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

1,3

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)

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A000217, A130518.

Sequence in context: A283421 A117512 A068127 * A322185 A171137 A144884

Adjacent sequences: A247244 A247245 A247246 * A247248 A247249 A247250

Keyword

nonn

Author

J. M. Bergot, Nov 28 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

Status

approved