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A280111 Indices of triangular numbers (A000217) that are also centered 10-gonal numbers (A062786).

%I A280111 #10 Dec 26 2016 13:46:04
%S A280111 1,58,2221,84358,3203401,121644898,4619302741,175411859278,
%T A280111 6661031349841,252943779434698,9605202587168701,364744754532975958,
%U A280111 13850695469665917721,525961683092771897458,19972693262055666185701,758436382275022543159198,28800609833188800973863841
%N A280111 Indices of triangular numbers (A000217) that are also centered 10-gonal numbers (A062786).
%C A280111 Also positive integers x in the solutions to x^2 - 10*y^2 + x + 10*y - 2 = 0, the corresponding values of y being A280112.
%H A280111 Colin Barker, <a href="/A280111/b280111.txt">Table of n, a(n) for n = 1..600</a>
%H A280111 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (39,-39,1).
%F A280111 a(n) = (-2 - (3+sqrt(10))*(19+6*sqrt(10))^(-n) + (-3+sqrt(10))*(19+6*sqrt(10))^n) / 4.
%F A280111 a(n) = 39*a(n-1) - 39*a(n-2) + a(n-3) for n>3.
%F A280111 G.f.: x*(1 + 19*x - 2*x^2) / ((1 - x)*(1 - 38*x + x^2)).
%e A280111 58 is in the sequence because the 58th triangular number is 1711, which is also the 19th centered 10-gonal number.
%t A280111 Table[Simplify[(-2 - (3 + #) (19 + 6 #)^(-n) + (-3 + #) (19 + 6 #)^n)/4] &@ Sqrt@ 10, {n, 17}] (* or *)
%t A280111 Rest@ CoefficientList[Series[x (1 + 19 x - 2 x^2)/((1 - x) (1 - 38 x + x^2)), {x, 0, 17}], x] (* _Michael De Vlieger_, Dec 26 2016 *)
%o A280111 (PARI) Vec(x*(1 + 19*x - 2*x^2) / ((1 - x)*(1 - 38*x + x^2)) + O(x^20))
%Y A280111 Cf. A000217, A062786, A280112, A280113.
%K A280111 nonn,easy
%O A280111 1,2
%A A280111 _Colin Barker_, Dec 26 2016