A252763 Numbers n such that the hexagonal number H(n) is equal to the sum of the pentagonal numbers P(m), P(m+1), P(m+2) and P(m+3) for some m.

8, 1480, 287064, 55688888, 10803357160, 2095795600104, 406573543062968, 78873171558615640, 15300988708828371144, 2968312936341145386248, 575837408661473376560920, 111709488967389493907432184, 21671065022264900344665282728, 4204074904830423277371157417000

Offset

1,1

Comments

Also positive integers y in the solutions to 12*x^2-4*y^2+32*x+2*y+36 = 0, the corresponding values of x being A252762.

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.: -8*x*(3*x^2-10*x+1) / ((x-1)*(x^2-194*x+1)).

a(n) = (6+(285-164*sqrt(3))*(97+56*sqrt(3))^n+(97+56*sqrt(3))^(-n)*(285+164*sqrt(3)))/24. - Colin Barker, Mar 02 2016

a(n) = 194*a(n-1)-a(n-2)-48. - Vincenzo Librandi, Mar 03 2016

Example

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

Mathematica

LinearRecurrence[{195, -195, 1}, {8, 1480, 287064}, 30] (* Vincenzo Librandi, Mar 03 2016 *)

Prog

(PARI) Vec(-8*x*(3*x^2-10*x+1)/((x-1)*(x^2-194*x+1)) + O(x^100))
(Magma) I:=[8, 1480]; [n le 2 select I[n] else 194*Self(n-1)- Self(n-2)-48: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016

Crossrefs

Cf. A000326, A000384, A252762.

Sequence in context: A201492 A172938 A252176 * A096970 A248386 A114617

Adjacent sequences: A252760 A252761 A252762 * A252764 A252765 A252766

Keyword

nonn,easy

Author

Colin Barker, Dec 21 2014

Status

approved