A387680 Number of length n words on the alphabet {a,b,c,d,e} that do not contain exactly one a and exactly one b.

1, 5, 23, 107, 517, 2585, 13195, 67919, 349801, 1795661, 9175135, 46662995, 236346157, 1193068193, 6006793363, 30182770295, 151439978065, 759036550421, 3801524968999, 19029320392379, 95220211854805, 476349008386985, 2382574896622363, 11915636016357407, 59587322430486457

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (14,-72,162,-135).

Formula

a(n) = 5^n - n*(n-1)*3^(n-2).

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

G.f.: (1 - x)*(1 - 8*x + 17*x^2)/((1 - 5*x)*(1 - 3*x)^3). - Robert Israel, Dec 02 2025

a(n) = 5^n - A006043(n-2).

Example

a(2) = 23 since there are 25 length-2 strings and all work except ab and ba.
a(4) = 517 since from the 625 words of length 4 we subtract the following 108 (number of permutations in parentheses): abcc (12), abdd (12), abee (12), abcd (24), abce (24), abde (24).

Mathematica

A387680[n_] := 5^n - n*(n - 1)*3^(n - 2); Array[A387680, 25, 0] (* Paolo Xausa, Jan 02 2026 *)

Crossrefs

Cf. A006043, A126473.

Sequence in context: A026894 A126473 A238112 * A109877 A336704 A179598

Adjacent sequences: A387677 A387678 A387679 * A387681 A387682 A387683

Keyword

nonn,easy

Author

Enrique Navarrete, Dec 01 2025

Status

approved