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A106853
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Expansion of 1/(1-x*(1-4*x)).
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11
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1, 1, -3, -7, 5, 33, 13, -119, -171, 305, 989, -231, -4187, -3263, 13485, 26537, -27403, -133551, -23939, 510265, 606021, -1435039, -3859123, 1881033, 17317525, 9793393, -59476707, -98650279, 139256549, 533857665, -23168531, -2158599191, -2065925067, 6568471697, 14832171965
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums of Riordan array (1,x(1-4x)). 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)).
For n>=1 a(n) equals the determinant of the n X n matrix with 2's along the superdiagonal and the subdiagonal, and 1's along the main diagonal, and 0's everywhere else. [John M. Campbell, June 4 2011]
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FORMULA
| G.f.: 1/(1-x+4*x^2).
a(n) = 2^n*(cos(2*n*atan(sqrt(15)/5))+sqrt(15)sin(2*n*atan(sqrt(15)/5))/15).
a(n) = ((1+sqrt(-15))^(n+1)-(1-sqrt(-15))^(n+1))/(2^(n+1)*sqrt(-15)).
a(n) = sum(k=0..n, (-1)^(n-k)*binomial(k, n-k)*4^(n-k) ).
a(n) = a(n-1)-4*a(n-2), a(0)=1, a(1)=1. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 21 2008]
a(n) = sum(k=0..n, A109466(n,k)*4^(n-k) ). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 25 2008]
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MATHEMATICA
| Join[{a=1, b=1}, Table[c=b-4*a; a=b; b=c, {n, 80}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 22 2011*)
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PROG
| (Sage) [lucas_number1(n, 1, 4) for n in xrange(1, 36)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A106852
Sequence in context: A161818 A161509 A108974 * A083778 A107785 A001663
Adjacent sequences: A106850 A106851 A106852 * A106854 A106855 A106856
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KEYWORD
| easy,sign
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 08 2005
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