A113807 -id:A113807 - cp's OEIS frontend

A113806 Numbers that are congruent to {6, 8} mod 14.

6, 8, 20, 22, 34, 36, 48, 50, 62, 64, 76, 78, 90, 92, 104, 106, 118, 120, 132, 134, 146, 148, 160, 162, 174, 176, 188, 190, 202, 204, 216, 218, 230, 232, 244, 246, 258, 260, 272, 274, 286, 288, 300, 302, 314, 316, 328, 330, 342, 344, 356, 358, 370

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Cf. A008589, A113801, A113802, A113803, A113804, A113805, A113807.

Flatten[# + {6, 8} &/@ (14 Range[0, 30])] (* Harvey P. Dale, Jan 11 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 6). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+5*(-1)^n)/2.
O.g.f.: 2*x*(3+x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*cosec(3*Pi/14)/4. (End)

A090243 Permutation of natural numbers generated by 4-rowed array shown below.

Original entry on oeis.org

1, 7, 2, 11, 6, 3, 13, 10, 5, 4, 19, 14, 9, 8, 23, 18, 15, 12, 25, 22, 17, 16, 31, 26, 21, 20, 35, 30, 27, 24, 37, 34, 29, 28, 43, 38, 33, 32, 47, 42, 39, 36, 49, 46, 41, 40, 55, 50, 45, 44, 59, 54, 51, 48, 61, 58, 53, 52, 67, 62, 57, 56, 71, 66, 63, 60, 73, 70, 65, 64, 79, 74

Offset: 1

Giovanni Teofilatto, Jan 24 2004

7 11 13 19 23 25 31 35 37 43 47 ...
6 10 14 18 22 26 30 34 38 42 46 ...
5  9 15 17 21 27 29 33 39 41 45 ...
8 12 16 20 24 28 32 36 40 44 48 ...

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Cf. A088520, A090298, A092260, A113807.

Corrected and extended by Ray Chandler, Jan 30 2004

A204453 Period length 14: [0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1] repeated.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5

Offset: 0

Wolfdieter Lang, Jan 17 2012

This sequence can be continued periodically for negative values of n, and then a(-n) = a(n).
This is the seventh 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, A203572, A for k=1..6, respectively.
See a comment on A203571 for the general case of the P_k sequences. For a(n)=P_7(n) the nonnegative members of the equivalence classes [0], [1],...,[6], defined by p==q iff P_7(p)=P_7(q), are found in the array A113807 if there the last class [7], starting with 7, is replaced by 0,7,14,..., to become the first class [0] (nonnegative part).

			a(16) = 16(mod 7) = 2 because 16\7 = floor(16/7)=2 is even; the sign is +1.
a(9) = (7-9)(mod 7) = 5 because 9\7 = floor(9/7)=1 is odd; the sign is -1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A203572 (k=6), A113807, A010876.

a(n) = n(mod 7) if (-1)^floor(n/7)=+1 else (7-n)(mod 7), n>=0. (-1)^floor(n/7) is the sign corresponding to the parity of the quotient floor(n/7). This quotient is sometimes denoted by n\7.
O.g.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+6*x^7+5*x^8+4*x^9+ 3*x^10+2*x^11+x^12)/(1-x^14).
a(n) = (7*m*(m^4-21*m^3+175*m^2-735*m+1624)*((-1)^floor(n/7)-1)-10908*(-1)^floor(n/7)+12348)*m/1440 where m = n-7*floor(n/7). - Luce ETIENNE, Oct 13 2017

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.

A337713 Irregular triangle T read by rows: row n gives the inverse elements of row n of A216319 Modd(n), for n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 5, 3, 1, 5, 3, 7, 1, 7, 5, 1, 7, 3, 9, 1, 7, 9, 3, 5, 1, 5, 7, 11, 1, 9, 5, 11, 3, 7, 1, 9, 11, 3, 5, 13, 1, 13, 11, 7, 1, 11, 13, 9, 7, 3, 5, 15, 1, 11, 7, 5, 15, 3, 13, 9, 1, 7, 5, 13, 11, 17, 1, 13, 15, 11, 17, 7, 3, 5, 9, 1, 13, 17, 9, 11, 3, 7, 19, 1, 17, 19, 13, 5, 11, 1, 15, 9, 19, 5, 17, 3, 13, 7, 21

Offset: 1

Wolfdieter Lang, Oct 20 2020

The length of row n is A055034(n), called here delta(n), for n >= 1.
For the modified modular equivalence relation Modd n see a comment in A203571, and the W. Lang link, Definition 4. p. 25. For Modd(a, n) one has to consider the parity of floor(a/n). If it is even then Modd(a, n) = mod(a, n), otherwise it is mod(-a, n).
The rows of A216319 are the smallest positive restricted residue system mod n with only odd members (RRSodd(n)). This is not a group mod n, but a group Modd n, called here G(rho(n)). This group is isomorphic to the Galois group Gal(Q(rho(n))/Q), where the algebraic number of degree delta(n) is rho(n) = 2*cos(Pi/n), for n >= 1. See A187360 for the minimal polynomials of rho(n), called C(n, x).

			The irregular triangle T(n, k) begins:
n\k 1  2  3  4  5  6  7  8 9 ...
1:  1
2:  1
3:  1
4:  1  3
5:  1  3
6:  1  5
7:  1  5  3
8:  1  5  3  7
9:  1  7  5
10: 1  7  3  9
11: 1  7  9  3  5
12: 1  5  7 11
13: 1  9  5 11  3  7
14: 1  9 11  3  5 13
15: 1 13 11  7
16: 1 11 13  9  7  3  5 15
17: 1 11  7  5 15  3 13  9
18: 1  7  5 13 11 17
19: 1 13 15 11 17  7  3  5 9
20: 1 13 17  9 11  3  7 19
...
T(7, 2) = 5 because A216319(7, 2) = 3 and Modd(3*5, 7) = 1 since floor(15/7) = 2 is even, hence Modd(3*5, 7) = mod(15, 7) = 1. The residue classes Modd 7 for  1, 3, 5 are shown in the array given in A113807 (including the negative numbers) [3]*[5] = [1] (Modd 7).
T(9, 2) = 7 because A216319(9, 2) = 5 and Modd(7*5, 9) = 1, since floor(35/9) = 3 is odd, hence Moddn(35, 9) = mod(-35, 9) = 1.

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012.

Cf. A055034, A203571, A187360, A216319.

rowa(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]); \\ A216319
Modd(x, n) = if ((x\n)%2, Mod(-x,n), Mod(x,n));
findinvm(k, n) = for (i=1, n, if (Modd(k*i, n) == 1, return(i)));
row(n) = my(ra=rowa(n)); vector(#ra, k, findinvm(ra[k], n)); \\ Michel Marcus, Sep 13 2023

T(n, k) = Inverse of A216319(n, k) (Modd n), for n >= 1. For Modd n see the comment above.

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A113806 Numbers that are congruent to {6, 8} mod 14.

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A090243 Permutation of natural numbers generated by 4-rowed array shown below.

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A204453 Period length 14: [0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1] repeated.

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

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A337713 Irregular triangle T read by rows: row n gives the inverse elements of row n of A216319 Modd(n), for n >= 1.

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