%I #49 Apr 25 2024 13:32:49
%S 1,2,5,14,45,164,661,2906,13829,70736,386397,2242118,13759933,
%T 88975628,604202693,4296191090,31904681877,246886692680,1986631886029,
%U 16592212576862,143589971363981,1285605080403332,11891649654471285,113491862722958474,1116236691139398565
%N Number of maximal independent vertex sets (and minimal vertex covers) in the n-triangular honeycomb bishop graph.
%C From _Andrew Howroyd_, Aug 09 2017: (Start)
%C See A146304 for algorithm and PARI code to produce this sequence.
%C The total number of independent vertex sets is given by Bell(n+1) where Bell=A000110.
%C A bishop can move along two axes in the triangular honeycomb grid.
%C Equivalently, the number of arrangements of non-attacking rooks on an n X n right triangular board with every square controlled by at least one rook. (End)
%H Andrew Howroyd, <a href="/A290615/b290615.txt">Table of n, a(n) for n = 1..150</a>
%H Max Alekseyev, <a href="https://mathoverflow.net/a/405996">Subsequences of odd powers</a>, answer to question on Mathoverflow.
%H Max Alekseyev, <a href="https://mathoverflow.net/a/406085">Generating function for partial sums of the sequence</a>, answer to question on Mathoverflow.
%H Peter Taylor, <a href="https://mathoverflow.net/a/404182">Subsequence of the cubes</a>, answer to question on Mathoverflow.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalVertexCover.html">Minimal Vertex Cover</a>
%F Conjecture: a(n) = Sum_{k=0..2^(n-1) - 1} b(k) for n > 0 where b(2n+1) = b(n - 2^f(n)), b(2n) = b(n) + b(2n - 2^f(n)) for n > 0 with b(0) = b(1) = 1 and where f(n) = A007814(n). Also b((4^n - 1)/3) = (floor((n+1)/2)!)^3. - _Mikhail Kurkov_, Sep 18 2021
%F a(n) = Sum_{m=0..n} Sum_{k=0..min(m,n-m)} k! * S(m,k) * S(n+1-m,k+1), where S(,) are Stirling numbers of second kind. - _Max Alekseyev_, Oct 14 2021
%t Table[Sum[k! StirlingS2[m, k] StirlingS2[n + 1 - m, k + 1], {m, 0, n}, {k, 0, Min[m, n - m]}], {n, 20}] (* _Eric W. Weisstein_, Feb 01 2024 *)
%o (PARI) { A290615(n) = sum(m=0, n, sum(k=0, min(m,n-m), k! * stirling(m,k,2) * stirling(n+1-m,k+1,2) )); } \\ _Max Alekseyev_, Oct 14 2021
%Y Row sums of A290724.
%Y Cf. A000110 (independent vertex sets), A007814, A146304.
%Y Similar recurrences: A124758, A243499, A284005, A329369, A341392.
%K nonn
%O 1,2
%A _Eric W. Weisstein_, Aug 07 2017
%E Terms a(10) and beyond from _Andrew Howroyd_, Aug 09 2017