A077835 Expansion of 1/(1 - 2*x - 2*x^2 - 2*x^3).

1, 2, 6, 18, 52, 152, 444, 1296, 3784, 11048, 32256, 94176, 274960, 802784, 2343840, 6843168, 19979584, 58333184, 170311872, 497249280, 1451788672, 4238699648, 12375475200, 36131927040, 105492203776, 307999212032, 899246685696, 2625476203008, 7665444201472

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

a(n) is the number of ways two opposing basketball teams could score a combined total of n points (counting one point free throws, two point field goals, and three point field goals) considering the order of the scoring as important. - Geoffrey Critzer, Feb 07 2009

Number of permutations of length a(n+1) avoiding the partially ordered pattern (POP) {1>3, 4>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third element, and the fourth element is larger than the second element. - Sergey Kitaev, Dec 08 2020

a(n) is the number of compositions of n into parts 1, 3, and 3, each part of two kinds. - Joerg Arndt, Jul 30 2023

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2149
• Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
• Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
• Index entries for linear recurrences with constant coefficients, signature (2,2,2).

Crossrefs

Cf. A071675.

Programs

Mathematica

LinearRecurrence[{2, 2, 2}, {1, 2, 6}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
m={{2/3,1/3,0,0},{2/3,0,1/3,0},{2/3,0,0,1/3},{0,0,0,0}};
initialState={{1,0,0,0}};
Table[(initialState.MatrixPower[m,n])[[1,4]]*3^n,{n,3,31}] (* Robert P. P. McKone, Jul 29 2023 *)

PARI

Vec(1/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

Formula

a(n) = Sum_{k=0..n} T(n-k, k)*2^(n-k), T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005

a(n) = Sum_{k=0..n} 2^k * Sum_{i=0..floor((n-k)/2)} C(n-k-i, i)*C(k, n-k-i). - Paul Barry, Apr 26 2005

a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3). - Geoffrey Critzer, Feb 07 2009