OFFSET
0,3
COMMENTS
A row of A099233.
Row sums of number triangle A116089. - Paul Barry, Feb 04 2006
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2382
Hùng Việt Chu, Yubo Geng, Julian King, Steven J. Miller, Garrett Tresch, and Zachary Louis Vasseur, Linear Recurrences from Counting Schreier-Type Multisets, Integers 26 (2026), Article A53. See p. 4. See also arXiv:2509.05158 [math.CO], 2025. See p. 3.
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 24A (2024), Art. No. A7. See p. 3.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (1,3,3,1).
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
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 *)
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Paul Barry, Oct 08 2004
STATUS
approved
