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A006342
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Coloring a circuit with 4 colors.
(Formerly M3398)
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2
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1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684, 435848050, 1307544151, 3922632451, 11767897354, 35303692060, 105911076181, 317733228541
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture. Ph.D. Dissertation, Kansas State Univ., 1974.
G. D. Birkhoff, D. C. Lewis, Chromatic polynomials. Trans. Amer. Math. Soc. 60, (1946). 355-451.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, Arxiv preprint arXiv:1104.4323, 2011.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| G.f.: (1 - 2 x ) / ( 1 - x^2 ) ( 1 - 3 x ).
Binomial transform of A002001 (with interpolated zeros). Partial sums of A054878. E.g.f.: exp(x)(3cosh(2x)+1)/4; a(n)=3*3^n/8+1/4+3(-1)^n/8=sum{k=0..n, (3^k+3(-1)^k)/4 }. - Paul Barry (pbarry(AT)wit.ie), Sep 03 2003
a(n) = 2*a(n-1) + 3*a(n-2) - 1, n>1 [From Gary Detlefs (gdetlefs(AT)aol.com), Jun 21 2010]
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MAPLE
| A006342:=-(-1+2*z)/(z-1)/(3*z-1)/(z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-1 od: seq(a[n], n=1..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 28 2008
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PROG
| (MAGMA) [3*3^n/8+1/4+3*(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Aug 20 2011
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CROSSREFS
| Sequence in context: A145453 A034730 A095127 * A135831 A015796 A034717
Adjacent sequences: A006339 A006340 A006341 * A006343 A006344 A006345
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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