A252003 Numbers n such that the sum of the octagonal numbers N(n) and N(n+1) is equal to the sum of the pentagonal numbers P(m) and P(m+1) for some m.

232, 267920, 309179640, 356793036832, 411738855324680, 475146282251644080, 548318397979541943832, 632758956122109151538240, 730203287046515981333185320, 842653960492723320349344321232, 972421940205315665167162013516600, 1122174076342973784879584614253835360

Offset

1,1

Comments

Also nonnegative integers x in the solutions to 6*x^2-3*y^2+2*x-2*y = 0, the corresponding values of y being A252004.

Links

Colin Barker, Table of n, a(n) for n = 1..326

Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

Formula

a(n) = 1155*a(n-1)-1155*a(n-2)+a(n-3).

G.f.: 8*x*(5*x-29) / ((x-1)*(x^2-1154*x+1)).

a(n) = (-2 - (sqrt(2)-1)*(577+408*sqrt(2))^(-n) + (sqrt(2)+1)*(577+408*sqrt(2))^n) / 12. - Colin Barker, May 30 2017

Example

232 is in the sequence because N(232)+N(233) = 161008+162401 = 323409 = 161212+162197 = P(328)+P(329).

Mathematica

LinearRecurrence[{1155, -1155, 1}, {232, 267920, 309179640}, 20] (* Harvey P. Dale, Sep 14 2020 *)

Prog

(PARI) Vec(8*x*(5*x-29)/((x-1)*(x^2-1154*x+1)) + O(x^100))

Crossrefs

Cf. A000326, A000567, A252004.

Sequence in context: A207130 A125850 A139673 * A151629 A301828 A132917

Adjacent sequences: A252000 A252001 A252002 * A252004 A252005 A252006

Keyword

nonn,easy

Author

Colin Barker, Dec 12 2014

Status

approved