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A159673 Expansion of 56*x^2/(-x^3 + 783*x^2 - 783*x + 1). 3
0, 56, 43848, 34289136, 26814060560, 20968561068840, 16397387941772376, 12822736401904929248, 10027363468901712899616, 7841385409944737582570520, 6131953363213315887857247080, 4795179688647403079566784646096, 3749824384568905994905337736000048 (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 (783,-783,1).

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... (A159669).

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... (this sequence).

G.f.: 56*x^2/(- x^3 + 783*x^2 - 783*x + 1). - Vincenzo Librandi, Feb 26 2014

a(n) = -((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390. - 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[56 x/(- x^3 + 783 x^2 - 783 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

LinearRecurrence[{783, -783, 1}, {0, 56, 43848}, 20] (* Harvey P. Dale, Jan 06 2019 *)

PROG

(PARI) Vec(56*x^2/(-x^3+783*x^2-783*x+1) + O(x^100)) \\ Colin Barker, Feb 24 2014

(PARI) a(n) = round(-((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390) \\ Colin Barker, Jul 25 2016

CROSSREFS

Cf. A157456, A159668, A159669.

Sequence in context: A202579 A090218 A184897 * A009837 A308390 A093256

Adjacent sequences:  A159670 A159671 A159672 * A159674 A159675 A159676

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Apr 19 2009

EXTENSIONS

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

STATUS

approved

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Last modified October 22 09:51 EDT 2021. Contains 348160 sequences. (Running on oeis4.)