A099234 A trisection of 1/(1-x-x^4).
1, 1, 4, 10, 26, 69, 181, 476, 1252, 3292, 8657, 22765, 59864, 157422, 413966, 1088589, 2862617, 7527704, 19795288, 52054840, 136886433, 359964521, 946583628, 2489191330, 6545722210, 17213011605, 45264335853, 119029728628
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..2382
- Hùng Việt Chu, Nurettin Irmak, Steven J. Miller, László Szalay, and Sindy Xin Zhang, Schreier Multisets and the s-step Fibonacci Sequences, arXiv:2304.05409 [math.CO], 2023. See also Integers (2024) Vol. 24A, Art. No. A7, p. 3.
- Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
- Index entries for linear recurrences with constant coefficients, signature (1,3,3,1).
Programs
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Mathematica
CoefficientList[Series[1/(1-x (1+x)^3),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,3,1},{1,1,4,10},30] (* Harvey P. Dale, Jun 05 2011 *)
Formula
G.f.: 1/(1-x*(1+x)^3).
a(n) = Sum_{k=0..n} binomial(3*(n-k), k).
a(n) = a(n-1)+3*a(n-2)+3*a(n-3)+a(n-4).
a(n) = A003269(3n).
a(n) = Sum_{k=0..n} C(3*k,n-k) = Sum_{k=0..n} C(n,k)*C(4*k,n)/C(4*k,k). - Paul Barry, Feb 04 2006
G.f.: 1/(G(0) - x) where G(k) = 1 - (2*k+3)*x/(2*k+1 - x*(k+2)*(2*k+1)/(x*(k+2) - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2012
Comments