A086224 a(n) = 7*2^n - 1.

6, 13, 27, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511, 234881023, 469762047, 939524095, 1879048191, 3758096383, 7516192767, 15032385535, 30064771071

Offset: 0

Author

Marco Matosic, Jul 27 2003

Keywords

• nonn
• easy

Comments

a(n) = A164874(n+2,2); subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-5). - Milan Janjic, Jan 27 2010

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3319
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-5, 14.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Other sequences with recurrence a(n+1) = 2*a(n) + 1 are:

a(0) = 2 gives A153893, a(0)=3 essentially A126646.

a(0) = 4 gives A153894, a(0)=5 essentially A153893.

a(0) = 7 gives essentially A000225.

a(0) = 8 gives A052996 except for some initial terms,

a(0) = 9 is essentially A153894.

a(0) = 10 gives A086225,

a(0) = 11 is essentially A153893.

a(0) = 13 is essentially A086224.

Cf. A030130, A052940, A164874.

Programs

Mathematica

7*2^Range[0,30]-1 (* Harvey P. Dale, May 09 2018 *)

PARI

a(n)=7<Charles R Greathouse IV, Sep 24 2015

Formula

a(n+1) = 2*a(n) + 1.

G.f.: (6-5*x)/((1-x)*(1-2*x)). - Jaume Oliver Lafont, Sep 14 2009

a(n-1)^2 + a(n) = (7*2^(n-1))^2. - Vincenzo Librandi, Aug 08 2010

a(n) = (A052940(n+1) + A000225(n+3))/2. - Gennady Eremin, Aug 31 2023

From Elmo R. Oliveira, Apr 22 2025: (Start)

E.g.f.: exp(x)*(7*exp(x) - 1).

a(n) = 3*a(n-1) - 2*a(n-2). (End)

Extensions

More terms from David Wasserman, Feb 22 2005