A159290 A generalized Jacobsthal sequence.

3, 5, 13, 25, 53, 105, 213, 425, 853, 1705, 3413, 6825, 13653, 27305, 54613, 109225, 218453, 436905, 873813, 1747625, 3495253, 6990505, 13981013, 27962025, 55924053, 111848105, 223696213, 447392425, 894784853, 1789569705, 3579139413

Offset: 0

Author

Creighton Dement, Apr 08 2009

Keywords

• easy
• nonn

Comments

Sequence generated by the floretion: X*Y with X = 0.5('i + 'j + 'k + 'ee') and Y = 0.5(i' + j' + k' + 'ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj' + 'ee')

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Creighton Dement, Online Floretion Multiplier [broken link]
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

A083943, A068156

Programs

Magma

[-1 + (2*(-1)^n + 5*2^(n+1))/3: n in [0..50]]; // G. C. Greubel, Jun 27 2018

Mathematica

LinearRecurrence[{2, 1, -2}, {3, 5, 13}, 50] (* or *) Table[-1 + (2*(-1)^n + 5*2^(n+1))/3, {n,0,30}] (* G. C. Greubel, Jun 27 2018 *)

PARI

x='x+O('x^50); Vec((3-x)/(-x^2+1-2*x+2*x^3)) \\ G. C. Greubel, Jun 27 2018

Formula

a(n) = -1 + (2*(-1)^n + 5*2^(n+1))/3.

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

a(n) = 3*A000975(n+1) - A000975(n). - R. J. Mathar, Sep 11 2019

a(n)+a(n+1) = A051633(n+1). - R. J. Mathar, Mar 23 2023