A107906 Expansion of (1+8*x)/(1-16*x^2).

1, 8, 16, 128, 256, 2048, 4096, 32768, 65536, 524288, 1048576, 8388608, 16777216, 134217728, 268435456, 2147483648, 4294967296, 34359738368, 68719476736, 549755813888, 1099511627776, 8796093022208, 17592186044416, 140737488355328, 281474976710656, 2251799813685248

Offset

0,2

Comments

Fifth binomial transform is A096053.

Links

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (0,16).

Formula

a(n) = ((1+sqrt(4))*(2*sqrt(4))^n + (1-sqrt(4))*(-2*sqrt(4))^n)/2;

a(n) = 3*4^n/2 - (-4)^n/2.

a(2n) = 16^n, a(2n+1) = 8*16^n.

From Amiram Eldar, Jan 25 2026: (Start)

a(n) = 2^A014601(n).

E.g.f.: (3*exp(4*x) - exp(-4*x)) / 2.

Sum_{n>=0} 1/a(n) = 6/5.

Sum_{n>=0} (-1)^n/a(n) = 14/15. (End)

Mathematica

LinearRecurrence[{0, 16}, {1, 8}, 27] (* Amiram Eldar, Jan 25 2026 *)

Crossrefs

Cf. A014601, A096053.

Sequence in context: A264478 A277364 A278312 * A328127 A323385 A061359

Adjacent sequences: A107903 A107904 A107905 * A107907 A107908 A107909

Keyword

easy,nonn

Author

Paul Barry, May 27 2005

Extensions

More terms from Amiram Eldar, Jan 25 2026

Status

approved