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
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
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