%I #12 Dec 10 2016 03:19:00

%S 238,685382117,380358424875,211083026086753,606761281164154030532,

%T 336727147248939746087250,186869490875990437334549968,

%U 537159115995469726115102627401247,298100855085679705519137567072187125,165433513379232813905174469119753565883

%N Numbers k such that 2*k+1 and 3*k+1 are both pentagonal numbers (A000326).

%H Colin Barker, <a href="/A279274/b279274.txt">Table of n, a(n) for n = 1..250</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,885289046402,-885289046402,0,-1,1).

%F G.f.: x*(238 +685381879*x +379673042758*x^2 +3874618202*x^3 +6994421*x^4 +2*x^5) / ((1 -x)*(1 -9602*x +x^2)*(1 +9602*x +92198403*x^2 +9602*x^3 +x^4)).

%e 238 is in the sequence because 2*238+1 = 477 and 3*238+1 = 715 are both pentagonal numbers.

%t Rest@ CoefficientList[Series[x (238 + 685381879 x + 379673042758 x^2 + 3874618202 x^3 + 6994421 x^4 + 2 x^5)/((1 - x) (1 - 9602 x + x^2) (1 + 9602 x + 92198403 x^2 + 9602 x^3 + x^4)), {x, 0, 10}], x] (* _Michael De Vlieger_, Dec 09 2016 *)

%o (PARI) isok(k) = ispolygonal(2*k+1, 5) & ispolygonal(3*k+1, 5)

%o (PARI) Vec(x*(238 +685381879*x +379673042758*x^2 +3874618202*x^3 +6994421*x^4 +2*x^5) / ((1 -x)*(1 -9602*x +x^2)*(1 +9602*x +92198403*x^2 +9602*x^3 +x^4)) + O(x^15))

%Y Cf. A000326, A279275, A279276.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Dec 09 2016