A252769 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 heptagonal number H(m) for some m.

94, 5908, 366282, 22703656, 1407260470, 87227445564, 5406694364578, 335127823158352, 20772518341453326, 1287561009346947940, 79808010061169319034, 4946809062783150832248, 306622353882494182280422, 19005639131651856150553996, 1178043003808532587152067410

Offset

1,1

Comments

Also positive integers x in the solutions to 12*x^2-5*y^2+32*x+3*y+36 = 0, the corresponding values of y being A252770.

Links

Colin Barker, Table of n, a(n) for n = 1..557

Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Formula

a(n) = 63*a(n-1)-63*a(n-2)+a(n-3).

G.f.: 2*x*(7*x-47) / ((x-1)*(x^2-62*x+1)).

a(n) = 2*(-2/3+1/240*(31+8*sqrt(15))^(-n)*(80-27*sqrt(15)+(31+8*sqrt(15))^(2*n)*(80+27*sqrt(15)))). - Colin Barker, Mar 03 2016

Example

94 is in the sequence because P(94)+P(95)+P(96)+P(97) = 13207+13490+13776+14065 = 54538 = H(148).

Mathematica

LinearRecurrence[{63, -63, 1}, {94, 5908, 366282}, 30] (* Harvey P. Dale, Mar 04 2015 *)

Prog

(PARI) Vec(2*x*(7*x-47)/((x-1)*(x^2-62*x+1)) + O(x^100))

Crossrefs

Cf. A000326, A000566, A252770.

Sequence in context: A220759 A218178 A084238 * A218519 A281138 A127457

Adjacent sequences: A252766 A252767 A252768 * A252770 A252771 A252772

Keyword

nonn,easy

Author

Colin Barker, Dec 21 2014

Status

approved