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

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5, -4).

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)

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 *)

Prog

(Magma) [0] cat [4^(n-1) +1: n in [1..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

Crossrefs

Cf. A052539, A123162.

Sequence in context: A150012 A150013 A052539 * A008932 A167809 A262449

Adjacent sequences: A123163 A123164 A123165 * A123167 A123168 A123169

Keyword

nonn

Author

Roger L. Bagula, Oct 02 2006

Extensions

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

Status

approved