%I A000931 M0284 N0102
%S A000931 1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,
%T A000931 200,265,351,465,616,816,1081,1432,1897,2513,3329,4410,5842,7739,
%U A000931 10252,13581,17991,23833,31572,41824,55405,73396,97229,128801,170625
%N A000931 Padovan sequence: a(n) = a(n-2) + a(n-3).
%C A000931 Number of compositions of n into parts congruent to 2 mod 3 (offset -1). 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 09 2005
%C A000931 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
%C A000931 Referred to as N0102 in R. K. Guy's "Anyone for Twopins". - Rainer Rosenthal 
               (r.rosenthal(AT)web.de), Dec 05 2006
%C A000931 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
%C A000931 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]
%C A000931 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]
%C A000931 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]
%C A000931 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]
%C A000931 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008: 
               (Start)
%C A000931 With offset 3: (1, 0, 1, 1, 1, 2, 2,...) convolved with the Tribonacci 
               numbers
%C A000931 prefaced with a "1": (1, 1, 1, 2, 4, 7, 13,...) = the Tribonacci numbers, 
               A000073. (Cf. triangle A153462). (End)
%C A000931 a(n+4)=Sum_{x=1..nth number of Padovan sequence}x [From Juri-Stepan Gerasimov 
               (2stepan(AT)rambler.ru), 17 Jul 2009]
%D A000931 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, p. 47, ex. 4.
%D A000931 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.
%D A000931 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.
%D A000931 T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 
               (1968), 13-21.
%D A000931 R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical 
               Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 10-11.
%D A000931 D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
%D A000931 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000931 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000931 I. Stewart, Math. Rec., Scientific American, No. 6, 1996 p 102.
%D A000931 I. Stewart, L'univers des nombres, "La sculpture et les nombres", pp. 
               19-20, Belin-Pour La Science, Paris 2000.
%D A000931 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.
%H A000931 T. D. Noe, <a href="b000931.txt">Table of n, a(n) for n=0..1000</a>
%H A000931 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000931 P. Chinn and S. Heubach, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Integer Sequences Related to Compositions without 
               2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000931 P. Flajolet and B. Salvy, <a href="http://www.expmath.org/expmath/volumes/
               7/7.html">Euler Sums and Contour Integral Representations</a>, Experimental 
               Mathematics Vol. 7 issue 1 (1998)
%H A000931 J. B. Gil, M. D. Weiner & C. Zara, <a href="http://arXiv.org/abs/math/
               0605348">Complete Padovan sequences in finite fields</a>
%H A000931 J. B. Gil, M. D. Weiner & C. Zara, <a href="http://math.aa.psu.edu/~juan/
               papers/padovan.pdf">Complete Padovan Sequences In Finite Fields</
               a>
%H A000931 Rachel Hall, <a href="http://www.sju.edu/~rhall/Rhythms/poets.pdf">Math 
               for Poets and Drummers</a>.
%H A000931 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=393">
               Encyclopedia of Combinatorial Structures 393</a>
%H A000931 I. Stewart, <a href="http://www.fortunecity.com/emachines/e11/86/padovan.html">
               Tales of a Neglected Number</a>
%H A000931 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000931 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000931 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PadovanSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A000931 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales 
               of Mt. Meru</a>
%H A000931 S. Zlobin, <a href="http://arXiv.org/abs/math/0601151">A note on arithmetic 
               properties of multiple zeta values</a>
%F A000931 G.f.: (1-x^2)/(1-x^2-x^3).
%F A000931 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
%F A000931 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).
%F A000931 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
%F A000931 G.f.: 1/(1-x^3-x^5-x^7-x^9-....) - Jon Perry (perry(AT)globalnet.co.uk), 
               Jul 04 2004
%F A000931 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
%F A000931 a(n)=sum{k=0..floor(n/2), binomial(k, n-2k)} - Paul Barry (pbarry(AT)wit.ie), 
               Sep 17 2004
%F A000931 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
%F A000931 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
%F A000931 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
%F A000931 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
%F A000931 a(n) = A023434(n-7)+1 for n>=7. - David Callan (callan(AT)stat.wisc.edu), 
               Jul 14 2006
%F A000931 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
%F A000931 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
%F A000931 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
%e A000931 When run backwards gives (-1)^n*A050935(n).
%e A000931 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]
%p A000931 A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 
               0 else A000931(n-2)+A000931(n-3); fi; end;
%p A000931 A000931:=-(1+z)/(-1+z^2+z^3); [S. Plouffe in his 1992 dissertation. Gives 
               sequence without five leading terms.]
%t A000931 CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 50}], x]
%t A000931 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)
%o A000931 (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]
%Y A000931 Essentially the same as (probably) A020720, A078027 and A096231.
%Y A000931 Cf. A020720, A078027, A001608, A096231.
%Y A000931 Cf. A103372-103380, A005682-A005691.
%Y A000931 Cf. A106510, A145462.
%Y A000931 A146973 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 03 2008]
%Y A000931 A153462, A000073 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 
               2008]
%Y A000931 Sequence in context: A018124 A124745 A133034 this_sequence A078027 A134816 
               A072493
%Y A000931 Adjacent sequences: A000928 A000929 A000930 this_sequence A000932 A000933 
               A000934
%K A000931 nonn,easy,nice
%O A000931 0,9
%A A000931 N. J. A. Sloane (njas(AT)research.att.com).
%E A000931 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009