Search: seq:1,1,1,1,2,2,2,3,4,4,5,7,8,9,12,15,17
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A079398
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a(0)=0, a(1)=1, a(2)=1, a(3)=1, a(n) = a(n-3) + a(n-4) for n > 3.
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+30
24
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0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 7, 8, 9, 12, 15, 17, 21, 27, 32, 38, 48, 59, 70, 86, 107, 129, 156, 193, 236, 285, 349, 429, 521, 634, 778, 950, 1155, 1412, 1728, 2105, 2567, 3140, 3833, 4672, 5707, 6973, 8505, 10379, 12680, 15478, 18884, 23059, 28158, 34362
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OFFSET
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0,6
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COMMENTS
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P(0)=P(1)=P(2)=P(3)=1, for m > 3: P(m) = P(m-3) + P(m-4) is the 3rd sequence in the series: Fibonacci sequence, Padovan sequence, ... The Padovan sequence (whose ratio of successive terms approaches the plastic constant) is similar to the Perrin sequence. - Jonathan Vos Post, Jan 23 2005
Binomial transform yields A079398 without the initial (0,1,1,1). - R. J. Mathar, Apr 09 2008
a(n+1) corresponds to the diagonal sums of "triangle": 1; 1; 1; 1,1; 1,1; 1,1; 1,2,1; 1,2,1; 1,2,1; 1,3,3,1; 1,3,3,1; 1,3,3,1; 1,4,6,4,1; ..., rows of Pascal's triangle (A007318) repeated three times. - Philippe Deléham, Dec 13 2008
a(n) is the number of pairs of rabbits living at month n with the following rules: a pair of rabbits born in month n begins to procreate in month n + 3, procreates again in month n + 4, and dies at the end of this month (each pair therefore gives birth to 2 pairs); warning! The first pair is born in month 2. - Robert FERREOL, Oct 24 2017
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart. 44 (2006), no. 4, 335-340.
Eric Weisstein's World of Mathematics, Padovan Sequence.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1).
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FORMULA
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a(0)=0, a(1)=1, a(2)=1, a(3)=1, a(n) = a(n-3) + a(n-4) for n > 3. - Colin Barker, Sep 18 2013
From Paul Barry, Jul 06 2004: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(floor((n-k-1)/3), k)} (offset 0).
a(n) = Sum_{k=0..floor(n/2)} binomial(floor((n-k-1)/3), k)}-0^n (offset 0). (End)
For n > 1, a(n) = P(n-2) where P(n) is defined by: P(0)=P(1)=P(2)=P(3)=1, for m > 3: P(m) = P(m-3) + P(m-4). - Jonathan Vos Post, Jan 23 2005
The same sequence may be constructed as follows: Let M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}}; v[1] = {1, 1, 1, 1}; v[n] = M.v[n - 1]. Then a(n) = v[n][[1]]. - Roger L. Bagula, Sep 16 2006
O.g.f.: -x^2*(1+x+x^2)/(-1+x^3+x^4). a(n) = A017817(n-1) + A017817(n-2) + A017817(n-3). - R. J. Mathar, Apr 09 2008
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MATHEMATICA
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CoefficientList[Series[x (1 + x + x^2)/(1 - x^3 - x^4), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 16 2014 *)
LinearRecurrence[{0, 0, 1, 1}, {0, 1, 1, 1}, 60] (* Jean-François Alcover, Dec 05 2017 *)
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PROG
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(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 1, 1, 0, 0]^n*[0; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(PARI) x='x+O('x^50); concat([0], Vec(x*(1+x+x^2)/(1-x^3-x^4))) \\ G. C. Greubel, Apr 30 2017
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CROSSREFS
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Cf. A000931.
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KEYWORD
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nonn,easy
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AUTHOR
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Benoit Cloitre, Feb 16 2003
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EXTENSIONS
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Recurrence corrected by Colin Barker, Sep 18 2013
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STATUS
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approved
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1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 7, 8, 9, 12, 15, 17, 21, 27, 32, 37, 47, 58, 68, 82, 103, 124, 147, 181, 223, 266, 321, 396, 480, 575, 701, 858, 1033, 1248, 1525, 1852, 2232, 2712, 3305, 3998, 4836, 5886, 7148, 8644, 10487, 12752, 15453, 18713, 22731, 27596
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OFFSET
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1,5
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COMMENTS
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The triangular array (g(n,k)) in A274196 is defined as follows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,4) for n > 0, k > 1.
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LINKS
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Table of n, a(n) for n=1..55.
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MATHEMATICA
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g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 4 k]];
z = 300; w = Reverse[Table[g[z, k], {k, 0, z}]];
Take[w, z/3]
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CROSSREFS
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Cf. A274196, A274199, A274200.
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Jun 16 2016
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STATUS
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approved
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