A102714 Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).

2, 5, 14, 36, 95, 248, 650, 1701, 4454, 11660, 30527, 79920, 209234, 547781, 1434110, 3754548, 9829535, 25734056, 67372634, 176383845, 461778902, 1208952860, 3165079679, 8286286176, 21693778850, 56795050373, 148691372270, 389279066436, 1019145827039

Offset: 0

Author

Creighton Dement, Feb 06 2005

Keywords

• easy
• nonn

Comments

A floretion-generated sequence relating Fibonacci numbers.

Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 2dia[I]forseq[ + .5'i + .5'ii' + .5'ij' + .5'ik' ], 2dia[J]forseq = 2dia[K]forseq = A001654, mixforseq = A001519, tesforseq = A099016, vesforseq = A000004. Identity used: dia[I] + dia[J] + dia[K] + mix + tes = ves

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A001519, A099016, A001654, A100545, A055849, A000004.

Programs

Mathematica

CoefficientList[Series[(x+2)/((x+1)(x^2-3x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-1},{2,5,14},30] (* Harvey P. Dale, Apr 22 2012 *)

PARI

a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

PARI

Vec((x+2)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), a(0) = 2, a(1) = 5, a(2) = 14.

a(n) + a(n+1) = A100545(n).

a(n) + 2*a(n+1) + a(n+2) = A055849(n+2).

a(n) + 2*A001654(n) - A099016(n+2) + 2*A001519(n) = 0.

a(n) = (2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5. - Colin Barker, Oct 01 2016

a(n) = (-1)^n +9*A001906(n+1) -A001906(n) . - R. J. Mathar, Sep 11 2019

Extensions

Corrected by T. D. Noe, Nov 02 2006, Nov 07 2006