A278620 Expansion of x/(1 - 99*x + 99*x^2 - x^3).

0, 1, 99, 9702, 950698, 93158703, 9128602197, 894509856604, 87652837344996, 8589083549953005, 841642535058049495, 82472379352138897506, 8081451533974553906094, 791899777950154143899707, 77598096787581131548265193, 7603821585405000737586089208, 745096917272902491151888477192

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Formula

O.g.f.: x/((1 - x)*(1 - 98*x + x^2)).

E.g.f.: ((5-2*sqrt(6))*exp((5-2*sqrt(6))^2*x) + (5+2*sqrt(6))*exp((5+2*sqrt(6))^2*x) - 10*exp(x))/960.

a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>2.

a(n) = 98*a(n-1) - a(n-2) + 1 for n>1.

a(n) = a(-n-1) = ((5+2*sqrt(6))^(2*n+1) + (5-2*sqrt(6))^(2*n+1))/960 - 1/96.

a(n) = floor((5+2*sqrt(6))^(2*n+1)/960).

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

Lim_{i -> infinity} a(i)/a(i-1) = (5 + 2*sqrt(6))^2.

From the closed form: a(n) + a(-n) = A108741(n).

a(n) = A200993(n)/10 = A200994(n)/15.

a(n) = A123479(n)/20 for n>0.

a(n) = A045502(n)/40.

Maple

P:=proc(q) local a, b, c, n; a:=0; b:=1; print(a); print(b); for n from 1 to q do
c:=98*b-a+1; a:=b; b:=c; print(b); od; end: P(100); # Paolo P. Lava, Nov 30 2016

Mathematica

CoefficientList[x/(1 - 99 x + 99 x^2 - x^3) + O[x]^20, x]
LinearRecurrence[{99, -99, 1}, {0, 1, 99}, 20] (* Harvey P. Dale, Aug 22 2020 *)

Prog

(PARI) concat(0, Vec(1/(1-99*x+99*x^2-x^3) + O(x^20)))
(Sage) gf = x/((1-x)*(1-98*x+x^2)); print(taylor(gf, x, 0, 20).list())
(Maxima) makelist(coeff(taylor(x/((1-x)*(1-98*x+x^2)), x, 0, n), x, n), n, 0, 20);

Crossrefs

First differences: A173205.

Cf. A045502, A108741, A123479, A200993, A200994.

Sequence in context: A093233 A213155 A046173 * A171415 A327926 A098609

Adjacent sequences: A278617 A278618 A278619 * A278621 A278622 A278623

Keyword

nonn,easy

Author

Bruno Berselli, Nov 24 2016

Status

approved