A092260 -id:A092260 - cp's OEIS frontend

A203572 Period length 12: 0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1 repeated.

0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 0

Offset: 0

Wolfdieter Lang, Jan 12 2012

This sequence can be continued periodically for negative values of n.
This is the sixth sequence of a k-family of sequences P_k, k>=1, which starts with A000007(n+1), n>=0, (the 0-sequence), A000035, A193680, A193682, A203571 for k=1,...,5, respectively.
See a comment on A203571 for the general case of the P_k sequences. For a(n)=P_6(n) the nonnegative members of the equivalence classes [0], [1],...,[5], defined by p==q iff P_6(p)=P_6(q), are found in the array A092260 if there class [6], starting with 6, is replaced by 0,6,12,..., which is class [0] (nonnegative part).

			a(14) = 14(mod 6) = 2 because 14\6 = floor(14/6)=2 is even; the sign is +1.
a(8) = (6-8)(mod 6) = 4 because 8\6 = floor(8/6)=1 is odd; the sign is -1.

Cf. A203571.

PadRight[{},120,{0,1,2,3,4,5,0,5,4,3,2,1}] (* Harvey P. Dale, Nov 28 2015 *)

a(n) = n(mod 6) if (-1)^floor(n/6)=+1 else (6-n)(mod 6), n>=0. (-1)^floor(n/6) is the sign corresponding to the parity of the quotient floor(n/6). This quotient is sometimes denoted by n\6.

O.g.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+5*x^6+4*x^7+3*x^8+2*x^9+ x^10)/(1-x^12).

A203575 Array of certain four complete residue classes (nonnegative members), read by SW-NE antidiagonals.

Original entry on oeis.org

0, 1, 4, 2, 7, 8, 3, 6, 9, 12, 5, 10, 15, 16, 11, 14, 17, 20, 13, 18, 23, 24, 19, 22, 25, 28, 21, 26, 31, 32, 27, 30, 33, 36, 29, 34, 39, 40, 35, 38, 41, 44, 37, 42, 47, 48, 43, 46

Offset: 1

Wolfdieter Lang, Jan 12 2012

See A193682 for the sequence called P_4, with period length 8, which defines the four complete residue classes [m], m = 0,1,2,3, via the equivalence relation  p==q iff P_4(p) = P_4(q).
See a comment on A203571 for the general P_k sequences, and the multiplicative (but not additive) structure of these residue classes.
The row length sequence of this tabf array is [1,2,3,4,4,4,...].
This array defines a certain permutation of the nonnegative integers.

			The array starts
n\m  1   2   3   4
1:   0
2:   1   4
3:   2   7   8
4:   3   6   9  12
5:   5  10  15  16
6:  11  14  17  20
7:  13  18  23  24
8:  19  22  25  28
9:  21  26  31  32
10: 27  30  33  36
...
The sequence P_4(n)=A193682(n), n>=0, is repeated 0, 1, 2, 3, 0, 3, 2, 1, with period length 8. P_4(6)=2, hence 6 belongs to class [2].
Multiplicative structure: 11*23 == 3*1 = 3. Indeed: P_4(11*23) = P_4(253) = P_(5), because 253==5(mod 8), and P_(5)= 3, hence 11*23 belongs to class 3. In general, P_4(p*q) = P_4(P_4(p)*P_4(q)).

Cf.A193682, A088520 (k=3), A090298 (k=5), A092260 (k=6), A113807 (k=7).

The nonnegative members of the four complete residue classes are (see a comment above for their definition):
[0]: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36,...  (A008586)
[1]: 1, 7, 9, 15, 17, 23, 25, 31, 33, 39,...  (A047522)
[2]: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38,... (A016825)
[3]: 3, 5, 11, 13, 19, 21, 27, 29, 35, 37,... (A047621)
In each class the corresponding negative numbers should be included.

cp's OEIS Frontend

A203572 Period length 12: 0, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1 repeated.

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