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A106852
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Expansion of 1/(1-x(1-3x)).
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17
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1, 1, -2, -5, 1, 16, 13, -35, -74, 31, 253, 160, -599, -1079, 718, 3955, 1801, -10064, -15467, 14725, 61126, 16951, -166427, -217280, 282001, 933841, 87838, -2713685, -2977199, 5163856, 14095453, -1396115, -43682474, -39494129, 91553293, 210035680, -64624199, -694731239, -500858642
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OFFSET
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0,3
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COMMENTS
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Row sums of Riordan array (1,x(1-3x)) In general, a(n)=sum{k=0..n,(-1)^(n-k)*binomial(k,n-k)*r^(n-k)} yields the row sums of the Riordan array (1,x(1-kx)).
Row sums of Riordan array (1/(1+3x^2),x/(1+3x^2)). - Paul Barry, Sep 10 2005
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LINKS
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Table of n, a(n) for n=0..38.
Index to sequences with linear recurrences with constant coefficients, signature (1,-3).
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FORMULA
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G.f.: 1/(1-x+3x^2); a(n)=2*sqrt(33)*3^(n/2)*cos((n+1)*atan(sqrt(11)/11)-pi*n/2)/11; a(n)=3^(n/2)(cos(-n*acot(sqrt(11)/11))-sqrt(11)*sin(-n*acot(sqrt(11)/11))/11); a(n)=((1+sqrt(-11))^(n+1)-(1-sqrt(-11))^(n+1))/(2^(n+1)sqrt(-11)); a(n)=sum{k=0..n, (-1)^(n-k)*binomial(k, n-k)*3^(n-k)} = sum{0<=k<=n} A109466(n,k)*3^(n-k).
a(n)=sum{k=0..n, C((n+k)/2, k)*(-3)^((n-k)/2)*(1+(-1)^(n-k))/2}; a(n)=sum{k=0..floor(n/2), C(n-k, k)(-3)^k}; - Paul Barry, Sep 10 2005
a(n)=a(n-1)-3*a(n-2), a(0)=1, a(1)=1. [From Philippe DELEHAM, Oct 21 2008]
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MATHEMATICA
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Join[{a=1, b=1}, Table[c=b-3*a; a=b; b=c, {n, 80}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 22 2011*)
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PROG
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(Sage) [lucas_number1(n, 1, +3) for n in xrange(1, 40)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
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Sequence in context: A197365 A121579 A214733 * A162975 A187244 A120294
Adjacent sequences: A106849 A106850 A106851 * A106853 A106854 A106855
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KEYWORD
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easy,sign
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AUTHOR
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Paul Barry, May 08 2005
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STATUS
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approved
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