A133215 Hexagonal numbers (A000384) which are sum of 2 other hexagonal numbers > 0.

276, 703, 861, 1225, 2850, 3003, 4560, 5151, 8128, 10878, 11781, 12090, 12720, 13366, 14706, 15400, 16110, 18721, 21115, 22366, 24090, 24531, 26796, 29161, 29646, 31125, 32131, 33153, 36315, 38503, 39621, 40186, 42486, 45451, 47895

Offset: 1

Author

Jonathan Vos Post, Dec 18 2007

Keywords

• nonn

Comments

This is to A136117 as A000384 is to A000326. Duke and Schulze-Pillot (1990) proved that every sufficiently large integer (and hence every sufficiently large hexagonal number) can be written as the sum of three hexagonal numbers.

Examples

			hex(19) = 703 = 378 + 325 = hex(14) + hex(13).
hex(21) = 861 = 630 + 231 = hex(18) + hex(11).
hex(25) = 1225 = 1035 + 190 = hex(23) + hex(10).
hex(38) = 2850 = 2415 + 435 = hex(35) + hex(15).
hex(39) = 3003 = 2850 + 153 = hex(38) + hex(9) = 2415 + 435 + 153 = hex(35) + hex(15) + hex(9).
hex(48) = 4560 = 2415 + 2145 = hex(35) + hex(33).
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• W. Duke and R. Schulze-Pilot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99(1990), 49-57.
• Eric Weisstein's World of Mathematics, Hexagonal Number.

Crossrefs

Cf. A000384, A136117.

Programs

Mathematica

With[{upto=60000},Select[Union[Total/@Subsets[Table[n(2n-1),{n, Ceiling[ (1+Sqrt[1+8upto])/4]}],{2}]],IntegerQ[(1+Sqrt[1+8#])/4]&&#<=upto&]] (* Harvey P. Dale, Jul 24 2011 *)

Formula

{x: x>0 and x in A000384 and x = A000384(i) + A000384(j) for i>0 and j>0}, where A000384 = {n*(2*n-1) for n > 0}.

Extensions

Added missing term 276 and a(8)-a(35) from Donovan Johnson, Sep 27 2008