0,9

Number of compositions of n into parts congruent to 2 mod 3 (offset -1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 09 2005

a(n) = number of compositions of n into parts that are odd and >=3. Example: a(10)=3 counts 3+7,5+5,7+3. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006

Referred to as N0102 in R. K. Guy's "Anyone for Twopins". - Rainer Rosenthal (r.rosenthal(AT)web.de), Dec 05 2006

Zagier conjectures that a(n+3) is the maximum number of multiple zeta values of weight n > 1 which are linearly independent over the rationals. - Jonathan Sondow and Sergey Zlobin (jsondow(AT)alumni.princeton.edu and sirg_zlobin(AT)mail.ru), Dec 20 2006

Starting with offset 6: (1, 1, 2, 2, 3, 4, 5,...) = INVERT transform of A106510: (1, 1, -1, 0, 1, -1, 0, 1, -1,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 10 2008]

Triangle A145462: right border = A000931 starting with offset 6. Row sums = Padovan sequence starting with offset 7. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 10 2008]

Starting with offset 3 = row sums of triangle A146973 and INVERT transform of [1, -1, 2, -2, 3, -3,...] [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 03 2008]

a(n+5) corresponds to the diagonal sums of "triangle" : 1 ; 1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,4,6,4,1 ; ..., rows of Pascal's triangle (A007318) repeated . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 12 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008: (Start)

With offset 3: (1, 0, 1, 1, 1, 2, 2,...) convolved with the Tribonacci numbers

prefaced with a "1": (1, 1, 1, 2, 4, 7, 13,...) = the Tribonacci numbers, A000073. (Cf. triangle A153462). (End)

a(n+4)=Sum_{x=1..nth number of Padovan sequence}x [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 17 Jul 2009]

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).

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 47, ex. 4.

Reinhardt Euler, The Fibonacci Number of a Grid Graph and a New Class of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.6.

Juan B. Gil, Michael D. Weiner and Catalin Zara, "Complete Padovan sequences in finite fields", The Fibonacci Quarterly, vol. 45 (Feb 2007 issue), pp. 64 - 75.

T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 (1968), 13-21.

R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 10-11.

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

I. Stewart, Math. Rec., Scientific American, No. 6, 1996 p 102.

I. Stewart, L'univers des nombres, "La sculpture et les nombres", pp. 19-20, Belin-Pour La Science, Paris 2000.

D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics (Paris, 1992), Vol. II, A. Joseph et. al. (eds.), Birkhaeuser, Basel, 1994, pp. 497-512.

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

Index entries for sequences related to linear recurrences with constant coefficients

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.

P. Flajolet and B. Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics Vol. 7 issue 1 (1998)

J. B. Gil, M. D. Weiner & C. Zara, Complete Padovan sequences in finite fields

J. B. Gil, M. D. Weiner & C. Zara, Complete Padovan Sequences In Finite Fields

Rachel Hall, Math for Poets and Drummers.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 393

I. Stewart, Tales of a Neglected Number

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

E. Wilson, The Scales of Mt. Meru

S. Zlobin, A note on arithmetic properties of multiple zeta values

G.f.: (1-x^2)/(1-x^2-x^3).

a(n) is asymptotic to r^n / (2*r+3) where r = 1.3247179572447..., the real root of x^3 = x + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 13 2004

a(n)^2+a(n+2)^2+a(n+6)^2 = a(n+1)^2+a(n+3)^2+a(n+4)^2+a(n+5)^2 (Barniville, Question 16884, Ed. Times 1911).

a(n) = central and lower right terms in the (n-3)-th power of the 3 X 3 matrix M = [0 1 0 / 0 0 1 / 1 1 0]. E.g. a(13) = 7. M^10 = [3 5 4 / 4 7 5 / 5 9 7]. - Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Feb 01 2004

