A137256 Binomial transform of 2^n, 2^n, 2^n.

1, 2, 4, 9, 21, 48, 108, 243, 549, 1242, 2808, 6345, 14337, 32400, 73224, 165483, 373977, 845154, 1909980, 4316409, 9754749, 22044960, 49819860, 112588947, 254442141, 575019162, 1299497904, 2936762649, 6636851721, 14998760928

Offset: 0

Author

Paul Curtz, Mar 11 2008

Keywords

• nonn

Comments

Sequence is identical to half its third differences.

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,3).

Crossrefs

Cf. A051164, A052103.

Programs

Magma

I:=[1,2,4]; [n le 3 select I[n] else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..30]]; // G. C. Greubel, Apr 10 2021

Maple

m:=30; S:=series( (1-x+x^2)/(1-3*x+3*x^2-3*x^3), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 10 2021

Mathematica

LinearRecurrence[{3, -3, 3},{1, 2, 4},30] (* Ray Chandler, Sep 23 2015 *)

Sage

def A137256_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2)/(1-3*x+3*x^2-3*x^3) ).list()
A137256_list(30) # G. C. Greubel, Apr 10 2021

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3).

O.g.f.: (1 -x +x^2)/(1 -3*x +3*x^2 -3*x^3). - R. J. Mathar, Apr 02 2008

Extensions

More terms from R. J. Mathar, Apr 02 2008