This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378203 #19 Nov 21 2024 22:27:54
%S A378203 1,1,1,5,7,61,91,1105,1695,26281,40951,771561,1214423,26916709,
%T A378203 42664987,1087101569,1732076671,49868399761,79771413871,2560599031177,
%U A378203 4108933742199,145477500542221,234040800869107,9059621800971105,14605723004036255,613627780919407801
%N A378203 Number of palindromic n-ary words of length n that include the last letter of their respective alphabet.
%F A378203 a(n) = n^h(n) - (n-1)^h(n) for n > 0, where h(n) = ceiling(n/2).
%F A378203 a(n) = A047969(n-1,h(n)-1) for n > 0.
%e A378203 a(0) = 1: ().
%e A378203 a(1) = 1: (a).
%e A378203 a(2) = 1: (b,b).
%e A378203 a(3) = 5: (a,c,a), (b,c,b), (c,a,c), (c,b,c), (c,c,c).
%p A378203 a:= n-> (h-> n^h-`if`(n=0, 0, (n-1)^h))(ceil(n/2)):
%p A378203 seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 21 2024
%t A378203 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 *)
%o A378203 (PARI)
%o A378203 h(n) = {ceil(n/2)}
%o A378203 a(n) = {n^h(n)-(n-1)^h(n)}
%o A378203 (Python)
%o A378203 def A378203(n): return n**(m:=n+1>>1)-(n-1)**m if n else 1 # _Chai Wah Wu_, Nov 21 2024
%Y A378203 Cf. A045531, A047969, A248828, A252764, A292845.
%K A378203 nonn,easy
%O A378203 0,4
%A A378203 _John Tyler Rascoe_, Nov 19 2024