A123166 Row sums of A123162.

1, 2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417, 70368744177665, 281474976710657, 1125899906842625, 4503599627370497, 18014398509481985

Offset: 0

Author

Roger L. Bagula, Oct 02 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5, -4).

Crossrefs

Cf. A052539, A123162.

Programs

Magma

[0] cat [4^(n-1) +1: n in [1..40]]; // G. C. Greubel, May 31 2022

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n-1] od: seq(a[n]+sum((k), k=0..1), n=0..20); # Zerinvary Lajos, Mar 20 2008

Mathematica

A123162[n_, k_]= If [k==0, 1, Binomial[2*n-1, 2*k-1]];
Table[Sum[A123162[n, k], {k,0,n}], {n,0,30}]
Table[4^(n-1) +1 -Boole[n==0]/4, {n,0,40}] (* G. C. Greubel, May 31 2022 *)

SageMath

[4^(n-1) +1 -bool(n==0)/4 for n in (0..40)] # G. C. Greubel, May 31 2022

Formula

a(n) = 1 + Sum_{k=0..n} binomial(2*n-1, 2*k-1), for n > 0. - Paul Barry, May 26 2008

a(n) = A052539(n-1), n > 0. - R. J. Mathar, Jun 18 2008

From Sergei N. Gladkovskii, Dec 20 2011: (Start)

G.f.: (1 - 3*x - x^2)/((1-x)*(1-4*x)).

E.g.f.: (exp(4*x) + 4*exp(x) - 1)/4 = (G(0) - 1)/4; G(k) = 1 + 4/(4^k-x*16^k/(x*4^k+(k+1)/G(k+1))); (continued fraction). (End)

Extensions

Edited by N. J. A. Sloane, Oct 04 2006