A253878 - OEIS

%I #12 Sep 10 2022 11:49:52

%S 1,22,358,5713,91057,1451206,23128246,368600737,5874483553,

%T 93623136118,1492095694342,23779907973361,378986431879441,

%U 6040003002097702,96261061601683798,1534136982624843073,24449930660395805377,389664753583708042966,6210186126678932882086

%N Indices of triangular numbers (A000217) which are also centered heptagonal numbers (A069099).

%C Also positive integers x in the solutions to x^2 - 7*y^2 + x + 7*y - 2 = 0, the corresponding values of y being A253879.

%H Colin Barker, <a href="/A253878/b253878.txt">Table of n, a(n) for n = 1..832</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,-17,1).

%F a(n) = 17*a(n-1)-17*a(n-2)+a(n-3).

%F G.f.: -x*(x^2+5*x+1) / ((x-1)*(x^2-16*x+1)).

%F a(n) = (-2+(8-3*sqrt(7))^n*(3+sqrt(7))-(-3+sqrt(7))*(8+3*sqrt(7))^n)/4. - _Colin Barker_, Mar 04 2016

%e 22 is in the sequence because the 22nd triangular number is 253, which is also the 9th centered heptagonal number.

%t LinearRecurrence[{17,-17,1},{1,22,358},20] (* _Harvey P. Dale_, Sep 10 2022 *)

%o (PARI) Vec(-x*(x^2+5*x+1)/((x-1)*(x^2-16*x+1)) + O(x^100))

%Y Cf. A000217, A069099, A253879, A253880.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 17 2015