A159665 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 11*n(j) + 1 = a(j)*a(j) and 13*n(j) + 1 = b(j)*b(j); with positive integer numbers.
0, 48, 27600, 15842400, 9093510048, 5219658925200, 2996075129554800, 1719741904705530048, 987128857225844692800, 566610244305730148137200, 325233293102631879186060048, 186683343630666392922650330400, 107155914010709406905722103589600
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..363
- Index entries for linear recurrences with constant coefficients, signature (575,-575,1).
Programs
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Magma
I:=[0,48,27600]; [n le 3 select I[n] else 575*Self(n-1) -575*Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Jun 26 2022
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Maple
for a from 1 by 2 to 100000 do b:=sqrt((13*a*a-2)/11): if (trunc(b)=b) then n:=(a*a-1)/11: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
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Mathematica
LinearRecurrence[{575,-575,1}, {0,48,27600}, 30] (* G. C. Greubel, Jun 26 2022 *) -
PARI
concat(0, Vec(-48*x^2/((x-1)*(x^2-574*x+1)) + O(x^30))) \\ Colin Barker, Sep 25 2015
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PARI
a(n) = round((-24+(12+sqrt(143))*(287+24*sqrt(143))^(-n)-(-12+sqrt(143))*(287+24*sqrt(143))^n)/286) \\ Colin Barker, Jul 26 2016
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SageMath
[(12/143)*(chebyshev_U(n,287) -573*chebyshev_U(n-1,287) -1) for n in (1..30)] # G. C. Greubel, Jun 26 2022
Formula
The a(j) recurrence is a(1)=1; a(2)=23; a(t+2) = 24*a(t+1) - a(t) resulting in terms 1, 23, 551, 13201, ... (A159664).
The b(j) recurrence is b(1)=1; b(2)=25; b(t+2) = 24*b(t+1) - b(t) resulting in terms 1, 25, 599, 14351, ... (A159661).
The n(j) recurrence is n(0)=n(1)=1; n(2)=48; n(t+3) = 575*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 48, 27600, 15842400 as listed above.
From Colin Barker, Sep 25 2015: (Start)
a(n) = 575*a(n-1) - 575*a(n-2) + a(n-3) for n > 3.
G.f.: 48*x^2 / ((1-x)*(1-574*x+x^2)). (End)
a(n) = (-24 + (12 + sqrt(143))*(287 + 24*sqrt(143))^(-n) - (-12 + sqrt(143))*(287 + 24*sqrt(143))^n)/286. - Colin Barker, Jul 26 2016
From G. C. Greubel, Jun 25 2022: (Start)
a(n) = (12/143)*(ChebyshevU(n, 287) - 573*ChebyshevU(n-1, 287) - 1).
E.g.f.: (12/143)*(exp(287*x)*( (sqrt(143)/12)*sinh(24*sqrt(143)*x) + cosh(24*sqrt(143)*x) ) - exp(x)). (End)