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A159681 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 5*n(j)+1=a(j)*a(j) and 7*n(j)+1=b(j)*b(j) with positive integer numbers. 2
0, 24, 3432, 487344, 69199440, 9825833160, 1395199109304, 198108447688032, 28130004372591264, 3994262512460271480, 567157146764985958920, 80532320578115545895184, 11435022364945642531157232, 1623692643501703123878431784, 230552920354876897948206156120 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Colin Barker, Table of n, a(n) for n = 1..450

Index entries for linear recurrences with constant coefficients, signature (143,-143,1).

FORMULA

The a(j) recurrence is a(1)=1; a(2)=11; a(t+2)=12*a(t+1)-a(t) resulting in terms 1, 11, 131, 1561.

The b(j) recurrence is b(1)=1; b(2)=13; b(t+2)=12*b(t+1)-b(t) resulting in terms 1, 13, 155, 1847.

The n(j) recurrence is n(0)=n(1)=0; n(2)=24; n(t+3)=143*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 24, 3432, 487344 as listed above.

G.f.: -24*x^2/((x-1)*(x^2-142*x+1)). - R. J. Mathar, Apr 20 2009

a(n) = (-12+(6+sqrt(35))*(71+12*sqrt(35))^(-n)-(-6+sqrt(35))*(71+12*sqrt(35))^n)/70. - Colin Barker, Jul 26 2016

MAPLE

for a from 1 by 2 to 100000 do b:=sqrt((7*a*a-2)/5): if (trunc(b)=b) then

n:=(a*a-1)/5: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: end if: end do:

MATHEMATICA

LinearRecurrence[{143, -143, 1}, {0, 24, 3432}, 30] (* or *) CoefficientList[Series[24*x^2/((1-x)*(x^2-142*x+1)), {x, 0, 30}], x] (* G. C. Greubel, Jun 03 2018 *)

PROG

(PARI) concat(0, Vec(-24*x^2/((x-1)*(x^2-142*x+1)) + O(x^20))) \\ Colin Barker, Jul 26 2016

(PARI) a(n) = round((-12+(6+sqrt(35))*(71+12*sqrt(35))^(-n)-(-6+sqrt(35))*(71+12*sqrt(35))^n)/70) \\ Colin Barker, Jul 26 2016

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients( R!(24*x^2/((1-x)*(x^2-142*x+1)))); // G. C. Greubel, Jun 03 2018

CROSSREFS

Cf. A157456, A077417, A077416.

Sequence in context: A088731 A299697 A072529 * A275569 A071639 A061530

Adjacent sequences:  A159678 A159679 A159680 * A159682 A159683 A159684

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Apr 19 2009

EXTENSIONS

More terms from R. J. Mathar, Apr 20 2009

STATUS

approved

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Last modified October 18 15:21 EDT 2019. Contains 328162 sequences. (Running on oeis4.)