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A006342
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Coloring a circuit with 4 colors.
(Formerly M3398)
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6
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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, 953199685624, 2859599056870, 8578797170611
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OFFSET
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0,3
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COMMENTS
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Also equal to the number of set partitions of {1,2,...,n+2} with at most 4 parts such that each part does not contain both i,i+1 for 1<=i<n+2 or both 1 and n+2. E.g. a(3)=10 and the set partitions of {1,2,3,4,5} with at most 4 parts with no {i,i+1} or {1,5} in the same part are {14|25|3}, {13|25|4}, {14|2|35}, {1|24|35}, {13|24|5}, {1|25|3|4}, {1|2|35|4}, {14|2|3|5}, {1|24|3|5}, {13|2|4|5}. - Mike Zabrocki, Sep 08 2020
Also a(n) equals the number of color-complete multipoles with n terminals (that is, having all the states allowed by the Parity Lemma). - Miquel A. Fiol, May 27 2024
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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)(3*cosh(2*x) + 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, Sep 03 2003
a(n) = 2*a(n-1) + 3*a(n-2) - 1, n > 1. - Gary Detlefs, Jun 21 2010
a(n) = (3^(n+1) + 5) / 8 for n even.
a(n) = (3^(n+1) - 1) / 8 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 2.
(End)
a(n) = 3*a(n-1) + (3*(-1)^n - 1)/2 for n > 0. - Yuchun Ji, Dec 05 2019
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MAPLE
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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, Apr 28 2008
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MATHEMATICA
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CoefficientList[Series[(1-2 x)/((1-x^2) (1-3 x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 1, -3}, {1, 1, 4}, 30] (* Harvey P. Dale, Aug 16 2016 *)
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PROG
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(PARI) Vec((1 - 2*x) / ((1 - x)*(1 + x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 07 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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