A029546 Expansion of 1/( (1-x)*(1-34*x+x^2) ).
1, 35, 1190, 40426, 1373295, 46651605, 1584781276, 53835911780, 1828836219245, 62126595542551, 2110475412227490, 71694037420192110, 2435486796874304251, 82734857056306152425, 2810549653117534878200, 95475953348939879706376, 3243371864210838375138585
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (35,-35,1).
Programs
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GAP
List([0..20], n-> (Lucas(2,-1, 4*n+6)[2] -6)/192 ); # G. C. Greubel, Jan 13 2020
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Magma
I:=[1,35,1190]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2) +Self(n-3): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011
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Maple
seq(coeff(series(1/( (1-x)*(1-34*x+x^2) ), x, n+1), x, n), n = 0..20); # G. C. Greubel, Jan 13 2020
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Mathematica
LinearRecurrence[{35,-35,1},{1,35,1190},20] (* Vincenzo Librandi, Nov 22 2011 *) Table[(Fibonacci[2*n+3, 2]^2 -1)/24, {n,0,20}] (* G. C. Greubel, Jan 13 2020 *) -
PARI
Vec(1/(1-35*x+35*x^2-x^3)+O(x^20)) \\ Charles R Greathouse IV, Sep 23 2012
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Sage
[(lucas_number2(4*n+6, 2,-1) -6)/192 for n in (0..20)] # G. C. Greubel, Jan 13 2020
Formula
From Colin Barker, Mar 02 2016: (Start)
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n > 2.
a(n) = ( (99-70*sqrt(2))*(17-12*sqrt(2))^n - 6 + (99+70*sqrt(2))*(17+12*sqrt(2))^n )/192. (End)
a(n) = ( Pell(2*n+3)^2 - 1)/24 = (Q(4*n+6) - 6)/192, where Q(n) = Pell-Lucas numbers. - G. C. Greubel, Jan 13 2020
Sum_{n>=0} 1/a(n) = 6*(3 - 2*sqrt(2)). - Amiram Eldar, Dec 04 2024
Comments