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A121262
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The characteristic function of the multiples of four.
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17
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1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Periodic 4: repeat [1, 0, 0, 0] (with offset 0).
This sequence can be used to produce a periodic sequence of 4 numbers b,c,d,e: a(n) = b*(1/4)*(2*cos(n*Pi/2)+1+(-1)^n)+c*(1/4)*(2*cos((n+3)*Pi/2)+ 1+(-1)^(n+3))+d*(1/4)*(2*cos((n+2)*Pi/2)+ 1+(-1)^(n+2))+ e*(1/4)* (2*cos((n+1)*Pi/2)+ 1+(-1)^(n+1)).
a(n) is also the number of partitions of n where each part is four (Since the empty partition has no parts, a(0)=1). Hence a(n) is also the number of 2-regular graphs on n vertices such that each component has girth exactly four. - Jason Kimberley, Oct 01 2011
This sequence is the Euler transformation of A185014. - Jason Kimberley, Oct 01 2011
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REFERENCES
| G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 82.
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LINKS
| Index entries for characteristic functions
Index to sequences with linear recurrences with constant coefficients, signature (0,0,0,1).
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FORMULA
| a(n) = (1/4)*(2*cos(n*Pi/2)+1+(-1)^n).
Additive with a(p^e) = 1 if p = 2 and e > 1, 0 otherwise.
Sequence shifted right by 2 is additive with a(p^e) = 1 if p = 2 and e = 1, 0 otherwise.
a(n) = 1-(C(n+1,n+(-1)^(n+1)) mod 2)
a(n) = 0^(n mod 4). - Reinhard Zumkeller, Sep 30 2008
a(n) = (1/24)*(-5*(n mod 4)+((n+1) mod 4)+((n+2) mod 4)+7*((n+3) mod 4)). - Paolo P. Lava, Feb 06 2009
a(n) = (1/4)*(1+I^n+(-1)^n+(-I)^n), with I=sqrt(-1). - Paolo P. Lava, May 04 2010
a(n) = ((n-1)^k mod 4 - (n-1)^(k-1) mod 4)/2, k>2. - Gary Detlefs, Feb 21 2011
a(n) = floor(1/2*cos(n*Pi/2)+1/2). -Gary Detlefs, May 16 2011
G.f.: 1/(1-x^4). a(n) = (1+(-1)^n)*(1+i^((n-1)*n))/4, where i=sqrt(-1). - Bruno Berselli, Sep 28 2011
a(n) = floor(((n+3) mod 4)/3). [From Gary Detlefs, Dec 29 2011]
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EXAMPLE
| a(0) = (1/4)*(2*cos(0)+1+1) = (1/4)*(2+2) = 1
a(1) = (1/4)*(2*cos(Pi/2)+1-1) = (1/4)*(0+0) = 0
a(2) = (1/4)*(2*cos(Pi)+1+1) = (1/4)*(-2+2) = 0
a(3) = (1/4)*(2*cos(3*Pi/2)+1-1) = (1/4)*(0+0) = 0
a(n) =!(n%4) [From Jaume Oliver Lafont, Mar 01 2009]
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PROG
| (Haskell)
a121262 n = a121262_list !! n
a121262_list = cycle [1, 0, 0, 0] -- Reinhard Zumkeller, Jan 06 2012
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CROSSREFS
| A011765 is another version of the same sequence.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), this sequence (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7). - Jason Kimberley, Oct 14 2011
Sequence in context: A015985 A015777 A014017 * A102243 A173859 A202108
Adjacent sequences: A121259 A121260 A121261 * A121263 A121264 A121265
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Paolo P. Lava and Giorgio Balzarotti (paoloplava(AT)gmail.com), Aug 23 2006, Aug 30 2007
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