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

%I A247247 #27 Dec 04 2014 06:09:22
%S A247247 0,1,3,21,120,300,2080,11781,29403,203841,1154440,2881200,19974360,
%T A247247 113123361,282328203,1957283461,11084934960,27665282700,191793804840,
%U A247247 1086210502741,2710915376403,18793835590881,106437544333680,265642041604800,1841604094101520
%N A247247 Triangular numbers that are the sum of 2 consecutive terms of A130518.
%C A247247 What will be the distribution of these triangular numbers?
%C A247247 Will they mostly be multiples of three?
%C A247247 From _Hiroaki Yamanouchi_, Dec 04 2014: (Start)
%C A247247 a(n) is some nonnegative x in the integer solutions (x, y) of
%C A247247 (1) (6*x + 3)^2 - 6*(6*y + 4)^2 = -15,
%C A247247 (2) (6*x + 3)^2 - 6*(6*y + 8)^2 = -15 or
%C A247247 (3) (2*x + 1)^2 - 6*(2*y + 2)^2 = 1.
%C A247247 (End)
%H A247247 Hiroaki Yamanouchi, <a href="/A247247/b247247.txt">Table of n, a(n) for n = 1..1000</a>
%F A247247 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
%e A247247 A130518(8)+A130518(9) = 9+12 = 21 = A000217(6), so 21 is in the sequence.
%p A247247 f:= proc(n)
%p A247247 local x;
%p A247247   x:= ceil((n^2+2*n)/3);
%p A247247 if issqr(1+8*x) then x else NULL fi
%p A247247 end proc:
%p A247247 seq(f(n),n=0..10^6); # _Robert Israel_, Nov 30 2014
%t A247247 a247247[n_Integer] := Module[{a130518, a000217, s},
%t A247247   a130518[m_] := Table[i, {i, 0, m}, {3}] // Flatten // Accumulate;
%t A247247   a000217[m_] := Accumulate[Range[m]];
%t A247247   s[m_] :=
%t A247247    a130518[m] + Most@PrependTo[a130518[m], 0] // DeleteDuplicates;
%t A247247 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 *)
%Y A247247 Cf. A000217, A130518.
%K A247247 nonn
%O A247247 1,3
%A A247247 _J. M. Bergot_, Nov 28 2014
%E A247247 a(7)-a(13) from _Michel Marcus_, Nov 28 2014
%E A247247 a(14)-a(24) from _Michael De Vlieger_, Nov 30 2014
%E A247247 a(25) from _Hiroaki Yamanouchi_, Dec 04 2014