A247247 Triangular numbers that are the sum of 2 consecutive terms of A130518.

0, 1, 3, 21, 120, 300, 2080, 11781, 29403, 203841, 1154440, 2881200, 19974360, 113123361, 282328203, 1957283461, 11084934960, 27665282700, 191793804840, 1086210502741, 2710915376403, 18793835590881, 106437544333680, 265642041604800, 1841604094101520

Offset: 1

Author

J. M. Bergot, Nov 28 2014

Keywords

• nonn

Comments

What will be the distribution of these triangular numbers?

Will they mostly be multiples of three?

From Hiroaki Yamanouchi, Dec 04 2014: (Start)

a(n) is some nonnegative x in the integer solutions (x, y) of

(1) (6*x + 3)^2 - 6*(6*y + 4)^2 = -15,

(2) (6*x + 3)^2 - 6*(6*y + 8)^2 = -15 or

(3) (2*x + 1)^2 - 6*(2*y + 2)^2 = 1.

(End)

Examples

			A130518(8)+A130518(9) = 9+12 = 21 = A000217(6), so 21 is in the sequence.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000217, A130518.

Programs

Maple

f:= proc(n)
local x;
  x:= ceil((n^2+2*n)/3);
if issqr(1+8*x) then x else NULL fi
end proc:
seq(f(n),n=0..10^6); # Robert Israel, Nov 30 2014

Mathematica

a247247[n_Integer] := Module[{a130518, a000217, s},
  a130518[m_] := Table[i, {i, 0, m}, {3}] // Flatten // Accumulate;
  a000217[m_] := Accumulate[Range[m]];
  s[m_] :=
   a130518[m] + Most@PrependTo[a130518[m], 0] // DeleteDuplicates;
Intersection[s[n], a000217[n]]]; a247247[50000000] (* Michael De Vlieger, Nov 30 2014 after Jean-François Alcover at A130518 and Harvey P. Dale at A000217 *)

Formula

Empirical G.f.: x^2*(x+1)*(x^4+2*x^3+19*x^2+2*x+1)/((1-x)*(x^2+x+1)*(x^6-98*x^3+1)). - Robert Israel, Nov 30 2014

Extensions

a(7)-a(13) from Michel Marcus, Nov 28 2014

a(14)-a(24) from Michael De Vlieger, Nov 30 2014

a(25) from Hiroaki Yamanouchi, Dec 04 2014