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.

0, 24, 3432, 487344, 69199440, 9825833160, 1395199109304, 198108447688032, 28130004372591264, 3994262512460271480, 567157146764985958920, 80532320578115545895184, 11435022364945642531157232, 1623692643501703123878431784, 230552920354876897948206156120

Offset: 1

Author

Paul Weisenhorn, Apr 19 2009

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..450
• Index entries for linear recurrences with constant coefficients, signature (143,-143,1).

Crossrefs

Cf. A077417, A077416, A157456.

Programs

Magma

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

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:
# Second program
seq((6/35)*(simplify(ChebyshevU(n,71) -141*ChebyshevU(n-1,71)) -1), n=1..30); # G. C. Greubel, Sep 27 2022

Mathematica

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

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

SageMath

[(6/35)*(-1 + chebyshev_U(n, 71) - 141*chebyshev_U(n-1, 71)) for n in range(1,30)] # G. C. Greubel, Sep 27 2022

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, ... (A077417).

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, ... (A077416).

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, ... (this sequence).

G.f.: 24*x^2/((1-x)*(1-142*x+x^2)). - 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

a(n) = (6/35)*(ChebyshevU(n, 71) - 141*ChebyshevU(n-1, 71) - 1). - G. C. Greubel, Sep 27 2022

Extensions

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