A076049 Numbers k such that the sum of the k-th triangular number and (k+2)-nd triangular number is a triangular number.

0, 3, 8, 25, 54, 153, 322, 899, 1884, 5247, 10988, 30589, 64050, 178293, 373318, 1039175, 2175864, 6056763, 12681872, 35301409, 73915374, 205751697, 430810378, 1199208779, 2510946900, 6989500983, 14634871028, 40737797125

Offset: 1

Author

Bruce Corrigan (scentman(AT)myfamily.com), Oct 29 2002

Keywords

• easy
• nonn

Comments

T(a(n)) + T(a(n)+2) = A069017(n+1) where T(k) = k*(k+1)/2.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

Crossrefs

Cf. A000217, A001109, A069017.

Formula

Let b(n) = A001109(n). Then we have a pair of recursion formulas:

a(2n+2) = 2*a(2n+1) - a(2n) + 2*b(n+1);

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

G.f.: x*(3 + 5*x - x^2 - x^3)/((1-x)*(1 - 6*x^2 + x^4)).

a(n) = -3 + (1/8)*(-1^n)((7 + 5*sqrt(2))*(-1 - sqrt(2))^n + (7 - 5*sqrt(2))*(-1 + sqrt(2))^n - (1 + sqrt(2))^n - (1 - sqrt(2))^n).

Extensions

Edited by Jon E. Schoenfield, Sep 02 2019