G.f.: 1/(1-x^3-x^5-x^7-x^9-....) - Jon Perry (perry(AT)globalnet.co.uk), Jul 04 2004

a(n+4)=sum{k=0..floor((n-1)/2), binomial(floor((n+k-2)/3), k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004

a(n)=sum{k=0..floor(n/2), binomial(k, n-2k)} - Paul Barry (pbarry(AT)wit.ie), Sep 17 2004

a(n+3) is diagonal sum of A026729 (as a number triangle), with formula a(n+3)=sum{k=0..floor(n/2), sum{i=0..n-k, (-1)^(n-k+i)C(n-k, i)C(i+k, i-k)}} - Paul Barry (pbarry(AT)wit.ie), Sep 23 2004

a(n) = a(n-1)+a(n-5) = A003520(n-4)+A003520(n-13) = A003520(n-3)-A003520(n-9). - Henry Bottomley (se16(AT)btinternet.com), Jan 30 2005

a(n+3)=sum{k=0..floor(n/2), C((n-k)/2, k)(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Sep 09 2005

The sequence 1/(1-x^2-x^3) (a(n+3)) is given by the diagonal sums of the Riordan array (1/(1-x^3), x/(1-x^3)). The row sums are A000930. - Paul Barry (pbarry(AT)wit.ie), Feb 25 2005

a(n) = A023434(n-7)+1 for n>=7. - David Callan (callan(AT)stat.wisc.edu), Jul 14 2006

a(n+5) corresponds to the diagonal sums of A030528. The binomial transform of a(n+5) is A052921. a(n+5)=sum{k=0..floor(n/2), sum{k=0..n, (-1)^(n-k+i)C(n-k, i)C(i+k+1, 2k+1)}}. - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004

r^(n-1) = (1/r)*a(n) + r*(n+1) + a(n+2); where r = 1.32471... is the real root of x^3 - x - 1 = 0. Example: r^8 = (1/r)*a(9) + r*a(10) + a(11) = ((1/r)*2 + r*3 + 4 = 9.483909... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 22 2006

a(n) = (r^n)/(2r+3) + (s^n)/(2s+3) + (t^n)/(2t+3) where r, s, t are the three roots of x^3-x-1 in any order. - Keith Schneider (schneidk(AT)email.unc.edu), Sep 07 2007

When run backwards gives (-1)^n*A050935(n).

sage: taylor( mul(1-x/(1-(1-x^2)-(1-x^3)) for i in xrange(1,2)),x,0,47)# solution>> 1 + x + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 9*x^12 + 12*x^13 +.....+ 73396*x^44 + 97229*x^45 + 128801*x^46 + 170625*x^47+etc... if sage: taylor( mul((1-x^2)/(1-x^2-x^3) for i in xrange(1,2)),x,0,47)#(N. J. A. Sloane)then solution >>1 + x^3 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 4*x^11 + 5*x^12 + 7*x^13 + 9*x^14 + 12*x^15 +....+ 73396*x^46 + 97229*x^47 + 128801*x^48 +170625*x^49+etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2009]

A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else A000931(n-2)+A000931(n-3); fi; end;

A000931:=-(1+z)/(-1+z^2+z^3); [S. Plouffe in his 1992 dissertation. Gives sequence without five leading terms.]

CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 50}], x]

a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 2] + a[n - 3]; Table[a[n], {n, 0, 51}] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 04 2006)

(Other) sage: taylor( mul(1-x/(1-(1-x^2)-(1-x^3)) for i in xrange(1, 2)), x, 0, 47)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2009]

Essentially the same as (probably) A020720, A078027 and A096231.

Cf. A020720, A078027, A001608, A096231.

Cf. A103372-103380, A005682-A005691.

Cf. A106510, A145462.

A146973 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 03 2008]

A153462, A000073 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]

Adjacent sequences: A000928 A000929 A000930 this_sequence A000932 A000933 A000934

Sequence in context: A124745 A133034 A143074 this_sequence A078027 A134816 A072493

nonn,easy,nice

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

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009