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A254627 Indices of centered pentagonal numbers (A005891) that are also triangular numbers (A000217).

%I A254627 #32 Sep 08 2022 08:46:11
%S A254627 1,2,11,28,189,494,3383,8856,60697,158906,1089155,2851444,19544085,
%T A254627 51167078,350704367,918155952,6293134513,16475640050,112925716859,
%U A254627 295643364940,2026369768941,5305104928862,36361730124071,95196245354568,652484772464329
%N A254627 Indices of centered pentagonal numbers (A005891) that are also triangular numbers (A000217).
%C A254627 Also positive integers y in the solutions to x^2 - 5*y^2 + x + 5*y - 2 = 0, the corresponding values of x being A254626.
%C A254627 Also indices of centered pentagonal numbers (A005891) that are also hexagonal numbers (A000384). - _Colin Barker_, Feb 11 2015
%H A254627 Colin Barker, <a href="/A254627/b254627.txt">Table of n, a(n) for n = 1..1000</a>
%H A254627 Hermann Stamm-Wilbrandt, <a href="/A254627/a254627_1.svg">6 interlaced bisections</a>
%H A254627 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,18,-18,-1,1).
%F A254627 a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).
%F A254627 G.f.: x*(1+x-9*x^2-x^3)/((1-x)*(1+4*x-x^2)*(1-4*x-x^2)).
%F A254627 a(n) = (10 - sqrt(5)*(2-sqrt(5))^n - 5*(-2+sqrt(5))^n - 2*sqrt(5)*(-2+sqrt(5))^n + sqrt(5)*(2+sqrt(5))^n + (-2-sqrt(5))^n*(-5+2*sqrt(5)))/20. - _Colin Barker_, Jun 06 2016
%F A254627 a(2*n+2) = A232970(2*n+1); a(2*n+1) = A110679(2*n). See "6 interlaced bisections" link. - _Hermann Stamm-Wilbrandt_, Apr 18 2019
%F A254627 a(n) = (2 +(1+2*(-1)^n)*Fibonacci(3*n) -(-1)^n*Lucas(3*n))/4. - _G. C. Greubel_, Apr 19 2019
%e A254627 2 is in the sequence because the 2nd centered pentagonal number is 6, which is also the 3rd triangular number.
%t A254627 CoefficientList[Series[x (x^3 + 9 x^2 - x - 1)/((x - 1) (x^2 - 4 x - 1) (x^2 + 4 x - 1)), {x, 0, 25}], x] (* _Michael De Vlieger_, Jun 06 2016 *)
%t A254627 LinearRecurrence[{1,18,-18,-1,1},{1,2,11,28,189},30] (* _Harvey P. Dale_, Apr 23 2017 *)
%o A254627 (PARI) Vec(x*(x^3+9*x^2-x-1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^30))
%o A254627 (PARI) {a(n) = (2 +(1+3*(-1)^n)*fibonacci(3*n) - 2*(-1)^n*fibonacci(3*n+1))/4}; \\ _G. C. Greubel_, Apr 19 2019
%o A254627 (Magma) [(2 +(1+2*(-1)^n)*Fibonacci(3*n) -(-1)^n*Lucas(3*n))/4 : n in [1..30]]; // _G. C. Greubel_, Apr 19 2019
%o A254627 (Sage) [(2 +(1+3*(-1)^n)*fibonacci(3*n) -2*(-1)^n*fibonacci(3*n+1))/4 for n in (1..30)] # _G. C. Greubel_, Apr 19 2019
%Y A254627 Cf. A000217, A005891, A254626, A254628.
%Y A254627 Cf. A000384, A254962.
%K A254627 nonn,easy
%O A254627 1,2
%A A254627 _Colin Barker_, Feb 03 2015