A279275 Numbers k such that 2*k+1 and 5*k+1 are both pentagonal numbers (A000326).

500, 721525, 1040439075, 1500312425150, 2163449476627750, 3119692644984790875, 4498594630618591814525, 6486970337659364411754700, 9354206728310172863158463400, 13488759615252931609310092468625, 19450782010987999070452290181294375

Offset

1,1

Links

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

Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Formula

a(n) = 5*((7-2*sqrt(10))*(721+228*sqrt(10))^(-n) + (7+2*sqrt(10))*(721+228*sqrt(10))^n - 14)/192.

a(n) = 1443*a(n-1) - 1443*a(n-2) + a(n-3) for n>3.

G.f.: 25*x*(20 + x) / ((1 - x)*(1 - 1442*x + x^2)).

Example

500 is in the sequence because 2*500+1 = 1001 and 5*500+1 = 2501 are both pentagonal numbers.

Mathematica

Table[Simplify[5 ((7 - 2 #) (721 + 228 #)^(-n) + (7 + 2 #) (721 + 228 #)^n - 14)/192 &@ Sqrt@ 10], {n, 11}] (* or *)
Rest@ CoefficientList[Series[25 x (20 + x)/((1 - x) (1 - 1442 x + x^2)), {x, 0, 11}], x] (* Michael De Vlieger, Dec 09 2016 *)
LinearRecurrence[{1443, -1443, 1}, {500, 721525, 1040439075}, 20] (* Harvey P. Dale, Apr 21 2020 *)

Prog

(PARI) isok(k) = ispolygonal(2*k+1, 5) & ispolygonal(5*k+1, 5)
(PARI) Vec(25*x*(20 + x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^30))

Crossrefs

Cf. A000326, A279274, A279276.

Sequence in context: A093250 A214242 A239046 * A097425 A320206 A320215

Adjacent sequences: A279272 A279273 A279274 * A279276 A279277 A279278

Keyword

nonn,easy

Author

Colin Barker, Dec 09 2016

Status

approved