A137256 - OEIS

%I #10 Jun 17 2024 22:45:38

%S 1,2,4,9,21,48,108,243,549,1242,2808,6345,14337,32400,73224,165483,

%T 373977,845154,1909980,4316409,9754749,22044960,49819860,112588947,

%U 254442141,575019162,1299497904,2936762649,6636851721,14998760928

%N Binomial transform of 2^n, 2^n, 2^n.

%C Sequence is identical to half its third differences.

%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.

%H G. C. Greubel, <a href="/A137256/b137256.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3).

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

%F O.g.f.: (1 -x +x^2)/(1 -3*x +3*x^2 -3*x^3). - _R. J. Mathar_, Apr 02 2008

%p m:=30; S:=series( (1-x+x^2)/(1-3*x+3*x^2-3*x^3), x, m+1):

%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Apr 10 2021

%t LinearRecurrence[{3, -3, 3},{1, 2, 4},30] (* _Ray Chandler_, Sep 23 2015 *)

%o (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

%o (Sage)

%o def A137256_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1-x+x^2)/(1-3*x+3*x^2-3*x^3) ).list()

%o A137256_list(30) # _G. C. Greubel_, Apr 10 2021

%Y Cf. A051164, A052103.

%K nonn

%O 0,2

%A _Paul Curtz_, Mar 11 2008

%E More terms from _R. J. Mathar_, Apr 02 2008