A363256 Number of length n strings on the alphabet {0,1,2,3} with digit sum at most 4.

1, 4, 13, 32, 66, 121, 204, 323, 487, 706, 991, 1354, 1808, 2367, 3046, 3861, 4829, 5968, 7297, 8836, 10606, 12629, 14928, 17527, 20451, 23726, 27379, 31438, 35932, 40891, 46346, 52329, 58873, 66012, 73781, 82216, 91354, 101233, 111892, 123371, 135711

Offset

0,2

Links

Table of n, a(n) for n=0..40.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (((n + 10)*n + 35)*n + 26)*n/24 + 1.

G.f.: -(x^4 - 3*x^3 + 3*x^2 - x + 1)/(x - 1)^5.

a(n) = 1 + A005718(n-1) for n>=1.

Example

For n=2, the 13 strings are all possible 2-character strings of '0', '1', '2' and '3' except the four strings '33', '32', '23'.

Mathematica

f[n_, r_, l_] := If[r < 0, 0, If[r==0, 1, If[l < 0, 0, If[l == 0, 1, Sum[f[n, r-j, l-1], {j, 0, n}]]]]]; Table[f[3, 4, x], {x, 0, 40}]

Crossrefs

Cf. A227259 (the same for {0,1,2} with digit sum <= 4).

Cf. A105163 (the same for {0,1,2} with digit sum <= 3, shifted by 2).

Cf. A005718.

Sequence in context: A212747 A011936 A037235 * A051912 A060099 A208638

Adjacent sequences: A363253 A363254 A363255 * A363257 A363258 A363259

Keyword

nonn,easy

Author

Daniel T. Martin, May 23 2023

Status

approved