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 A335259 #28 Jul 02 2020 04:56:40
%S A335259 1,1,2,1,4,3,1,4,9,4,1,8,9,16,5,1,8,9,16,25,6,1,16,27,16,25,36,7,1,16,
%T A335259 27,16,25,36,49,8,1,32,27,64,25,36,49,64,9,1,32,81,64,25,36,49,64,81,
%U A335259 10,1,64,81,64,125,36,49,64,81,100,11,1,64,81,64,125,36,49,64,81,100,121,12
%N A335259 Triangle read by rows: T(n,k) = k^ceiling(n/k) for 1 <= k <= n.
%C A335259 T(n,k) is the number of functions f:[n]->[k] such that f(x)=f(y) whenever i*k-k+1<=x<=y<=i*k where 1<=i<=ceiling(n/k). An example of such a function is f:[8]->[3] defined by f(1)=f(2)=f(3)=2, f(4)=f(5)=f(6)=3, and f(7)=f(8)=2. To count all functions of this type when n=8 and k=3, we note that there are 3 possible values for f(1), f(4), and f(7). Hence T(8,3)=3^3 or, equivalently, 3^ceiling(8/3). A proof of the general result follows the same approach. We also note the following: (i) T(n,1)=1 for all n; (ii) T(n,n)=n for all n; T(n,k)=k^2 when ceiling(n/2)<=k<n.
%F A335259 G.f. for fixed k: k*x^k*(1+k*x+k*x^2+...+k*x^(k-1))/(1-k*x^k).
%F A335259 For n>1, T(n,2) = A016116(n).
%F A335259 For n>2, T(n,3) = A127975(n).
%e A335259 Triangle T(n,k):
%e A335259 1;
%e A335259 1, 2;
%e A335259 1, 4, 3;
%e A335259 1, 4, 9, 4;
%e A335259 1, 8, 9, 16, 5;
%e A335259 1, 8, 9, 16, 25, 6;
%e A335259 1, 16, 27, 16, 25, 36, 7;
%e A335259 1, 16, 27, 16, 25, 36, 49, 8;
%e A335259 1, 32, 27, 64, 25, 36, 49, 64, 9;
%e A335259 1, 32, 81, 64, 25, 36, 49, 64, 81, 10;
%e A335259 ...
%e A335259 T(8,3) counts the 27 functions from [8] to [3] where f(1)=f(2)=f(3), f(4)=f(5)=f(6), and f(7)=f(8). Letting f be defined by its vector of images <f(1), ...,f(8)>, the 27 functions are <1,1,1,1,1,1,1,1>, <1,1,1,1,1,1,2,2>, <1,1,1,1,1,1,3,3>, <1,1,1,2,2,2,1,1>, <1,1,1,2,2,2,2,2>, <1,1,1,2,2,2,3,3>, <1,1,1,3,3,3,1,1>, <1,1,1,3,3,3,2,2>, <1,1,1,3,3,3,3,3>, <2,2,2,1,1,1,1,1>, <2,2,2,1,1,1,2,2>, <2,2,2,1,1,1,3,3>, <2,2,2,2,2,2,1,1>, <2,2,2,2,2,2,2,2>, <2,2,2,2,2,2,3,3>, <2,2,2,3,3,3,1,1>, <2,2,2,3,3,3,2,2>, <2,2,2,3,3,3,3,3>, <3,3,3,1,1,1,1,1>, <3,3,3,1,1,1,2,2>, <3,3,3,1,1,1,3,3>, <3,3,3,2,2,2,1,1>, <3,3,3,2,2,2,2,2>, <3,3,3,2,2,2,3,3>, <3,3,3,3,3,3,1,1>, <3,3,3,3,3,3,2,2>, and <3,3,3,3,3,3,3,3>.
%p A335259 seq(seq(k^ceil(n/k),k=1..n),n=1..20);
%t A335259 Table[k^Ceiling[n/k], {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Jun 28 2020 *)
%Y A335259 Cf. A016116, A127975.
%K A335259 nonn,tabl,easy
%O A335259 1,3
%A A335259 _Dennis P. Walsh_, May 28 2020