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A126569
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Top-left "head" entry of the n-th power of the E8 Cartan matrix.
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3
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1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63442, 232828, 867146, 3269060, 12446684, 47771496, 184544427, 716658870, 2794956099, 10938266562, 42930256917, 168890693650, 665739119129, 2628578437646, 10393091551794, 41141896235012, 163028816478833
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..26.
Wikipedia, E8
Index entries for linear recurrences with constant coefficients, signature (16,-105,364,-714,784,-440,96,-1).
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FORMULA
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a(n) = leftmost term in M^n * [1,0,0,0,0,0,0,0], where M = the 8x8 matrix [2,-1,0,0,0,0,0,0; -1,2,-1,0,0,0,0,0; 0,-1,2,-1,0,0,0,-1; 0,0,-1,2,-1,0,0,0; 0,0,0,-1,2,-1,0,0; 0,0,0,0,-1,2,-1,0; 0,0,0,0,0,-1,2,0; 0,0,-1,0,0,0,0,2].
a(n) = 16*a(n-1)-105*a(n-2)+364*a(n-3)-714*a(n-4)+784*a(n-5)-440*a(n-6)+96*a(n-7) -a(n-8). - R. J. Mathar, May 08 2009 [Corrected by Georg Fischer, Mar 12 2020]
G.f.: -(2*x-1)*(2*x^2-4*x+1)*(x^4-16*x^3+20*x^2-8*x+1) / (1-16*x +105*x^2 -364*x^3+714*x^4-784*x^5+440*x^6-96*x^7+x^8). - R. J. Mathar, May 08 2009
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EXAMPLE
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a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0,0].
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MAPLE
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E8 := matrix(8, 8, [ [2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, -1 ], [ 0, 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, 0, -1, 2, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 2 ] ]) ;
printf("1, ") ; for n from 1 to 20 do T := evalm(E8^n) ; printf("%a, ", T[1, 1]) ; od: # R. J. Mathar, May 08 2009
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CROSSREFS
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Cf. A126566, A126567, A126568.
Sequence in context: A057413 A126567 A125501 * A162748 A061815 A340361
Adjacent sequences: A126566 A126567 A126568 * A126570 A126571 A126572
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson, Dec 28 2006
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EXTENSIONS
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Edited by R. J. Mathar, May 08 2009
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STATUS
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approved
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