A364135 Let d_r d_{r-1} ... d_1 d_0 be the decimal expansion of n; a(n) is the number of nonnegative integer solutions x_r ... x_0 to the Diophantine equation d_r*x_r + ... + d_0*x_0 = n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 7, 5, 4, 4, 3, 3, 3, 3, 1, 11, 12, 4, 7, 3, 5, 2, 4, 2, 1, 11, 6, 12, 3, 3, 7, 2, 2, 5, 1, 11, 11, 4, 12, 3, 4, 2, 7, 2, 1, 11, 6, 4, 3, 12, 2, 2, 2, 2, 1, 11, 11, 11, 6, 3, 12, 2, 3, 4, 1, 11, 6, 4, 3, 3, 2, 12, 2, 2, 1, 11, 11

Offset: 1

Author

Ctibor O. Zizka, Jul 10 2023

Keywords

• base
• look
• nonn

Comments

For a formula for a(n), please see the Samsonadze article in Links section. a(n) = P(b) if n = b AND the nonzero digits of b are the coefficients a_i (in the article).

a(n) is the number of partitions of n into parts that are nonzero digits of n. - Stefano Spezia, Feb 17 2024

Examples

			For n = 10: 1*x_1 + 0*x_0 = 10, the solution is x_1 = 10, thus a(10) = 1.
For n = 22: 2*x_1 + 2*x_0 = 22, the solutions are (0,11), (2,10), ..., (11,0), thus a(22) = 12.
		

Links

• Robert P. P. McKone, Table of n, a(n) for n = 1..636
• Eteri Samsonadze, On the number of integer non-negative solutions of a linear Diophantine equation, arXiv:2108.04756 [math.NT], 2021.

Crossrefs

Cf. A007954, A034838, A052423, A055642.

Programs

Mathematica

a[n_Integer] := Module[{ds = IntegerDigits[n], p, t, v}, p = Table[If[d == 0, {0}, Range[0, Quotient[n, d]]], {d, ds}]; t = Tuples[p]; v = Select[t, ds . # == n &]; Length[v]]; Table[a[n], {n, 1, 82}] (* Robert P. P. McKone, Aug 25 2023 *)
a[n_]:=Length[IntegerPartitions[n, All,DeleteCases[ IntegerDigits[n],0]]]; Array[a,82] (* Stefano Spezia, Feb 17 2024 *)

Formula

a(n) = 1 for n = (d 0 ... 0), the digit d >= 1, the number of zeros >= 0.

a(n) = (1 ... 1) + 1 for n = (d ... d), the digit d >= 1, n >= 10.