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A099087
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G.f.: 1/(1-2*x+2*x^2).
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16
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1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216
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OFFSET
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0,2
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COMMENTS
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Yet another variation on A009545.
Row sums of Krawtchouk triangle A098593. Partial sums of e.g.f. exp(x)cos(x), or 2^(n/2)cos(pi*n/2). See A009116.
Binomial transform of A057077. [R. J. Mathar, Nov 04 2008]
Partial sums of A146559. [Philippe Deléham, Dec 01 2008]
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4,... - R. J. Mathar, Aug 10 2012
Also the inverse Catalan transform of A000079. - Arkadiusz Wesolowski, Oct 26 2012
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LINKS
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Table of n, a(n) for n=0..48.
Index entries for linear recurrences with constant coefficients, signature (2,-2)
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FORMULA
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E.g.f.: exp(x)(cos(x)+sin(x)); a(n)=2^(n/2)(cos(pi*n/4)+sin(pi*n/4)); a(n)=sum{k=0..n, sum{i=0..k, C(n-k, k-i)C(n, i)(-1)^(k-i)}}; a(n)=2a(n-1)-2a(n-2).
a(n) = (1-I)^(n-1)+(1+I)^(n-1) where I=sqrt(-1). a(n) = 2 sum_{k=0,1,2,..(n-1)/2} (-1)^k*binomial(n-1,2k) if n>0. - R. J. Mathar, Apr 18 2008
a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*2^k. [Philippe Deléham, Oct 28 2008]
E.g.f.: (cos(x)+sin(x))*exp(x)=G(0); G(k)=1+2*x/(4*k+1-x*(4*k+1)/(2*(2*k+1)+x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
G.f.: U(0) where U(k)= 1 + x*(k+3) - x*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
a(n) = Re((1+i)^n) + Im((1+i)^n) where i = sqrt(-1) = A146559(n) + A009545(n). - Philippe Deléham, Feb 13 2013
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PROG
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(Sage) [lucas_number1(n, 2, 2) for n in xrange(1, 50)] # [From Zerinvary Lajos, Apr 23 2009]
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CROSSREFS
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Cf. A009545, A146559.
Sequence in context: A180813 A194656 A108520 * A009545 A084102 A221609
Adjacent sequences: A099084 A099085 A099086 * A099088 A099089 A099090
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KEYWORD
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easy,sign
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AUTHOR
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Paul Barry, Sep 24 2004
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EXTENSIONS
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Signs added by N. J. A. Sloane, Nov 14, 2006
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STATUS
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approved
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