information Systems, The University of Electro-Communications (2004)

I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.

M. A. Sainte-Lagu\"{e}, Les R\'{e}seaux (ou Graphes), M\'{e}morial des Sciences Math\'{e}matiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.

R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 91-94 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.

M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.

Amazing Mathematical Object Factory, Information on the n Queens problem [Link corrected by Gerry Myerson, Apr 08 2009]

Anonymous, N Queens Problem

D. Bill, Durango Bill's The N-Queens Problem [Broken link?]

Patrick GUILLEMIN, N-Queens Challenge [Broken link?]

Patrick GUILLEMIN, N-Queens Challenge [Broken link?]

Patrick GUILLEMIN, N-Queens Challenge [Broken link?]

Kenji KISE, 24-queens.

W. Kosters, n-Queens (Extensive Bibliography)

NQuens@home, Home Page

Objectweb ProActive INRIA Team, Home Page

Objectweb ProActive INRIA Team, Solve the N Queens challenge with ProActive !

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

Queens(AT)TUD project website. [From Thomas B. Preusser (thomas.preusser(AT)tu-dresden.de), Jul 11 2009]

Strong conjecture : there is a constant c around 2.54 such that a(n) is asymptotic to n!/c^n; weak conjecture : lim n -> infinity (1/n) * ln(n!/a(n)) = constant =0.90.... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

a(2) = a(3) = 0, since on 2 X 2 and 3 X 3 chessboards there are no solutions.

See A140393 for another version. Cf. A002562, A065256.

Sequence in context: A029673 A054790 A140393 this_sequence A038216 A145911 A027626

Adjacent sequences: A000167 A000168 A000169 this_sequence A000171 A000172 A000173

nonn,hard,nice

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

Terms for n=21-23 computed by Sylvain PION (Sylvain.Pion(AT)sophia.inria.fr) and Joel-Yann FOURRE (Joel-Yann.Fourre(AT)ens.fr).

a(24) from Kenji KISE (kis(AT)is.uec.ac.jp), Sep 01 2004

a(25) from Objectweb ProActive INRIA Team (proactive(AT)objectweb.org), Jun 11 2005 [Communicated by Alexandre Di Costanzo (Alexandre.Di_Costanzo(AT)sophia.inria.fr)]. This calculation took about 53 years of CPU time.

a(25) has been confirmed by the NTU 25Queen Project at National Taiwan University and Ming Chuan University, led by Yuh-Pyng (Arping) Shieh, Jul 26 2005. This computation took 26613 days CPU time.

Some of the links may be broken. I would appreciate receiving updates to them. - N. J. A. Sloane (njas(AT)research.att.com), May 01 2006

The NQueens-at-Home web site gives a different value for a(24), 226732487925864. Thanks to Goran Fagerstrom for pointing this out. I do not know which value is correct. I have therefore created a new entry, A140393, which gives the NQueens-at-home version of the sequence. - N. J. A. Sloane (njas(AT)research.att.com), Jun 18 2008

It now appears that this sequence (A000170) is correct and A140393 is wrong. - N. J. A. Sloane (njas(AT)research.att.com), Nov 08 2008

Added a(26) as calculated by Queens(AT)TUD [http://queens.inf.tu-dresden.de/]. Thomas B. Preusser (thomas.preusser(AT)tu-dresden.de), Jul 11 2009

0,6

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

G.f.: x^3(1 + x^3)/((1 - x)(1 - x^5)).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A061375 A029920 A100719 this_sequence A097508 A109964 A025778

Adjacent sequences: A057351 A057352 A057353 this_sequence A057355 A057356 A057357

nonn,easy

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

0,5

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

G.f.: (1+x^2+x^3)*x^2/((1-x)*(1-x^5)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

for all m>=0 a(5m)=0 mod 3; a(5m+1)=0 mod 3; a(5m+2)= 1 mod 3; a(5m+3) = 1 mod 3; a(5m+4) = 2 mod 3

a(n)=-1+Sum{k=0..n}{(1/50)*(3*(k mod 5)-7*((k+1) mod 5)+13*((k+2) mod 5)-7*((k+3) mod 5)+13*((k+4) mod 5)} [From Paolo P. Lava (ppl(AT)spl.at), Nov 17 2008]

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A156261 A071823 A139338 this_sequence A160511 A055930 A079952

Adjacent sequences: A057352 A057353 A057354 this_sequence A057356 A057357 A057358

nonn,easy

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)

0,8

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site

G.f.: x^4(1 + x^4)/((1 - x)(1 - x^7)).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Sequence in context: A034973 A066927 A060065 this_sequence A020913 A025787 A057810

Adjacent sequences: A057353 A057354 A057355 this_sequence A057357 A057358 A057359

nonn,easy

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu)