A159680 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 9*n(j) + 1 = a(j)*a(j) and 11*n(j) + 1 = b(j)*b(j) with positive integer numbers.
0, 40, 15960, 6352080, 2528111920, 1006182192120, 400457984351880, 159381271589856160, 63433345634778399840, 25246312181370213280200, 10047968814839710107119800, 3999066341994023252420400240, 1591618356144806414753212175760, 633460106679290959048526025552280
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..350
- Index entries for linear recurrences with constant coefficients, signature (399,-399,1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!(40*x^2/((1-x)*(1-398*x+x^2)))); // G. C. Greubel, Jun 03 2018 -
Maple
for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
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Mathematica
LinearRecurrence[{399,-399,1}, {0,40,15960}, 50] (* G. C. Greubel, Jun 03 2018 *) -
PARI
a(n) = round((-20+(10+3*sqrt(11))*(199+60*sqrt(11))^(-n)+(10-3*sqrt(11))*(199+60*sqrt(11))^n)/198) \\ Colin Barker, Jul 26 2016
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PARI
concat(0, Vec(-40*x^2/((x-1)*(x^2-398*x+1)) + O(x^20))) \\ Colin Barker, Jul 26 2016
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SageMath
[(10/99)*(chebyshev_U(n, 199) -397*chebyshev_U(n-1, 199) -1) for n in (1..30)] # G. C. Greubel, Jun 26 2022
Formula
The a(j) recurrence is a(1)=1; a(2)=19; a(t+2) = 20*a(t+1) - a(t) resulting in terms 1, 19, 379, 7561, ... (A075839).
The b(j) recurrence is b(1)=1; b(2)=21; b(t+2) = 20*b(t+1) - b(t) resulting in terms 1, 21, 419, 8359, ... (A083043).
The n(j) recurrence is n(0)=n(1)=0; n(2)=40; n(t+3) = 399*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 40, 15960, 6352080 as listed above
G.f.: 40*x^2/((1-x)*(1-398*x+x^2)). - R. J. Mathar, Apr 20 2009
a(n) = (-20 + (10 + 3*sqrt(11))*(199 + 60*sqrt(11))^(-n) + (10 - 3*sqrt(11))*(199 + 60*sqrt(11))^n)/198. - Colin Barker, Jul 26 2016
From G. C. Greubel, Jun 26 2022: (Start)
a(n) = (10/99)*( ChebyshevU(n, 199) - 397*ChebyshevU(n-1, 199) - 1 ).
E.g.f.: (10/99)*(exp(199*x)*( (3*sqrt(11)/10)*sinh(60*sqrt(11)*x) + cosh(60*sqrt(11)*x) ) - exp(x)). (End)
Extensions
More terms from R. J. Mathar, Apr 20 2009