OFFSET
0,3
COMMENTS
0,1,3,2,2,3,1,0 repeated indefinitely.
If N is any power of 2 then n(n+1)/2 mod N is a repeating pattern of length 2N. Moreover, the first N digits form a permutation P of A={0,1,...,N-1}. The subsequent N digits are P in the reversed order. The technique is useful for the generation of arbitrarily large pseudo-random permutations.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..8191
O. Y. Takeshita and D. J. Costello, Jr., New Deterministic Interleaver Designs for Turbo-Codes, IEEE Trans. Inform. Theory, vol. 46, no. 6, pp. 1988-2006, Sept. 2000.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).
FORMULA
From Paul Barry, Jul 26 2005: (Start)
G.f.: (x + 2x^2 + 2x^4 + x^5)/(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7).
a(n) = cos(3*Pi*n/4 + Pi/4)/2 + (1/2 - sqrt(2)/2)*sin(3*Pi*n/4 + Pi/4) - (1/2 + sqrt(2)/2)*cos(Pi*n/4 + Pi/4) - sin(Pi*n/4 + Pi/4)/2 - cos(Pi*n/2)/2 + sin(Pi*n/2)/2 + 3/2. (End)
a(n) = (((n+1)^5 - n^5 - 1) mod 120)/30. - Gary Detlefs, Mar 25 2012
a(n) = -ceiling(n/2)*(-1)^n mod 4. - Wesley Ivan Hurt, Jul 13 2014
MAPLE
for n from 0 to 300 do printf(`%d, `, n*(n+1)/2 mod 4) od: # James A. Sellers, Apr 21 2005
MATHEMATICA
Table[Mod[-Ceiling[n/2] (-1)^n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 13 2014 *)
PROG
(Magma) [ -Ceiling(n/2)*(-1)^n mod 4 : n in [0..100]]; // Wesley Ivan Hurt, Jul 13 2014
(PARI) Vec((x+2*x^2+2*x^4+x^5)/(1-x+x^2-x^3+x^4-x^5+x^6-x^7) + O(x^90)) \\ Michel Marcus, Jul 13 2014
(Scheme) (define (A105198 n) (modulo (* 1/2 n (+ 1 n)) 4)) ;; Antti Karttunen, Aug 10 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Oscar Takeshita, Apr 11 2005
EXTENSIONS
More terms from James A. Sellers, Apr 21 2005
STATUS
approved