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A159669 Expansion of x*(x + 1)/(x^2 - 28*x + 1). 3
1, 29, 811, 22679, 634201, 17734949, 495944371, 13868707439, 387827863921, 10845311482349, 303280893641851, 8481019710489479, 237165271000063561, 6632146568291290229, 185462938641156062851, 5186330135384078469599, 145031780852113041085921 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j)+1=a(j)*a(j) and 15*n(j)+1=b(j)*b(j) with positive integer numbers.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (28,-1).

FORMULA

The a(j) recurrence is a(1)=1; a(2)=27; a(t+2)=28*a(t+1)-a(t)

resulting in terms 1, 27, 755, 21113... (A159668)

The b(j) recurrence is b(1)=1; b(2)=29; b(t+2)=28*b(t+1)-b(t)

resulting in terms 1, 29, 811, 22679... (this sequence)

The n(j) recurrence is n(0)=n(1)=0; n(2)=56; n(t+3)=783*(n(t+2)-n(t+1))+n(t)

resulting in terms 0, 0, 56, 43848, 34289136... (A159673)

G.f.: x*(x+1)/(x^2-28*x+1). - Vincenzo Librandi, Feb 26 2014

a(n) = (14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26. - Colin Barker, Jul 25 2016

MAPLE

for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then

n:=(a*a-1)/13: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: endif: enddo:

MATHEMATICA

CoefficientList[Series[(x + 1)/(x^2 - 28 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

PROG

(PARI) Vec(x*(x+1)/(x^2-28*x+1) + O(x^100)) \\ Colin Barker, Feb 24 2014

(PARI) a(n) = round((14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26) \\ Colin Barker, Jul 25 2016

CROSSREFS

A157456, A159668, A159673.

Sequence in context: A135995 A046850 A180844 * A162831 A163207 A163549

Adjacent sequences:  A159666 A159667 A159668 * A159670 A159671 A159672

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Apr 19 2009

EXTENSIONS

More terms from, and new name by Colin Barker, Feb 24 2014

STATUS

approved

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Last modified December 8 02:07 EST 2016. Contains 278902 sequences.