A378203 Number of palindromic n-ary words of length n that include the last letter of their respective alphabet.

1, 1, 1, 5, 7, 61, 91, 1105, 1695, 26281, 40951, 771561, 1214423, 26916709, 42664987, 1087101569, 1732076671, 49868399761, 79771413871, 2560599031177, 4108933742199, 145477500542221, 234040800869107, 9059621800971105, 14605723004036255, 613627780919407801

Offset

0,4

Links

Table of n, a(n) for n=0..25.

Formula

a(n) = n^h(n) - (n-1)^h(n) for n > 0, where h(n) = ceiling(n/2).

a(n) = A047969(n-1,h(n)-1) for n > 0.

Example

a(0) = 1: ().
a(1) = 1: (a).
a(2) = 1: (b,b).
a(3) = 5: (a,c,a), (b,c,b), (c,a,c), (c,b,c), (c,c,c).

Maple

a:= n-> (h-> n^h-`if`(n=0, 0, (n-1)^h))(ceil(n/2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 21 2024

Mathematica

h[n_] := Ceiling[n/2]; a[n_] := n^h[n] - (n - 1)^h[n]; Join[{1}, Table[a[n], {n, 25}]] (* James C. McMahon, Nov 21 2024 *)

Prog

(PARI)
h(n) = {ceil(n/2)}
a(n) = {n^h(n)-(n-1)^h(n)}
(Python)
def A378203(n): return n**(m:=n+1>>1)-(n-1)**m if n else 1 # Chai Wah Wu, Nov 21 2024

Crossrefs

Cf. A045531, A047969, A248828, A252764, A292845.

Sequence in context: A260831 A268433 A201109 * A177336 A108200 A077780

Adjacent sequences: A378200 A378201 A378202 * A378204 A378205 A378206

Keyword

nonn,easy

Author

John Tyler Rascoe, Nov 19 2024

Status

approved