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A240607 a(n) = 2*a(n-2) + a(n-3) + a(n-4) for n>=4, a(n) = binomial(n,3) for n<4.

%I A240607 #37 Feb 09 2024 12:39:21
%S A240607 0,0,0,1,0,2,1,5,4,13,14,35,45,97,139,274,420,784,1253,2262,3710,6561,
%T A240607 10935,19094,32141,55684,94311,162603,276447,475201,809808,1389452,
%U A240607 2371264,4063913,6941788,11888542,20318753,34782785,59467836,101772865,174037210
%N A240607 a(n) = 2*a(n-2) + a(n-3) + a(n-4) for n>=4, a(n) = binomial(n,3) for n<4.
%C A240607 a(n) = term (4,1) in the 4 X 4 matrix [0,1,1,1; 1,0,1,0; 0,1,0,0; 0,0,1,0]^n.  There are 96 ways to define the sequence as an element of the n-th power of a 4 X 4 {0,1}-matrix.  These are listed in the second Gribble link.
%H A240607 Alois P. Heinz, <a href="/A240607/b240607.txt">Table of n, a(n) for n = 0..1000</a> (first 50 terms from Christopher Hunt Gribble)
%H A240607 Christopher Hunt Gribble, <a href="/A240607/a240607_1.txt">Matrix elements</a>
%H A240607 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,1).
%F A240607 G.f.: -x^3 / (x^4+x^3+2*x^2-1). - _Colin Barker_, Apr 20 2014
%p A240607 a:= proc(n) option remember; `if`(n<4, binomial(n, 3),
%p A240607        2*a(n-2) +a(n-3) +a(n-4))
%p A240607     end:
%p A240607 seq(a(n), n=0..50);
%p A240607 # second Maple program using the {0,1}-matrix:
%p A240607 a:= n-> (<<0|1|1|1>, <1|0|1|0>, <0|1|0|0>, <0|0|1|0>>^n)[4, 1]:
%p A240607 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 26 2014
%t A240607 LinearRecurrence[{0,2,1,1},{0,0,0,1,0,2,1,5,4},50] (* _Harvey P. Dale_, Jul 01 2015 *)
%o A240607 (PARI) concat([0,0,0], Vec(-x^3/(x^4+x^3+2*x^2-1) + O(x^100))) \\ _Colin Barker_, Apr 20 2014
%Y A240607 Cf. A239748.
%K A240607 nonn,easy
%O A240607 0,6
%A A240607 _Christopher Hunt Gribble_, Apr 09 2014
%E A240607 More terms from _Colin Barker_, Apr 20 2014