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A141384
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Traces of the powers of an order-8 adjacency matrix.
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2
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8, 8, 32, 158, 828, 4408, 23564, 126106, 675076, 3614144, 19349432, 103593806, 554625900, 2969386480, 15897666068, 85113810058, 455687062276, 2439682811480, 13061709929936, 69930511268510, 374397872321628
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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.
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.
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.
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LINKS
| G. P. Michon, A screaming game for short-sighted people.
G. P. Michon, Silent circles, enumerated by Max Alekseyev.
G. P. Michon, Brocoum's Screaming Circles.
Index to sequences with linear recurrences with constant coefficients, signature (8,-16,10,-1).
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FORMULA
| 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).
G.f.: 2*(4-28*x+48*x^2-25*x^3+2*x^4)/((1-x)*(1-7*x+9*x^2-x^3)). [Colin Barker, Jan 20 2012]
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EXAMPLE
| a(0) = 8 because the trace of the order-8 identity matrix is 8.
a(1) = 8 because all diagonal elements of the adjacency matrix are 1 (there's a loop at each vertex).
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CROSSREFS
| Cf. A141221.
Sequence in context: A143336 A122858 A053596 * A183400 A111218 A188275
Adjacent sequences: A141381 A141382 A141383 * A141385 A141386 A141387
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KEYWORD
| easy,nonn
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AUTHOR
| Gerard P. Michon (g.michon(AT)att.net), Jun 29 2008
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