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The sequence T_{3,1}(n,3).
1

%I #18 Jun 22 2024 18:20:25

%S 0,1,2,3,7,20,51,121,290,711,1747,4268,10407,25405,62066,151611,

%T 370255,904196,2208267,5393233,13171682,32168415,78563131,191870876,

%U 468596895,1144430581,2794984562,6826049523,16670917207,40714541300

%N The sequence T_{3,1}(n,3).

%D Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

%F a(n) = Sum_{j=0..n-1} Sum_{k=0..n-j-1} binomial(n - j - 1, 3*k)*3^k. a(n+1) = a(n) + A097122(n). - _Detlef Meya_, Jun 22 2024

%t a[n_] := Sum[Sum[Binomial[n - j - 1, 3*k]*3^k, {k,0,n-j-1}], {j,0,n-1}]; Table[a[n], {n,0,29}] (* _Detlef Meya_, Jun 22 2024 *)

%Y Cf. A097122, A372077.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 17 2024

%E a(11) and beyond from _Detlef Meya_, Jun 22 2024