A253120 Numbers n such that the sum of the octagonal numbers O(n), O(n+1), O(n+2) and O(n+3) is equal to the hexagonal number H(m) for some m.

6, 705, 69196, 6780615, 664431186, 65107475725, 6379868189976, 625161975142035, 61259493695729566, 6002805220206355545, 588213652086527113956, 57638935099259450812255, 5648027426075339652487146, 553449048820284026492928165, 54232358756961759256654473136

Offset

1,1

Comments

Also positive integers x in the solutions to 24*x^2-4*y^2+56*x+2*y+60 = 0, the corresponding values of y being A253121.

Links

Colin Barker, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Formula

a(n) = 99*a(n-1)-99*a(n-2)+a(n-3).

G.f.: x*(5*x^2-111*x-6) / ((x-1)*(x^2-98*x+1)).

a(n) = -7/6+1/48*(49+20*sqrt(6))^(-n)*(-92-39*sqrt(6)+(49+20*sqrt(6))^(2*n)*(-92+39*sqrt(6))). - Colin Barker, Mar 03 2016

Example

6 is in the sequence because O(6)+O(7)+O(8)+O(9) = 96+133+176+225 = 630 = H(18).

Prog

(PARI) Vec(x*(5*x^2-111*x-6)/((x-1)*(x^2-98*x+1)) + O(x^100))

Crossrefs

Cf. A000384, A000567, A253121.

Sequence in context: A116298 A112637 A289366 * A354250 A218559 A201391

Adjacent sequences: A253117 A253118 A253119 * A253121 A253122 A253123

Keyword

nonn,easy

Author

Colin Barker, Dec 27 2014

Status

approved