A252762 Numbers n such that the sum of the pentagonal numbers P(n), P(n+1), P(n+2) and P(n+3) is equal to the hexagonal number H(m) for some m.

3, 853, 165735, 32151993, 6237321163, 1210008153885, 234735344532783, 45537446831206273, 8834029949909484435, 1713756272835608774373, 332459882900158192744183, 64495503526357853783597385, 12511795224230523475825148763, 2427223777997195196456295262893

Offset

1,1

Comments

Also positive integers x in the solutions to 12*x^2-4*y^2+32*x+2*y+36 = 0, the corresponding values of y being A252763.

Links

Colin Barker, Table of n, a(n) for n = 1..437

Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Formula

a(n) = 195*a(n-1)-195*a(n-2)+a(n-3).

G.f.: x*(15*x^2-268*x-3) / ((x-1)*(x^2-194*x+1)).

a(n) = -4/3+1/24*(97+56*sqrt(3))^(-n)*(-164-95*sqrt(3)+(97+56*sqrt(3))^(2*n)*(-164+95*sqrt(3))). - Colin Barker, Mar 02 2016

a(n) = 194*a(n-1)-a(n-2)+256. - Vincenzo Librandi, Mar 03 2016

Example

3 is in the sequence because P(3)+P(4)+P(5)+P(6) = 12+22+35+51 = 120 = H(8).

Mathematica

LinearRecurrence[{195, -195, 1}, {3, 853, 165735}, 30] (* Vincenzo Librandi, Mar 03 2016 *)

Prog

(PARI) Vec(x*(15*x^2-268*x-3)/((x-1)*(x^2-194*x+1)) + O(x^100))
(Magma) I:=[3, 853]; [n le 2 select I[n] else  194*Self(n-1) - Self(n-2)+256: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016

Crossrefs

Cf. A000326, A000384, A252763.

Sequence in context: A332183 A000723 A020525 * A341574 A283017 A093189

Adjacent sequences: A252759 A252760 A252761 * A252763 A252764 A252765

Keyword

nonn,easy

Author

Colin Barker, Dec 21 2014

Status

approved