A253458 - OEIS

%I #20 Oct 13 2022 15:47:05

%S 1,13,325,8425,218713,5678101,147411901,3827031313,99355402225,

%T 2579413426525,66965393687413,1738520822446201,45134575989913801,

%U 1171760454915312613,30420637251808214125,789764808092098254625,20503464373142746406113,532300308893619308304301

%N Indices of centered heptagonal numbers (A069099) which are also centered hexagonal numbers (A003215).

%C Also positive integers y in the solutions to 6*x^2 - 7*y^2 - 6*x + 7*y = 0, the corresponding values of x being A253457.

%H Colin Barker, <a href="/A253458/b253458.txt">Table of n, a(n) for n = 1..708</a>

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2016volume16/FG2016volume16.pdf#page=423">Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences</a>, Forum Geometricorum, Volume 16 (2016) 419-427.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (27,-27,1).

%F a(n) = 27*a(n-1)-27*a(n-2)+a(n-3).

%F G.f.: -x*(x^2-14*x+1) / ((x-1)*(x^2-26*x+1)).

%F a(n) = 1/2+(13+2*sqrt(42))^(-n)*(7+sqrt(42)-(-7+sqrt(42))*(13+2*sqrt(42))^(2*n))/28. - _Colin Barker_, Mar 03 2016

%e 13 is in the sequence because the 13th centered heptagonal number is 547, which is also the 14th centered hexagonal number.

%t LinearRecurrence[{27,-27,1},{1,13,325},20] (* _Harvey P. Dale_, Oct 13 2022 *)

%o (PARI) Vec(-x*(x^2-14*x+1)/((x-1)*(x^2-26*x+1)) + O(x^100))

%Y Cf. A003215, A069099, A253457, A253546.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 01 2015