A194455 a(n) = 2^n + 3n + 1.

2, 6, 11, 18, 29, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742, 4294967393, 8589934692, 17179869287

Offset: 0

Author

Bruno Berselli, Sep 01 2011

Keywords

• nonn
• easy

Comments

Inverse binomial transform of this sequence: 2,4,1,1 (1 continued).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000051, A005126, A176691; A120845.

Cf. A062709 (first differences), A000079 (second and successive differences).

Cf. A146529 (differences between alternate terms, for n>2).

Cf. A086653, A115067.

Programs

Magma

[2^n+3*n+1: n in [0..31]];

Mathematica

Table[2^n + 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{4,-5,2},{2,6,11},40] (* Harvey P. Dale, Oct 01 2014 *)

PARI

for(n=0, 31, print1(2^n+3*n+1", "));

Formula

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

a(n) = A086653(n) - 1 for n > 0.

Sum_{i=0..n} a(i) = A115067(n+1) + 2^(n+1).

a(n) = 3*a(n-1) - 2*a(n-2) - 3 for n > 1.

a(n)^2 = 2^(n+1)*(a(n-1) + 3) + (3*n + 1)^2 for n > 2.

E.g.f.: exp(x)*(1 + exp(x) + 3*x). - Stefano Spezia, May 06 2023