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 A344638 #22 Jan 05 2025 19:51:42
%S A344638 15,1548,168386,18328142,1994963186,217145777610,23635668646510,
%T A344638 2572671863723654,280027640317060130,30480171391948784938,
%U A344638 3317675523140039250350,361119061152982241895174,39306730094143339494849314,4278420047285488959291378858,465693230069569504343096792622
%N A344638 Number of compositions of graph K_4 X P_n.
%H A344638 Liam Buttitta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Buttitta/but3.html">On the Number of Compositions of Km X Pn</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.1.
%H A344638 J. N. Ridley and M. E. Mays, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-3/Ridley-Mays-scanned.pdf">Compositions of unions of graphs</a>, Fib. Quart. 42 (2004), 222-230.
%F A344638 a(n) = 112*a(n-1) - 346*a(n-2) + 306*a(n-3) - 57*a(n-4) + 2*a(n-5) for n >= 6.
%F A344638 G.f.: (-15 + 132*x - 200*x^2 + 72*x^3 - 5*x^4)/(-1 + 112*x - 346*x^2 + 306*x^3 - 57*x^4 + 2*x^5).
%F A344638 For n>1, a(n) = z * M^(n-1) * z^T, where z is the 1 X 15 row vector [1,1,1,...,1], z^T is its transpose (a 15 X 1 column vector of 1's), and M is the 15 X 15 matrix
%F A344638 [[16, 12, 12, 12, 12, 12, 12, 9, 9, 9, 8, 8, 8, 8, 5],
%F A344638 [12, 8, 10, 10, 9, 10, 10, 6, 8, 8, 6, 6, 7, 7, 4],
%F A344638 [12, 10, 8, 9, 10, 10, 10, 8, 6, 8, 6, 7, 6, 7, 4],
%F A344638 [12, 10, 9, 8, 10, 10, 10, 8, 6, 8, 7, 6, 7, 6, 4],
%F A344638 [12, 9, 10, 10, 8, 10, 10, 6, 8, 8, 7, 7, 6, 6, 4],
%F A344638 [12, 10, 10, 10, 10, 8, 9, 8, 8, 6, 7, 6, 6, 7, 4],
%F A344638 [12, 10, 10, 10, 10, 9, 8, 8, 8, 6, 6, 7, 7, 6, 4],
%F A344638 [ 9, 6, 8, 8, 6, 8, 8, 4, 7, 7, 5, 5, 5, 5, 3],
%F A344638 [ 9, 8, 6, 6, 8, 8, 8, 7, 4, 7, 5, 5, 5, 5, 3],
%F A344638 [ 9, 8, 8, 8, 8, 6, 6, 7, 7, 4, 5, 5, 5, 5, 3],
%F A344638 [ 8, 6, 6, 7, 7, 7, 6, 5, 5, 5, 4, 5, 5, 5, 3],
%F A344638 [ 8, 6, 7, 6, 7, 6, 7, 5, 5, 5, 5, 4, 5, 5, 3],
%F A344638 [ 8, 7, 6, 7, 6, 6, 7, 5, 5, 5, 5, 5, 4, 5, 3],
%F A344638 [ 8, 7, 7, 6, 6, 7, 6, 5, 5, 5, 5, 5, 5, 4, 3],
%F A344638 [ 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2]].
%e A344638 Here are the a(1) = 15 compositions of the graph K_4 x P_1 = K_4, where the first block represents all four vertices of K_4 in the same partition (called "a"), the second block shows three vertices in partition "a" and the fourth vertex in its own partition (called "b"), and so on, up to the last block which shows all four vertices each in its own partition:
%e A344638 aa aa aa ba ab bb ab ab aa ba cb ac ab ba ab
%e A344638 aa ab ba aa aa aa ab ba bc ca aa ab ca ac cd
%t A344638 M = {{16, 12, 12, 12, 12, 12, 12, 9, 9, 9, 8, 8, 8, 8, 5},
%t A344638 {12, 8, 10, 10, 9, 10, 10, 6, 8, 8, 6, 6, 7, 7, 4},
%t A344638 {12, 10, 8, 9, 10, 10, 10, 8, 6, 8, 6, 7, 6, 7, 4},
%t A344638 {12, 10, 9, 8, 10, 10, 10, 8, 6, 8, 7, 6, 7, 6, 4},
%t A344638 {12, 9, 10, 10, 8, 10, 10, 6, 8, 8, 7, 7, 6, 6, 4},
%t A344638 {12, 10, 10, 10, 10, 8, 9, 8, 8, 6, 7, 6, 6, 7, 4},
%t A344638 {12, 10, 10, 10, 10, 9, 8, 8, 8, 6, 6, 7, 7, 6, 4},
%t A344638 {9, 6, 8, 8, 6, 8, 8, 4, 7, 7, 5, 5, 5, 5, 3},
%t A344638 {9, 8, 6, 6, 8, 8, 8, 7, 4, 7, 5, 5, 5, 5, 3},
%t A344638 {9, 8, 8, 8, 8, 6, 6, 7, 7, 4, 5, 5, 5, 5, 3},
%t A344638 {8, 6, 6, 7, 7, 7, 6, 5, 5, 5, 4, 5, 5, 5, 3},
%t A344638 {8, 6, 7, 6, 7, 6, 7, 5, 5, 5, 5, 4, 5, 5, 3},
%t A344638 {8, 7, 6, 7, 6, 6, 7, 5, 5, 5, 5, 5, 4, 5, 3},
%t A344638 {8, 7, 7, 6, 6, 7, 6, 5, 5, 5, 5, 5, 5, 4, 3},
%t A344638 {5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2}};
%t A344638 w = Table[1, {15}]; Join[{15}, Table[Transpose[w] . MatrixPower[M, n, w], {n, 1, 40}]]
%Y A344638 Cf. A108808, A346273, A000110.
%K A344638 nonn,easy
%O A344638 1,1
%A A344638 _Liam Buttitta_ and _Greg Dresden_, Jul 15 2021