A251991 Numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to the sum of the hexagonal numbers H(m) and H(m+1) for some m.

60, 11704, 2270580, 440480880, 85451020204, 16577057438760, 3215863692099300, 623860979209825504, 121025814103014048540, 23478384075005515591320, 4554685484736967010667604, 883585505654896594553923920, 171411033411565202376450572940

Offset

1,1

Comments

Also nonnegative integers y in the solutions to 4*x^2-3*y^2+2*x-2*y = 0, the corresponding values of x being A251990.

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.: -4*x*(x+15) / ((x-1)*(x^2-194*x+1)).

a(n) = (-4-(-2+sqrt(3))*(97+56*sqrt(3))^(-n)+(2+sqrt(3))*(97+56*sqrt(3))^n)/12. - Colin Barker, Mar 02 2016

a(n) = 194*a(n-1)-a(n-2)+64. - Vincenzo Librandi, Mar 03 2016

Example

60 is in the sequence because P(60)+P(61) = 5370+5551 = 10921 = 5356+5565 = H(52)+H(53).

Mathematica

LinearRecurrence[{195, -195, 1}, {60, 11704, 2270580}, 30] (* Vincenzo Librandi, Mar 03 2016 *)

Prog

(PARI) Vec(-4*x*(x+15)/((x-1)*(x^2-194*x+1)) + O(x^100))
(Magma) I:=[60, 11704]; [n le 2 select I[n] else 194*Self(n-1) - Self(n-2)+64: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016

Crossrefs

Cf. A000326, A000384, A251990.

Sequence in context: A309996 A146513 A269883 * A145411 A248708 A184890

Adjacent sequences: A251988 A251989 A251990 * A251992 A251993 A251994

Keyword

nonn,easy

Author

Colin Barker, Dec 12 2014

Status

approved