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A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).

%I A253621 #30 Sep 08 2022 08:46:10
%S A253621 1,6,66,781,9301,110826,1320606,15736441,187516681,2234463726,
%T A253621 26626048026,317278112581,3780711302941,45051257522706,
%U A253621 536834378969526,6396961290111601,76226701102369681,908323451938324566,10823654722157525106,128975533213951976701
%N A253621 Indices of centered heptagonal numbers (A069099) which are also centered pentagonal numbers (A005891).
%C A253621 Also positive integers y in the solutions to 5*x^2 - 7*y^2 - 5*x + 7*y = 0, the corresponding values of x being A133272.
%H A253621 Colin Barker, <a href="/A253621/b253621.txt">Table of n, a(n) for n = 1..930</a>
%H A253621 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 A253621 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13,-13,1).
%F A253621 a(n) = 13*a(n-1)-13*a(n-2)+a(n-3).
%F A253621 G.f.: -x*(x^2-7*x+1) / ((x-1)*(x^2-12*x+1)).
%F A253621 a(n) = (14-(-7+sqrt(35))*(6+sqrt(35))^n+(6-sqrt(35))^n*(7+sqrt(35)))/28. - _Colin Barker_, Mar 05 2016
%F A253621 a(n) = 12*a(n-1) - a(n-2) - 5. - _Vincenzo Librandi_, Mar 05 2016
%F A253621 a(n) = (5*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - _Seiichi Manyama_, Aug 11 2016
%e A253621 6 is in the sequence because the 6th centered heptagonal number is 106, which is also the 7th centered pentagonal number.
%t A253621 RecurrenceTable[{a[1] == 1, a[2] == 6, a[n] == 12 a[n-1] - a[n-2] - 5}, a, {n, 20}] (* _Vincenzo Librandi_, Mar 05 2016 *)
%o A253621 (PARI) Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-12*x+1)) + O(x^100))
%o A253621 (Magma) I:=[1,6]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2)-5: n in [1..20]]; // _Vincenzo Librandi_, Mar 05 2016
%Y A253621 Cf. A005891, A069099, A133272, A253622.
%K A253621 nonn,easy
%O A253621 1,2
%A A253621 _Colin Barker_, Jan 06 2015