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Expansion of 1/(1 - 2*x - 21*x^2).
0

%I #33 Jan 09 2018 01:44:11

%S 1,2,25,92,709,3350,21589,113528,680425,3744938,21778801,122201300,

%T 701757421,3969742142,22676390125,128717365232,733638923089,

%U 4170342516050,23747102416969,135071397670988,768831946098325,4374163243287398,24893797354639621,141645022818314600

%N Expansion of 1/(1 - 2*x - 21*x^2).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,21).

%F a(0)=1, a(1)=2, a(n) = 2*a(n-1) + 21*a(n-2) for n>1. - _Philippe Deléham_, Sep 19 2009

%F a(n) = (1/2 + sqrt(22)/44)*(1 + sqrt(22))^n + (1/2 - sqrt(22)/44)*(1 - sqrt(22))^n. - _Antonio Alberto Olivares_, Jun 08 2011

%p A:= gfun:-rectoproc({a(n)=2*a(n-1)+21*a(n-2), a(0)=1,a(1)=2},a(n),remember):

%p map(A, [$0..30]); # _Robert Israel_, Jan 28 2015

%t l = 2; m = 7; k = 3; p[x_] := -k/m - l*x/m + x^2 q[x_] := ExpandAll[x^2*p[1/x]] Table[ SeriesCoefficient[Series[x/q[x], {x, 0, 30}], n]*m^(n - 1), {n, 0, 30}] f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == l*a[n - 1]/m + k*a[n - 2]/m, a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] a = Table[Rationalize[N[f[n]*m^(n - 1), 100], 0], {n, 0, 25}]

%t Join[{a=1,b=2},Table[c=2*b+21*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)

%o (PARI) Vec(1/(1-2*x-21*x^2) + O(x^30)) \\ _Michel Marcus_, Jan 28 2015

%Y Cf. A002532.

%K nonn,easy

%O 0,2

%A _Roger L. Bagula_, Sep 23 2006

%E Edited by _N. J. A. Sloane_, Sep 26 2006