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A280112 Indices of centered 10-gonal numbers (A062786) that are also triangular numbers (A000217).

%I A280112 #9 Dec 26 2016 13:46:13
%S A280112 1,19,703,26677,1013005,38467495,1460751787,55470100393,2106403063129,
%T A280112 79987846298491,3037431756279511,115342418892322909,
%U A280112 4379974486151991013,166323688054883335567,6315920171599414760515,239838642832722877563985,9107552507471869932670897
%N A280112 Indices of centered 10-gonal numbers (A062786) that are also triangular numbers (A000217).
%C A280112 Also positive integers y in the solutions to x^2 - 10*y^2 + x + 10*y - 2 = 0, the corresponding values of x being A280111.
%H A280112 Colin Barker, <a href="/A280112/b280112.txt">Table of n, a(n) for n = 1..600</a>
%H A280112 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (39,-39,1).
%F A280112 a(n) = 1/2 + (19 + 6*sqrt(10))^(-n)*(10+3*sqrt(10) + (10-3*sqrt(10))*(19+6*sqrt(10))^(2*n)) / 40.
%F A280112 a(n) = 39*a(n-1) - 39*a(n-2) + a(n-3) for n>3.
%F A280112 G.f.: x*(1 - 20*x + x^2) / ((1 - x)*(1 - 38*x + x^2)).
%e A280112 19 is in the sequence because the 19th centered 10-gonal number is 1711, which is also the 58th triangular number.
%t A280112 Table[Simplify[1/2 + (19 + 6 #)^(-n) (10 + 3 # + (10 - 3 #) (19 + 6*#)^(2 n))/40] &@ Sqrt@ 10, {n, 17}] (* or *)
%t A280112 Rest@ CoefficientList[Series[x (1 - 20 x + x^2)/((1 - x) (1 - 38 x + x^2)), {x, 0, 17}], x] (* _Michael De Vlieger_, Dec 26 2016 *)
%o A280112 (PARI) Vec(x*(1 - 20*x + x^2) / ((1 - x)*(1 - 38*x + x^2)) + O(x^20))
%Y A280112 Cf. A000217, A062786, A280111, A280113.
%K A280112 nonn,easy
%O A280112 1,2
%A A280112 _Colin Barker_, Dec 26 2016