A253447 - OEIS

%I #17 Oct 07 2020 12:10:16

%S 1,15,435,13021,390181,11692395,350381655,10499757241,314642335561,

%T 9428770309575,282548466951675,8467025238240661,253728208680268141,

%U 7603379235169803555,227847648846413838495,6827826086157245351281,204606934935870946699921

%N Indices of centered octagonal numbers (A016754) which are also centered heptagonal numbers (A069099).

%C Also positive integers y in the solutions to 7*x^2 - 8*y^2 - 7*x + 8*y = 0, the corresponding values of x being A253446.

%H Colin Barker, <a href="/A253447/b253447.txt">Table of n, a(n) for n = 1..678</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 (31,-31,1).

%F a(n) = 31*a(n-1)-31*a(n-2)+a(n-3).

%F G.f.: -x*(x^2-16*x+1) / ((x-1)*(x^2-30*x+1)).

%F a(n) = (8+(4+sqrt(14))*(15+4*sqrt(14))^(-n)-(-4+sqrt(14))*(15+4*sqrt(14))^n)/16. - _Colin Barker_, Mar 03 2016

%e 15 is in the sequence because the 15th centered octagonal number is 841, which is also the 16th centered heptagonal number.

%o (PARI) Vec(-x*(x^2-16*x+1)/((x-1)*(x^2-30*x+1)) + O(x^100))

%Y Cf. A016754, A069099, A253446, A253514.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 01 2015