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

Author

John Tyler Rascoe, Nov 19 2024

Keywords

• nonn
• easy

Examples

			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).
		

Crossrefs

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

Programs

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 *)

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

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.