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, ... (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
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 *)
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) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients( R!(24*x^2/((1-x)*(1-142*x+x^2)))); // G. C. Greubel, Jun 03 2018
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Apr 19 2009
EXTENSIONS
More terms from R. J. Mathar, Apr 20 2009
STATUS
approved