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Number of compositions of graph C_3 X P_n.
1

%I #23 Jan 05 2025 19:51:42

%S 5,114,2712,64518,1534872,36514338,868669752,20665502358,491628707832,

%T 11695761476178,278240131889112,6619284357957798,157471623931541592,

%U 3746222552567209218,89121983141955313272,2120196482644091472438,50439105667748418772152

%N Number of compositions of graph C_3 X P_n.

%H 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 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.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (24,-5).

%F a(n) = 24*a(n-1) - 5*a(n-2) for n >= 4.

%F G.f.: x*(5 - 6*x + x^2)/(1 - 24*x + 5*x^2).

%F For n>1, a(n) = z * M^(n-1) * z^T, where z is the 1 X 5 row vector [1,1,1,1,1], z^T is its transpose (a 5 X 1 column vector of 1's), and M is the 5 X 5 matrix

%F [[8, 6, 6, 6, 4],

%F [6, 4, 5, 5, 3],

%F [6, 5, 4, 5, 3],

%F [6, 5, 5, 4, 3],

%F [4, 3, 3, 3, 2]].

%e For n=1 the a(1)=5 solutions are given here, where the first picture has all three vertices in the same partition (called A), the next three pictures have two vertices in the partition A and one in the partition B, and the last picture has all three vertices in their own partitions.

%e A A B A A

%e / \ / \ / \ / \ / \

%e A___A B___A A___A A___B B___C

%p a:= n-> ceil((<<0|1>, <-5|24>>^n. <<6/25, 24/5>>)[1$2]):

%p seq(a(n), n=1..21); # _Alois P. Heinz_, Jul 14 2021

%t M = {{8, 6, 6, 6, 4}, {6, 4, 5, 5, 3}, {6, 5, 4, 5, 3}, {6, 5, 5, 4,

%t 3}, {4, 3, 3, 3, 2}}; w = {1, 1, 1, 1, 1}; Join[{5},Table[Transpose[w] . MatrixPower[M, n, w], {n, 1, 40}]]

%Y Cf. A108808.

%K nonn,easy

%O 1,1

%A _Liam Buttitta_ and _Greg Dresden_, Jul 12 2021