A279276 Numbers k such that 2*k+1 and 7*k+1 are both pentagonal numbers (A000326).

25, 2093, 2413024782, 199383164500, 16474611689525, 1361262526857873, 1569151855418042668762, 129655718749849826609000, 10713179445632171628299025, 885207493668292813536022453, 1020394636386389128112999131619942, 84313063475888056056234492629533500

Offset

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..300

Index entries for linear recurrences with constant coefficients, signature (1,0,0,650284185602,-650284185602,0,0,-1,1).

Formula

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)).

Example

25 is in the sequence because 2*25+1 = 51 and 7*25+1 = 176 are both pentagonal numbers.

Mathematica

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 *)
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 *)

Prog

(PARI) isok(k) = ispolygonal(2*k+1, 5) & ispolygonal(7*k+1, 5)
(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))

Crossrefs

Cf. A000326, A279274, A279275.

Sequence in context: A051112 A061843 A173948 * A197408 A197430 A195272

Adjacent sequences: A279273 A279274 A279275 * A279277 A279278 A279279

Keyword

nonn,easy

Author

Colin Barker, Dec 09 2016

Status

approved