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

0,2

Comments

Numbers m such that r = 24*m+1 and 2*r-1 are both squares. - Bruno Berselli, Jul 17 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Formula

a(n) = A029549(n+1)/6 = A075528(n+1)/3.

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

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

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 *)

Prog

(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
(PARI) Vec(1/(1-35*x+35*x^2-x^3)+O(x^20)) \\ Charles R Greathouse IV, Sep 23 2012
(Sage) [(lucas_number2(4*n+6, 2, -1) -6)/192 for n in (0..20)] # G. C. Greubel, Jan 13 2020
(GAP) List([0..20], n-> (Lucas(2, -1, 4*n+6)[2] -6)/192 ); # G. C. Greubel, Jan 13 2020

Crossrefs

Cf. A029547 (first differences), A245031 (see Comments line).

Cf. A000129, A002203.

Sequence in context: A046176 A352183 A162847 * A305539 A163218 A163600

Adjacent sequences: A029543 A029544 A029545 * A029547 A029548 A029549

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved