A319602 Numbers with at least two representations as truncated triangular numbers.

36, 75, 91, 102, 127, 153, 168, 190, 192, 201, 213, 231, 267, 270, 300, 322, 333, 348, 351, 361, 388, 397, 420, 426, 432, 435, 465, 487, 498, 531, 543, 546, 558, 582, 586, 595, 621, 627, 630, 657, 663, 673, 685, 696, 712, 717, 738, 762, 768, 777, 811, 816, 817

Offset: 1

Author

Allan C. Wechsler, Nov 15 2018

Keywords

• easy
• nonn

Comments

A truncated triangular number is a figurate number, the number of dots in a hexagonal diagram where the side lengths alternate between two values. Include a number in this list if there are two different side-length pairs that give the same count.

The underlying quadratic form is (4ab + a(a-3) + b(b-3) + 2)/2; n is in the list if n can be expressed in this form in two different ways, where a <= b. (That is, exchanging a and b is not considered different.)

A number occurs at least three times in A008867 if and only if it occurs in this sequence.

Examples

			75 is in the list because there are 75 dots in both the (2,10) hexagon and the (5,6) hexagon.
Table of solutions for the smallest 10 examples:
36: (1,8) (3,5)
75: (2,10) (5,6)
91: (1,13) (6,6)
102: (2,12) (4,9)
127: (3,12) (7,7)
153: (1,17) (4,12)
168: (2,16) (7,9)
190: (1,19) (7,10)
192: (4,14) (8,9)
201: (3,16) (5,13)
		

Crossrefs

Cf. A008912 (all truncated triangular numbers), A008867 (see comments).

Extensions

More terms from Alois P. Heinz, Nov 15 2018