A168614 -id:A168614 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jan 18 2016

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.

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

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.

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

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

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

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

A376324 (1/6) times obverse convolution (4)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 7, 63, 819, 17199, 636363, 43909047, 5839903251, 1524214748511, 788019024980187, 810871576704612423, 1664719346974569304419, 6827014041942708717422319, 55961034101804383356710748843, 917145387894472038833132462787927

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000051, A010709, A168614, A374848, A375053, A375054.

s[n_] := 4; t[n_] := 2^n + 1;
u[n_] := (1/6) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2 - 1) * QPochhammer[-5, 1/2, n+1]/3, {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

a(n) = a(n-1)*A168614(n) for n>=1.

cp's OEIS Frontend

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

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