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A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)). 2
1, 1, -1, -2, 2, 4, -4, -8, 8, 16, -16, -32, 32, 64, -64, -128, 128, 256, -256, -512, 512, 1024, -1024, -2048, 2048, 4096, -4096, -8192, 8192, 16384, -16384, -32768, 32768, 65536, -65536, -131072, 131072, 262144, -262144, -524288, 524288 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums of A116949.

From Paul Curtz, Oct 24 2012: (Start)

b(n)=abs (a(n))=A158780(n+1)=1,1,1,2,2,4,4,8,8,8,... .

Consider the autosequence (that is a sequence whose inverse binomial transform is equal to the signed sequence) of the first kind of the example. Its numerator is A046978(n), its denominator is b(n). The numerator of the first column is A075553(n).

The denominator corresponding to the 0's is a choice.

The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)

LINKS

Table of n, a(n) for n=0..40.

Index entries for linear recurrences with constant coefficients, signature (0,-2).

FORMULA

a(n) = a(n-1)-2*a(n-2)+2*a(n-3), n>=3;

a(n) = (cos(pi*/2)+sin(pi*n/2)) * (2^((n-1)/2)*(1-(-1)^n)/2 + 2^((n-2)/2)*(1+(-1)^n)/2 + 0^n/2).

a(n+1) = Sum_{k, 0<=k<=n} A122016(n,k)*(-1)^k. - Philippe Deléham, Jan 31 2012

EXAMPLE

0/1,  1/1    1/1,  1/2,   0/2,  -1/4,  -1/4,  -1/8,...

1/1,   0/1,  -1/2, -1/2,  -1/4,   0/4,   1/8,   1/8,...

-1/1, -1/2,   0/2,  1/4,   1/4,   1/8,   0/8,  -1/16,...

1/2,   1/2,   1/4,  0/4   -1/8,  -1/8,  -1/16,  0/16,...

0/2,  -1/4,  -1/4, -1/8,   0/8,   1/16,  1/16,  1/32,...

-1/4,  0/4,   1/8,  1/8,   1/16,  0/16, -1/32, -1/32,...

1/4,   1/8,   0/8, -1/16, -1/16, -1/32,  0/32,  1/64,...

-1/8, -1/8,  -1/16, 0/16,  1/32,  1/32,  1/64,  0/64. - Paul Curtz, Oct 24 2012

PROG

(PARI) a(n)=if(n, (-1)^(n\2)<<((n-1)\2), 1) \\ Charles R Greathouse IV, Jan 31 2012

CROSSREFS

Cf. A016116, A191754/A192366.

Sequence in context: A082267 A076939 A158780 * A131572 A152166 A016116

Adjacent sequences:  A117572 A117573 A117574 * A117576 A117577 A117578

KEYWORD

easy,sign

AUTHOR

Paul Barry, Mar 29 2006

STATUS

approved

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Last modified December 6 06:58 EST 2016. Contains 278775 sequences.