A212335 - OEIS

%I #24 Mar 29 2024 05:43:35

%S 1,22,462,9681,202840,4249960,89046321,1865722782,39091132102,

%T 819048051361,17160917946480,359560228824720,7533603887372641,

%U 157846121406000742,3307234945638642942,69294087737005501041,1451868607531476878920

%N Expansion of 1/(1-22*x+22*x^2-x^3).

%C Partial sums of A092499 (after 0).

%H Bruno Berselli, <a href="/A212335/b212335.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (22,-22,1).

%F G.f.: 1/((1-x)*(1-21*x+x^2)).

%F a(n) = (((230-11*sqrt(437))*(21-sqrt(437))^n+(230+11*sqrt(437))*(21+sqrt(437))^n)/2^n-23)/437.

%F a(n) = a(-n-3) = 23*a(n-1)-23*a(n-2)+a(n-3).

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

%p a:= n-> (<<0|1|0>, <0|0|1>, <1|-22|22>>^n. <<1, 22, 462>>)[1, 1]:

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 15 2012

%t CoefficientList[Series[1/(1 - 22 x + 22 x^2 - x^3), {x, 0, 16}], x]

%t LinearRecurrence[{22,-22,1},{1,22,462},20] (* _Harvey P. Dale_, Nov 04 2017 *)

%o (PARI) Vec(1/(1-22*x+22*x^2-x^3)+O(x^17))

%o (Maxima) makelist(coeff(taylor(1/(1-22*x+22*x^2-x^3), x, 0, n), x, n), n, 0, 16);

%o (Magma) m:=17; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-22*x+22*x^2-x^3)));

%Y Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Jun 12 2012