A254136 - OEIS

%I #9 Mar 24 2024 16:59:18

%S 1,73,889,84049,1025713,96992281,1183671721,111929008033,

%T 1365956140129,129165978277609,1576312202036953,149057427003352561,

%U 1819062915194503441,172012141595890577593,2099197027822254933769,198501862344230723189569,2422471551043966999065793

%N Indices of pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

%C Also positive integers x in the solutions to 3*x^2 - 6*y^2 - x + 6*y - 2 = 0, the corresponding values of y being A254137.

%H Colin Barker, <a href="/A254136/b254136.txt">Table of n, a(n) for n = 1..653</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1154,-1154,-1,1).

%F a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).

%F G.f.: -x*(x^4+72*x^3-338*x^2+72*x+1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).

%e 73 is in the sequence because the 73rd pentagonal number is 7957, which is also the 52nd centered hexagonal number.

%t LinearRecurrence[{1,1154,-1154,-1,1},{1,73,889,84049,1025713},20] (* _Harvey P. Dale_, Mar 24 2024 *)

%o (PARI) Vec(-x*(x^4+72*x^3-338*x^2+72*x+1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

%Y Cf. A000326, A003215, A254137, A254138.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 26 2015