A253442 Expansion of x * (96 - 816*x) / ((1 - x) * (1 - 1442*x + x^2)) in powers of x.

96, 137712, 198579888, 286352060064, 412919472031680, 595429592317621776, 858609059202538568592, 1238113667940468298287168, 1785359050561096083591526944, 2574486512795432612070683565360, 3712407766091963265509842109721456

Offset: 1

Author

Michael Somos, Dec 31 2014

Keywords

• nonn
• easy

Comments

The continued fraction convergents of sqrt(10) are 3/1, 19/6, 117/37, 721/228, ...

Examples

			G.f. = 96*x + 137712*x^2 + 198579888*x^3 + 286352060064*x^4 + ...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..300
• Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Crossrefs

Cf. A253410.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(48*x*(2 - 17*x)/((1 - x)*(1 - 1442*x + x^2)))); // G. C. Greubel, Aug 03 2018

Mathematica

CoefficientList[Series[48*x*(2-17*x)/((1-x)*(1-1442*x+x^2)), {x,0,30}], x] (* G. C. Greubel, Aug 03 2018 *)
LinearRecurrence[{1443,-1443,1},{96,137712,198579888},20] (* Harvey P. Dale, Aug 23 2020 *)

PARI

{a(n) = my(t=(721 - 228*quadgen(40))^n); (1 - real(t) - 4*imag(t)) / 2};

PARI

Vec(48*x*(2 - 17*x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^20)) \\ Colin Barker, Nov 24 2017

Formula

G.f.: x * (96 - 816*x) / ((1 - x) * (1 - 1442*x + x^2)).

a(n) = A253410(2*n) for all n in Z.

1 - a(-n) = A253410(2*n + 1) for all n in Z.

From Colin Barker, Nov 24 2017: (Start)

a(n) = (1/2 - (5+2*sqrt(10))/20*(721+228*sqrt(10))^(-n) + (-1/4 + 1/sqrt(10))*(721+228*sqrt(10))^n).

a(n) = 1443*a(n-1) - 1443*a(n-2) + a(n-3) for n>3.

(End)