A251924 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to a hexagonal number H(m) for some m.

0, 34, 1188, 40390, 1372104, 46611178, 1583407980, 53789260174, 1827251437968, 62072759630770, 2108646576008244, 71631910824649558, 2433376321462076760, 82663163018885960314, 2808114166320660573948, 95393218491883573553950, 3240561314557720840260384

Offset

1,2

Comments

Also nonnegative integers x in the solutions to 2*x^2-4*y^2+4*x+2*y+2 = 0, the corresponding values of y being A008844.

First bisection of A076708. [Bruno Berselli, Dec 11 2014]

Links

Colin Barker, Table of n, a(n) for n = 1..653

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Formula

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).

G.f.: 2*x^2*(x-17) / ((x-1)*(x^2-34*x+1)).

a(n) = (-8-(4+3*sqrt(2))*(17+12*sqrt(2))^(-n)+(-4+3*sqrt(2))*(17+12*sqrt(2))^n)/8. - Colin Barker, Mar 02 2016

Example

34 is in the sequence because T(34)+T(35) = 595+630 = 1225 = H(25).

Mathematica

LinearRecurrence[{35, -35, 1}, {0, 34, 1188}, 20] (* Harvey P. Dale, Feb 04 2019 *)

Prog

(PARI) concat(0, Vec(2*x^2*(x-17)/((x-1)*(x^2-34*x+1)) + O(x^100)))

Crossrefs

Cf. A000217, A000384, A008844, A076708.

Sequence in context: A041545 A189434 A167258 * A158731 A297693 A093550

Adjacent sequences: A251921 A251922 A251923 * A251925 A251926 A251927

Keyword

nonn,easy

Author

Colin Barker, Dec 11 2014

Status

approved