Search: id:A141384
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%I A141384
%S A141384 8,8,32,158,828,4408,23564,126106,675076,3614144,19349432,103593806,
%T A141384 554625900,2969386480,15897666068,85113810058,455687062276,
%U A141384 2439682811480,13061709929936,69930511268510,374397872321628
%N A141384 Traces of the powers of an order-8 adjacency matrix.
%C A141384 a(n) is the trace of the n-th power of the adjacency matrix of order 
               8 whose rows are (up to simultaneous permutations of the rows and 
               columns): 10111010 01111001 01111001 10111010 00111000 11111111 11111111 
               11111111.
%C A141384 For n>2, this is also the number of ways to mark one edge at every vertex 
               of a regular n-gonal prism so that no edge is marked at both extremities.
%C A141384 Remarkably, for n>1, a(n)=A141221(n)+2. The fourth-order linear recurrence 
               established by Max Alekseyev for A141221, based on the characteristic 
               polynomial of the above (singular) matrix, namely x^4(x-1)(x^3-7x^2+9x-1) 
               = x^4(x^4-8x^3+16x^2-10x+1). Because of the x^4 factor, the recurrence 
               is guaranteed a priori to hold for corresponding elements of the 
               successive powers (or sums thereof, including matrix traces) only 
               if n is 4 or more. It happens to have a greater validity downward 
               the case of this sequence (and A141221 as well). The recurrence would 
               be valid down to n=0 if we had a(0)=4, which is not the case.
%H A141384 G. P. Michon, <a href="http://www.numericana.com/answer/graphs.htm#prisms">
               A screaming game for short-sighted people</a>.
%H A141384 G. P. Michon, <a href="http://www.numericana.com/answer/graphs.htm#alekseyev">
               Silent circles</a>, enumerated by Max Alekseyev.
%H A141384 G. P. Michon, <a href="http://www.numericana.com/answer/counting.htm#scream">
               Brocoum's Screaming Circles</a>.
%F A141384 For positive values of n: a(n) = (5.3538557854308)^n + (1.5235479602692)^n 
               + 1 + (0.1225962542999)^n. The dominant term in the above is the 
               n-th power of (7+2*sqrt(22)*cos(atan(sqrt(5319)/73)/3))/3. For n>
               0 we have: a(n+4) = 8*a(n+3)-16*a(n+2)+10*a(n+1)-a(n)
%e A141384 a(0) = 8 because the trace of the order-8 identity matrix is 8.
%e A141384 a(1) = 8 because all diagonal elements of the adjacency matrix are 1 
               (there's a loop at each vertex).
%Y A141384 Cf. A141221.
%Y A141384 Sequence in context: A143336 A122858 A053596 this_sequence A111218 A103744 
               A151782
%Y A141384 Adjacent sequences: A141381 A141382 A141383 this_sequence A141385 A141386 
               A141387
%K A141384 easy,nonn
%O A141384 0,1
%A A141384 Gerard P. Michon (g.michon(AT)att.net), Jun 29 2008

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