A333868 The number of ways to write n as the difference of two k-simplex numbers for k >= 2.

1, 3, 3, 4, 5, 4, 3, 6, 7, 4, 4, 4, 5, 9, 4, 4, 5, 5, 7, 9, 4, 4, 4, 6, 4, 7, 7, 4, 7, 5, 3, 6, 6, 11, 9, 4, 4, 6, 4, 4, 6, 4, 5, 11, 5, 4, 4, 6, 6, 6, 5, 4, 7, 12, 8, 6, 4, 4, 6, 4, 4, 8, 5, 8, 9, 4, 4, 7, 8, 4, 5, 4, 5, 8, 4, 8, 9, 4, 5, 8, 4, 6, 10, 7, 4, 6

Offset: 2

Author

Peter Kagey, Apr 08 2020

Keywords

• nonn

Comments

a(n) >= A001227(n) + A307666(n).

a(n) >= A003016(n) + A003016(n+1) - 2.

Records occur at indices 2, 3, 5, 6, 9, 10, 15, 35, 55, 105, 210, 1365, 2925, 3003,...

Examples

			The a(9) = 6 ways to write 9 as the difference of k-simplex numbers for k > 2 are:
C(5,  2) - C(2, 2) = 10 -  1,
C(6,  2) - C(4, 2) = 15 -  6,
C(10, 2) - C(9, 2) = 45 - 36,
C(5,  3) - C(3, 3) = 10 -  1,
C(9,  8) - C(7, 8) =  9 -  0, and
C(10, 9) - C(9, 9) = 10 -  1,
where C(n,k) = binomial(n,k) = A007318(n,k).
		

Links

• Peter Kagey, Table of n, a(n) for n = 2..5000

Crossrefs

Cf. A001227, A003016, A007318, A307666.

Cf. A333822, A333836.

The k-simplex numbers for 2 <= k <= 6 are A000217 (k=2), A000292 (k=3), A000332 (k=4), A000389 (k=5), and A000579 (k=6).