A385589 a(n) = 2^(n-2)*(3*binomial(n,3) + 6*binomial(n,2) + 6*n + 4).

1, 5, 22, 86, 304, 992, 3040, 8864, 24832, 67328, 177664, 458240, 1159168, 2883584, 7069696, 17113088, 40960000, 97058816, 227934208, 530972672, 1227882496, 2820669440, 6440353792, 14623440896, 33034338304, 74272735232, 166262210560, 370675810304, 823291543552, 1822139875328

Offset

0,2

Comments

a(n) is the number of words of length n defined on 5 letters that contain zero or one a's, zero or one b's, zero or one c's, and any number of d's and e's.

Links

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Formula

E.g.f.: exp(2*x)*(1+x)^3.

G.f.: (1 - 3*x + 6*x^2 - 2*x^3)/(1 - 2*x)^4. - Stefano Spezia, Jul 03 2025

Example

a(1) = 5 since the words are a, b, c, d, e.
a(2) = 22 since the words are ab, ba, ac, ca, ad, da, ae, ea, bc, cb, bd, db, be, eb, cd, dc, ce, ec, de, ed, dd, ee.

Mathematica

LinearRecurrence[{8, -24, 32, -16}, {1, 5, 22, 86}, 30] (* Amiram Eldar, Jul 03 2025 *)

Crossrefs

Cf. A385407.

Sequence in context: A216041 A122058 A387843 * A191008 A006148 A262293

Adjacent sequences: A385586 A385587 A385588 * A385590 A385591 A385592

Keyword

nonn,easy

Author

Enrique Navarrete, Jul 03 2025

Status

approved