A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

19, 90, 162, 299, 509, 816, 1248, 1837, 2619, 3634, 4926, 6543, 8537, 10964, 13884, 17361, 21463, 26262, 31834, 38259, 45621, 54008, 63512, 74229, 86259, 99706, 114678, 131287, 149649, 169884, 192116, 216473, 243087, 272094, 303634, 337851, 374893, 414912, 458064, 504509

Offset: 3

Author

Ilya Gutkovskiy, Nov 05 2024

Keywords

• nonn

Comments

From David A. Corneth, Nov 06 2024: (Start)

a(n) <= (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. Proof:

A polygonal number is of the form P(m, n) = m/2 * ((n - 2) * m - n + 4).

We have P(n - 5, n) + P(n - 4, n) + P(n, n) = P(n - 6, n) + P(n - 2, n) + P(n - 1, n) = (3*n^3 - 18*n^2 + 21*n) / 2.

This lets us find the upper bound on a(n) by making two lists from 1 through n + 3. From one of them we remove n-2, n-1 and n + 3 and from the other we remove n-3, n+1 and n+2. The sum for remaining polygonal numbers is the same giving an upper bound on a(n) which turns out to be (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 (End)

Examples

			a(3) = 19 = 1 + 3 + 15 = 3 + 6 + 10.
a(4) = 90 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2.
		

Links

• Eric Weisstein's World of Mathematics, Polygonal Number

Crossrefs

Cf. A006484, A025377, A057145, A350405, A374144, A374256, A374287.

Formula

From David A. Corneth, Nov 06 2024: (Start)

a(n) >= A006484(n).

Conjecture: a(n) = (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. (End)

Conjectured g.f.: x^3*(19 - 5*x - 98*x^2 + 199*x^3 - 171*x^4 + 72*x^5 - 12*x^6) / (1 - x)^5.

Extensions

a(12)-a(36) from Michael S. Branicky, Nov 06 2024

More terms from David A. Corneth, Nov 10 2024