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%I #13 May 02 2019 18:51:43
%S 25,2093,2413024782,199383164500,16474611689525,1361262526857873,
%T 1569151855418042668762,129655718749849826609000,
%U 10713179445632171628299025,885207493668292813536022453,1020394636386389128112999131619942,84313063475888056056234492629533500
%N Numbers k such that 2*k+1 and 7*k+1 are both pentagonal numbers (A000326).
%H Colin Barker, <a href="/A279276/b279276.txt">Table of n, a(n) for n = 1..300</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,650284185602,-650284185602,0,0,-1,1).
%F G.f.: x*(25 +2068*x +2413022689*x^2 +196970139718*x^3 +18123884975*x^4 +219343412*x^5 +2687111*x^6 +2*x^7) / ((1 -x)*(1 -898*x +x^2)*(1 +898*x +x^2)*(1 +806402*x^2 +x^4)).
%e 25 is in the sequence because 2*25+1 = 51 and 7*25+1 = 176 are both pentagonal numbers.
%t Rest@CoefficientList[Series[x (25 + 2068 x + 2413022689 x^2 + 196970139718 x^3 + 18123884975 x^4 + 219343412 x^5 + 2687111 x^6 + 2 x^7)/((1 - x) (1 - 898 x + x^2) (1 + 898 x + x^2) (1 + 806402 x^2 + x^4)), {x, 0, 12}], x] (* _Michael De Vlieger_, Dec 09 2016 *)
%t LinearRecurrence[{1,0,0,650284185602,-650284185602,0,0,-1,1}, {25,2093,2413024782,199383164500,16474611689525,1361262526857873,1569151855418042668762,129655718749849826609000,10713179445632171628299025},13] (* _Harvey P. Dale_, May 02 2019 *)
%o (PARI) isok(k) = ispolygonal(2*k+1, 5) & ispolygonal(7*k+1, 5)
%o (PARI) Vec(x*(25 +2068*x +2413022689*x^2 +196970139718*x^3 +18123884975*x^4 +219343412*x^5 +2687111*x^6 +2*x^7) / ((1 -x)*(1 -898*x +x^2)*(1 +898*x +x^2)*(1 +806402*x^2 +x^4)) + O(x^15))
%Y Cf. A000326, A279274, A279275.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Dec 09 2016
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