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A003688
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a(n) = 3*a(n-1) + a(n-2), with a(1)=1 and a(2)=4.
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24
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1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564, 21932293, 72437443, 239244622, 790171309, 2609758549, 8619446956, 28468099417, 94023745207, 310539335038, 1025641750321, 3387464586001, 11188035508324, 36951571110973
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OFFSET
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1,2
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COMMENTS
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Number of 2-factors in K_3 X P_n.
Form the graph with matrix [1,1,1,1;1,1,1,0;1,1,0,1;1,0,1,1]. The sequence 1,1,4,13,... with g.f. (1-2*x)/(1-3*x-x^2) counts closed walks of length n at the vertex of degree 5. - Paul Barry, Oct 02 2004
a(n) is term (1,1) in M^n, where M is the 3x3 matrix [1,1,2; 1,1,1; 1,1,1]. - Gary W. Adamson, Mar 12 2009
Row sums of triangle
m/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....3
.2..|..1.....3.....9
.3..|..1.....6.....9.....27
.4..|..1.....6....27.....27...81
.5..|..1.....9....27....108...81...243
.6..|..1.....9....54....108..405...243...729
.7..|..1....12....54....270..405..1458...729..2187
which is the triangle for numbers 3^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
Pisano period lengths: 1, 3, 1, 6, 12, 3, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12, ... - R. J. Mathar, Aug 10 2012
a(n-1) is the number of length-n strings of 4 letters {0,1,2,3} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012
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REFERENCES
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F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
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LINKS
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FORMULA
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a(n) = ((13 - sqrt(13))/26)*((3 + sqrt(13))/2)^n + ((13 + sqrt(13))/26)*((3 - sqrt(13))/2)^n. - Paul Barry, Oct 02 2004
Starting (1, 1, 4, 13, 43, 142, 469, ...), row sums (unsigned) of triangle A136159. - Gary W. Adamson, Dec 16 2007
For n>=2, a(n) = F_n(3) + F_(n+1)(3), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} binomial(n-i-1,i) * x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
G.f.: G(0)*(1+x)/(2-3*x), where G(k) = 1 + 1/(1 - (x*(13*k-9))/( x*(13*k+4) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
a(n)^2 is the denominator of continued fraction [3,3,...,3, 5, 3,3,...,3], which has n-1 3's before, and n-1 3's after, the middle 5. - Greg Dresden, Sep 18 2019
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EXAMPLE
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G.f. = x + 4*x^2 + 13*x^3 + 43*x^4 + 142*x^5 + 469*x^6 + 1549*x^7 + 5116*x^8 + ...
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MAPLE
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with(combinat): a:=n->fibonacci(n, 3)-2*fibonacci(n-1, 3): seq(a(n), n=2..25); # Zerinvary Lajos, Apr 04 2008
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MATHEMATICA
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a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}] (* Robert G. Wilson v, Jan 13 2005 *)
LinearRecurrence[{3, 1}, {1, 4}, 30] (* Harvey P. Dale, Mar 15 2015 *)
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PROG
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(Magma) [n le 2 select 4^(n-1) else 3*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 19 2011
(SageMath)
@CachedFunction
if (n<3): return 4^(n-1)
else: return 3*a(n-1) + a(n-2)
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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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EXTENSIONS
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STATUS
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approved
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