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A095897
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Sequence generated from the Z/4Z addition table considered as a matrix.
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6
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1, 8, 48, 320, 1888, 11648, 69504, 419840, 2515456, 15116288, 90667008, 544194560, 3264913408, 19591036928, 117544157184, 705277460480, 4231648116736, 25389989101568, 152339800915968, 914039609753600, 5484236586876928
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Garrity, p. 218, states "Since the integers form an Abelian group, every subgroup including nZ, is normal and thus Z/nZ will form a group. It is common to represent each coset in Z/nZ by an integer between 0 and (n-1): Z/nZ = {0,1,2,...,(n-1}." Performing the analogous operation with the Z/3Z addition table (as a matrix), generates A007070: 1, 4, 14, 48, 164, 560... The recursion multipliers 4, 20, -32, -96 are present with changed signs in the characteristic polynomial of M: x^4 - 4x^3 - 20x^2 + 32x + 96.
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REFERENCES
| Thomas A. Garrity, "All the Mathematics You Missed But Need to Know for Graduate School", Cambridge University Press, 2002.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index to sequences with linear recurrences with constant coefficients, signature (4,20,-32,-96).
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FORMULA
| Let M = the Z/4Z = {0, 1, 2, 3} addition table considered as a matrix = [0 1 2 3 / 1 2 3 0 / 2 3 0 1 / 3 0 1 2]. Then a(n) = 2nd term from left in M^n * [1 0 0 0].
The recursion operation (n>4) is a(n+4) = 4*a(n+3) + 20*a(n+2) - 32*a(n+1) - 96*a(n).
Contribution by Bruno Berselli, Aug 04 2011: (Start)
G.f.: x*(1+4*x-4*x^2)/((1+2*x)*(1-6*x)*(1-8*x^2)).
a(n) = 2^(n-2)*(3^n-1+((-1)^n-1)*(sqrt(2)^(n-1)-1)).
a(2k+1) = 2^(2k-1)*(3*9^k-2*2^k+1), a(2k) = 4^(k-1)*(9^k-1). (End)
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EXAMPLE
| a(3) = 48 since M^3 * [1 0 0 0] = [44 48 60 64] (a(3) = 2nd term from left.
a(8) = 419840 = 4*69504 + 20*11648 - 32*1888 - 96*320.
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MATHEMATICA
| a[n_] := (MatrixPower[{{0, 1, 2, 3}, {1, 2, 3, 0}, {2, 3, 0, 1}, {3, 0, 1, 2}}, n].{{1}, {0}, {0}, {0}})[[2, 1]]; Table[ a[n], {n, 22}] (from Robert G. Wilson v Jun 16 2004)
LinearRecurrence[{4, 20, -32, -96}, {1, 8, 48, 320}, 30] (* From Harvey P. Dale, June 11 2011 *)
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PROG
| (MAGMA) [Floor(2^(n-2)*(3^n-1+((-1)^n-1)*(Sqrt(2)^(n-1)-1))): n in [1..30]]; // Vincenzo Librandi, Aug 05 2011
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CROSSREFS
| Cf. A007070.
Sequence in context: A037507 A037690 A180031 * A025013 A200161 A172111
Adjacent sequences: A095894 A095895 A095896 * A095898 A095899 A095900
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2004
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EXTENSIONS
| Edited, corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 16 2004
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