A335979 Number of partitions of n into exactly two parts with no decimal carries.

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 3, 6, 10, 13, 17, 20, 24, 27, 31, 34, 3, 7, 11, 15, 19, 23

Offset: 0

Author

Jason Zimba, Jul 02 2020

Keywords

• nonn
• base
• easy

Comments

a(m) = a(n) if m and n have the same nonzero digits, irrespective of order. For example, a(6044005) = a(45604).

Examples

			a(31) = 3 because there are three partitions of 31 into exactly two parts with no decimal carries: 30 + 1, 21 + 10, and 20 + 11.
a(100) = 0 because every partition of 100 into exactly two parts has at least one decimal carry.
		

Crossrefs

Cf. A088512 (analogous sequence for base 2), A089898.

Programs

Mathematica

Ceiling[(1/2) Times @@ (IntegerDigits[n, 10] + 1)] - 1

Formula

If n has digits n_1, n_2, ..., n_k and all digits n_i are even, then a(n) = (1/2)(n_1 + 1)(n_2 + 1)...(n_k + 1) - 1/2. Otherwise, a(n) = (1/2)(n_1 + 1)(n_2 + 1)...(n_k + 1) - 1. Equivalently, a(n) = ceiling((1/2)(n_1 + 1)(n_2 + 1)...(n_k + 1)) - 1 for all n.

a(n) = ceiling((1/2)*A089898(n)) - 1.