A278620 - OEIS

%I #53 Aug 22 2020 19:36:37

%S 0,1,99,9702,950698,93158703,9128602197,894509856604,87652837344996,

%T 8589083549953005,841642535058049495,82472379352138897506,

%U 8081451533974553906094,791899777950154143899707,77598096787581131548265193,7603821585405000737586089208,745096917272902491151888477192

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

%H Colin Barker, <a href="/A278620/b278620.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).

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

%F 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.

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

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

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

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

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

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

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

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

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

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

%p P:=proc(q) local a,b,c,n; a:=0; b:=1; print(a); print(b);for n from 1 to q do

%p c:=98*b-a+1; a:=b; b:=c; print(b); od; end: P(100); # _Paolo P. Lava_, Nov 30 2016

%t CoefficientList[x/(1 - 99 x + 99 x^2 - x^3) + O[x]^20, x]

%t LinearRecurrence[{99,-99,1},{0,1,99},20] (* _Harvey P. Dale_, Aug 22 2020 *)

%o (PARI) concat(0, Vec(1/(1-99*x+99*x^2-x^3) + O(x^20)))

%o (Sage) gf = x/((1-x)*(1-98*x+x^2)); print(taylor(gf, x, 0, 20).list())

%o (Maxima) makelist(coeff(taylor(x/((1-x)*(1-98*x+x^2)), x, 0, n), x, n), n, 0, 20);

%Y First differences: A173205.

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

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Nov 24 2016