A202564 - OEIS

%I #15 Mar 28 2024 09:02:10

%S 1,91,8901,872191,85465801,8374776291,820642610701,80414601072391,

%T 7879810262483601,772140991122320491,75661937319724924501,

%U 7414097716341920280591,726505914264188462573401,71190165500174127411912691,6975909713102800297904870301

%N Indices of pentagonal numbers which are also decagonal.

%C As n increases, this sequence is approximately geometric with common ratio r = lim(n->oo, a(n)/a(n-1)) = (sqrt(3)+sqrt(2))^4 = 49+20*sqrt(6).

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

%F G.f.: x*(1+x)*(1-9*x) / ((1-x)*(1-98*x+x^2)).

%F a(n) = 98*a(n-1)-a(n-2)-16.

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

%F a(n) = 1/24*(5*sqrt(2)*((sqrt(3)+sqrt(2))^(4n-3)-(sqrt(3)-sqrt(2))^(4n-3))+4).

%F a(n) = ceiling(5/24*sqrt(2)*(sqrt(3)+sqrt(2))^(4n-3)).

%e The second pentagonal number that is also decagonal is A000326(91) = 12376. Hence a(2)=91.

%t LinearRecurrence[{99, -99, 1}, {1, 91, 8901}, 15]

%Y Cf. A202563, A202565, A000326, A001107.

%K nonn,easy

%O 1,2

%A _Ant King_, Dec 22 2011