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A113435 a(n) = a(n-1) + Sum_{k=0..n/3} a(n-3k) with a(0)=1. 7
1, 1, 1, 2, 3, 4, 7, 11, 16, 26, 41, 62, 98, 154, 237, 371, 581, 901, 1406, 2197, 3418, 5329, 8317, 12956, 20196, 31501, 49096, 76532, 119338, 186029, 289997, 452141, 704861, 1098826, 1713111, 2670692, 4163483, 6490879, 10119152, 15775426 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
If presented in three rows a(3n), a(3n+1) and a(3n+2) each term is the sum of the previous term in the sequence and the partial sum of its row.
LINKS
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19.
FORMULA
a(n) = a(n-1) + 2*a(n-3) - a(n-4) = 7*a(n-3) - 5*a(n-6) + 11*a(n-9) - a(n-12).
G.f.: (1-x^3)/(1-x-2*x^3+x^4).
G.f.: 1/(1-x) + x^3*Q(0)/(2-2*x) , where Q(k) = 1 + 1/(1 - x*(4*k+1 + 2*x^2 - x^3)/( x*(4*k+3 + 2*x^2 - x^3 ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 11 2013
MATHEMATICA
CoefficientList[Series[(1 - x^3)/(1 - x - 2*x^3 + x^4), {x, 0, 50}], x] (* G. C. Greubel, Mar 10 2017 *)
LinearRecurrence[{1, 0, 2, -1}, {1, 1, 1, 2}, 40] (* Harvey P. Dale, Dec 17 2023 *)
PROG
(PARI) x='x+O(x^50); Vec((1 - x^3)/(1 - x - 2*x^3 + x^4)) \\ G. C. Greubel, Mar 10 2017
CROSSREFS
Partial sums of A176848.
Sequence in context: A120415 A023361 A210518 * A367667 A222022 A025048
KEYWORD
nonn
AUTHOR
Floor van Lamoen, Nov 04 2005
STATUS
approved

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Last modified May 18 08:32 EDT 2024. Contains 372618 sequences. (Running on oeis4.)