A077938 Expansion of 1/(1 - 2*x - x^2 - 2*x^3).
1, 2, 5, 14, 37, 98, 261, 694, 1845, 4906, 13045, 34686, 92229, 245234, 652069, 1733830, 4610197, 12258362, 32594581, 86667918, 230447141, 612751362, 1629285701, 4332217046, 11519222517, 30629233482, 81442123573, 216551925662, 575804441861, 1531045056530
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..2354
- Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
- Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; also arXiv preprint, arXiv:1302.2274 [math.CO], 2013.
- Index entries for linear recurrences with constant coefficients, signature (2,1,2)
Crossrefs
Absolute values of A077987.
Programs
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Mathematica
LinearRecurrence[{2, 1, 2}, {1, 2, 5}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *) CoefficientList[Series[1/(1-2x-x^2-2x^3),{x,0,30}],x] (* Harvey P. Dale, Jul 22 2024 *) -
Maxima
a(n):=sum(sum(binomial(j,n-3*k+2*j)*binomial(k,j)*2^(-n+4*k-2*j),j,0,k),k,1,n); /* Vladimir Kruchinin, May 05 2011 */
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PARI
Vec(1/(1-2*x-x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
Formula
a(n)=sum(k=1..n, sum(j=0..k, binomial(j,n-3*k+2*j)* binomial(k,j)*2^(-n+4*k-2*j))), n>0, a(0)=1. [Vladimir Kruchinin, May 05 2011]
a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3). [Stefan Schuster, Apr 24 2015]
Comments