A267615 a(n) = 2^n + 11.

12, 13, 15, 19, 27, 43, 75, 139, 267, 523, 1035, 2059, 4107, 8203, 16395, 32779, 65547, 131083, 262155, 524299, 1048587, 2097163, 4194315, 8388619, 16777227, 33554443, 67108875, 134217739, 268435467, 536870923, 1073741835, 2147483659, 4294967307, 8589934603, 17179869195, 34359738379

Offset: 0

Author

Ilya Gutkovskiy, Jan 18 2016

Keywords

• nonn
• easy

Comments

Recurrence relation b(n) = 3*b(n - 1) - 2*b(n - 2) for n>1, b(0) = k, b(1) = k + 1, gives the closed form b(n) = 2^n + k - 1.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. sequences with closed form 2^n + k - 1: A168616 (k=-4), A028399 (k=-3), A036563 (k=-2), A000918 (k=-1), A000225 (k=0), A000079 (k=1), A000051 (k=2), A052548 (k=3), A062709 (k=4), A140504 (k=5), A168614 (k=6), A153972 (k=7), A168415 (k=8), A242475 (k=9), A188165 (k=10), A246139 (k=11), this sequence (k=12).

Cf. A156940.

Programs

Magma

[2^n+11: n in [0..30]]; // Vincenzo Librandi, Jan 19 2016

Mathematica

Table[2^n + 11, {n, 0, 35}]
LinearRecurrence[{3, -2}, {12, 13}, 40] (* Vincenzo Librandi, Jan 19 2016 *)

PARI

a(n) = 2^n + 11; \\ Altug Alkan, Jan 18 2016

Formula

G.f.: (12 - 23*x)/(1 - 3*x + 2*x^2).

a(n) = 3*a(n - 1) - 2*a(n - 2) for n>1, a(0)=12, a(1)=13.

a(n) = A000079(n) + A010850(n).

Sum_{n>=0} 1/a(n) = 0.367971714327125...

Lim_{n->oo} a(n + 1)/a(n) = 2.

E.g.f.: exp(2*x) + 11*exp(x). - Elmo R. Oliveira, Nov 08 2023