A103454 a(n) = 0^n + 4^n - 1.

1, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663

Offset: 0

Author

Paul Barry, Feb 06 2005

Keywords

• easy
• nonn

Comments

A transform of 4^n under the matrix A103452.

The square of the cotangent of the arcsin of 1/(2^n). - Al Hakanson (hawkuu(AT)excite.com), Feb 23 2006

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Cf. A024036, A103452.

Programs

Magma

[0^n+4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Mathematica

Table[Boole[n==0] +4^n -1, {n,0,40}] (* G. C. Greubel, Jun 21 2021 *)

Sage

[1]+[4^n -1 for n in [1..40]] # G. C. Greubel, Jun 21 2021

Formula

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

a(n) = Sum_{k=0..n} A103452(n, k)*4^k;

a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))4^k.

a(n) = A024036(n), n > 0. - R. J. Mathar, Aug 30 2008

E.g.f.: 1 - exp(x) + exp(4*x). - G. C. Greubel, Jun 21 2021

a(n) = 5*a(n-1) - 4*a(n-2). - Wesley Ivan Hurt, Mar 17 2023