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A000931
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Padovan sequence: a(n) = a(n-2) + a(n-3) with a(0)=1, a(1)=a(2)=0.
(Formerly M0284 N0102)
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189
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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, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625
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listen;
history;
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internal format)
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OFFSET
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0,9
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COMMENTS
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Number of compositions of n into parts congruent to 2 mod 3 (offset -1). - Vladeta Jovovic, 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, Jul 14 2006
Referred to as N0102 in R. K. Guy's "Anyone for Twopins". - Rainer Rosenthal, 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, Oct 10 2008]
Triangle A145462: right border = A000931 starting with offset 6. Row sums = Padovan sequence starting with offset 7. [From Gary W. Adamson, 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, 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, Dec 12 2008]
Contribution from Gary W. Adamson, 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)
Contribution from Toby Gottfried, Mar 02 2010: a(n) is also the number of strings of length (n-8) from an alphabet {A, B} with no more than one A or 2 B's consecutively. (E.g., n = 4: {ABAB,ABBA,BABA,BABB,BBAB} and a(4+8)= 5)
p(n):=A000931(n+3), n>=1, is the number of partitions of the numbers {1,2,3,...,n} into lists of length two or three containing neighboring numbers. The 'or' is inclusive. For n=0 one takes p(0)=1. For details see the W. Lang link. There the explicit formula for p(n)(analogon of the Binet-de Moivre formula for Fibonacci numbers)is also given. Padovan sequences with different inputs are also considered there. [From Wolfdieter Lang, Jun 15 2010]
Equals the INVERTi transform of Fibonacci numbers prefaced with three 1's, i.e. (1 + x + x^2 + x^3 + x^4 + 2x^5 + 3x^6 + 5x^7 + 8x^8 + 13x^9 + ...) - Gary W. Adamson, Apr 1 2011.
When run backwards gives (-1)^n*A050935(n).
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REFERENCES
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J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
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.
A. Ilic, S. Klavzar and Y. Rho, Parity index of binary words and powers of prime words, http://www.fmf.uni-lj.si/~klavzar/preprints/BalancedFibo-submit.pdf, 2012. - From N. J. A. Sloane, Sep 27 2012
M. Janjic, Recurrence Relations and Determinants, Arxiv preprint arXiv:1112.2466, 2011
M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. - From N. J. A. Sloane, Sep 16 2012
D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.
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).
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.
M. Waldschmidt, Lectures on Multiple Zeta Values (IMSC 2011), http://www.math.jussieu.fr/~miw/articles/pdf/MZV2011IMSc.pdf
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
F. Brown, On the decomposition of motivic multiple zeta values
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)
D. Gerdemann Sums of Padovan numbers equal to sums of powers of plastic number (YouTube video)
D. Gerdemann Tuba Fantasy (music generated from Padovan numbers)
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
W. Lang: Padovan combinatorics, explicit formula, and sequences with various inputs. [From Wolfdieter Lang, Jun 15 2010]
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
I. Stewart, Tales of a Neglected Number
Eric Weisstein's World of Mathematics, Padovan Sequence
E. Wilson, The Scales of Mt. Meru
S. Zlobin, A note on arithmetic properties of multiple zeta values
Index to sequences with linear recurrences with constant coefficients, signature (0,1,1).
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FORMULA
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G.f.: (1-x^2)/(1-x^2-x^3).
a(n) is asymptotic to r^n / (2*r+3) where r = 1.3247179572447.. = A060006, the real root of x^3 = x + 1 . - Philippe Deléham, 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+5) = a(0) + a(1) + ... + a(n)
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, Feb 01 2004
G.f.: 1/(1-x^3-x^5-x^7-x^9-....) - Jon Perry, Jul 04 2004
a(n+4)=sum{k=0..floor((n-1)/2), binomial(floor((n+k-2)/3), k)}. - Paul Barry, Jul 06 2004
a(n)=sum{k=0..floor(n/2), binomial(k, n-2k)} - Paul Barry, 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, 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, Jan 30 2005
a(n+3)=sum{k=0..floor(n/2), C((n-k)/2, k)(1+(-1)^(n-k))/2}; - Paul Barry, 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, Feb 25 2005
a(n) = A023434(n-7)+1 for n>=7. - David Callan, 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, 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, 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. - Keith Schneider (schneidk(AT)email.unc.edu), Sep 07 2007
a(n)=-k*a(n-1)+a(n-2)+(k+1)a(n-2)+k*a(n-4),n> 3, for any value of k [From Gary Detlefs, Sep 13 2010]
Contribution from Francesco Daddi, Aug 04 2011: (Start)
a(0)+a(2)+a(4)+a(6)+...+a(2*n) = a(2*n+3).
a(0)+a(3)+a(6)+a(9)+...+a(3*n) = a(3*n+2)+1.
a(0)+a(5)+a(10)+a(15)+...+a(5*n) = a(5*n+1)+1.
a(0)+a(7)+a(14)+a(21)+...+a(7*n) = (a(7*n)+a(7*n+1)+1)/2. (End)
a(n+3) = sum{k=0..floor((n+1)/2), C((n+k)/3,k)}, where C((n+k)/3,k)=0 for noninteger (n+k)/3. - Nikita Gogin, Dec 07 2012
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EXAMPLE
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1 + x^3 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 4*x^11 + ...
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MAPLE
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A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else procname(n-2)+procname(n-3); fi; end;
A000931:=-(1+z)/(-1+z^2+z^3); [Simon Plouffe in his 1992 dissertation. Gives sequence without five leading terms.]
a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to 50 do a[n]:=a[n-2]+a[n-3]; end do; [From Francesco Daddi, Aug 04 2011]
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MATHEMATICA
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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, May 04 2006)
LinearRecurrence[{0, 1, 1}, {1, 0, 0}, 50] (* From Harvey P. Dale, Jan 10 2012 *)
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PROG
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(Haskell)
a000931 n = a000931_list !! n
a000931_list = 1 : 0 : 0 : zipWith (+) a000931_list (tail a000931_list)
-- Reinhard Zumkeller, Feb 10 2011
(PARI) Vec((1-x^2)/(1-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 11 2011
(PARI) {a(n) = if( n<0, polcoeff( 1 / (1 + x - x^3) + x * O(x^-n), -n), polcoeff( (1 - x^2) / (1 - x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Sep 18 2012 */
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CROSSREFS
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The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097 and probably A020720. However, each one has its own special features and deserves its own entry.
Closely related to A001608.
Cf. A103372-A103380, A005682-A005691, A106510, A145462, A146973, A153462, A000073
Sequence in context: A018124 A124745 A133034 * A078027 A134816 A182097
Adjacent sequences: A000928 A000929 A000930 * A000932 A000933 A000934
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Removed attribute "conjectured" from Simon Plouffe g.f R. J. Mathar, Mar 11 2009
Edited by Charles R Greathouse IV, Mar 17 2010
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STATUS
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approved
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