A372076 The sequence T_{3,1}(n,3).

0, 1, 2, 3, 7, 20, 51, 121, 290, 711, 1747, 4268, 10407, 25405, 62066, 151611, 370255, 904196, 2208267, 5393233, 13171682, 32168415, 78563131, 191870876, 468596895, 1144430581, 2794984562, 6826049523, 16670917207, 40714541300

Offset

0,3

References

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.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,4).

Formula

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

G.f.: (x-x^2)/(1-3*x+3*x^2-4*x^3). - Georg Fischer, Apr 10 2025, from the reference, p. 108.

Mathematica

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 *)
LinearRecurrence[{3, -3, 4}, {0, 1, 2}, 40] (* Harvey P. Dale, Sep 11 2025 *)

Crossrefs

Cf. A097122, A372077.

Sequence in context: A293683 A006073 A052402 * A300041 A222939 A222867

Adjacent sequences: A372073 A372074 A372075 * A372077 A372078 A372079

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 17 2024

Extensions

a(11) and beyond from Detlef Meya, Jun 22 2024

Status

approved