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

%I A077835 #62 May 02 2025 14:29:27
%S A077835 1,2,6,18,52,152,444,1296,3784,11048,32256,94176,274960,802784,
%T A077835 2343840,6843168,19979584,58333184,170311872,497249280,1451788672,
%U A077835 4238699648,12375475200,36131927040,105492203776,307999212032,899246685696,2625476203008,7665444201472
%N A077835 Expansion of 1/(1 - 2*x - 2*x^2 - 2*x^3).
%C A077835 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
%C A077835 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
%C A077835 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
%H A077835 Michael De Vlieger, <a href="/A077835/b077835.txt">Table of n, a(n) for n = 0..2149</a>
%H A077835 Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A077835 Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H A077835 Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H A077835 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
%H A077835 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2).
%F A077835 a(n) = Sum_{k=0..n} T(n-k, k)*2^(n-k), T(n, k) = trinomial coefficients (A027907). - _Paul Barry_, Feb 15 2005
%F A077835 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
%F A077835 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3). - _Geoffrey Critzer_, Feb 07 2009
%t A077835 LinearRecurrence[{2, 2, 2}, {1, 2, 6}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jul 03 2011 *)
%t A077835 m={{2/3,1/3,0,0},{2/3,0,1/3,0},{2/3,0,0,1/3},{0,0,0,0}};
%t A077835 initialState={{1,0,0,0}};
%t A077835 Table[(initialState.MatrixPower[m,n])[[1,4]]*3^n,{n,3,31}] (* _Robert P. P. McKone_, Jul 29 2023 *)
%o A077835 (PARI) Vec(1/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%Y A077835 Cf. A071675.
%K A077835 nonn,easy
%O A077835 0,2
%A A077835 _N. J. A. Sloane_, Nov 17 2002