%I #15 Nov 19 2020 22:34:14

%S 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,

%T 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,

%U 7,8,4,5,4,5,8,4,8,9,4,5,8,4,6,10,7,4,6

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

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

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

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

%H Peter Kagey, <a href="/A333868/b333868.txt">Table of n, a(n) for n = 2..5000</a>

%e The a(9) = 6 ways to write 9 as the difference of k-simplex numbers for k > 2 are:

%e C(5, 2) - C(2, 2) = 10 - 1,

%e C(6, 2) - C(4, 2) = 15 - 6,

%e C(10, 2) - C(9, 2) = 45 - 36,

%e C(5, 3) - C(3, 3) = 10 - 1,

%e C(9, 8) - C(7, 8) = 9 - 0, and

%e C(10, 9) - C(9, 9) = 10 - 1,

%e where C(n,k) = binomial(n,k) = A007318(n,k).

%Y Cf. A001227, A003016, A007318, A307666.

%Y Cf. A333822, A333836.

%Y 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).

%K nonn

%O 2,2

%A _Peter Kagey_, Apr 08 2020