A263657 -id:A263657 - cp's OEIS frontend

A263656 Number of length-2n central circular binary strings without zigzags (see reference for precise definition).

0, 0, 4, 6, 12, 30, 70, 168, 412, 1014, 2514, 6270, 15702, 39468, 99516, 251586, 637500, 1618638, 4117102, 10488684, 26758762, 68354250, 174810354, 447533586, 1146836662, 2941443180, 7550434480, 19395863358, 49859516292, 128252962434, 330101861850

Offset: 0

Felix Fröhlich, Oct 23 2015

nonn

See page 6 in the reference.

A zigzag is a substring which is either 010 or 101. The central binary strings are those that contain an equal number of 0's and 1's.

			For n=3 the 6 strings are 000111, 001110, 011100, 111000, 110001, 100011.

Andrew Howroyd, Table of n, a(n) for n = 0..100
E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Main diagonal of A263655.

Cf. A007039, A263657, A263658, A263659.

a[n_ /; n < 6] := {0, 0, 4, 6, 12, 30}[[n + 1]]; a[n_] := a[n] = (-(3*(n - 6)*a[n - 6]) + (7*n - 37)*a[n - 5] - 6*a[n - 4] + (7*n - 27)*a[n - 3] - 4*(n - 4)*a[n - 2] + 3*(n - 1)*a[n - 1])/n;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

a(n) = (1/n)*(3*(n-1)*a(n-1) - 4*(n-4)*a(n-2) + (7*n-27)*a(n-3) - 6*a(n-4) + (7*n-37)*a(n-5) - 3*(n-6)*a(n-6)) for n >= 6. - Andrew Howroyd, Feb 26 2017

corrected a(1) and a(17)-a(30) from Andrew Howroyd, Feb 26 2017

A263655 Table T(m, n) of number of circular binary strings with m ones and n zeros without zigzags, read by antidiagonals (see reference for precise definition).

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 4, 0, 1, 1, 0, 5, 5, 0, 1, 1, 0, 6, 6, 6, 0, 1, 1, 0, 7, 7, 7, 7, 0, 1, 1, 0, 8, 8, 12, 8, 8, 0, 1, 1, 0, 9, 9, 18, 18, 9, 9, 0, 1, 1, 0, 10, 10, 25, 30, 25, 10, 10, 0, 1, 1, 0, 11, 11, 33, 44, 44, 33, 11, 11, 0, 1

Offset: 0

Felix Fröhlich, Oct 23 2015

See page 5, figure 1 in the reference.

A zigzag is a substring which is either 010 or 101.

			Table starts:
0 1  1  1  1   1   1   1   1    1    1    1    1 ...
1 0  0  0  0   0   0   0   0    0    0    0    0 ...
1 0  4  5  6   7   8   9  10   11   12   13   14 ...
1 0  5  6  7   8   9  10  11   12   13   14   15 ...
1 0  6  7 12  18  25  33  42   52   63   75   88 ...
1 0  7  8 18  30  44  60  78   98  120  144  170 ...
1 0  8  9 25  44  70 104 147  200  264  340  429 ...
1 0  9 10 33  60 104 168 255  368  510  684  893 ...
1 0 10 11 42  78 147 255 412  629  918 1292 1765 ...
1 0 11 12 52  98 200 368 629 1014 1558 2300 3283 ...
1 0 12 13 63 120 264 510 918 1558 2514 3885 5786 ...

Andrew Howroyd, Table of n, a(n) for n = 0..819
E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Main diagonal is A263656. Antidiagonal sums are A007039.

Cf. A263657, A263658, A263659.

max = 11;
U[r_, k_] := Binomial[r - k + 2*Floor[k/3], Floor[k/3]];
V[r_, k_] := Binomial[r - Ceiling[k/2] - 1, Floor[k/2]];
T[0, 0] = T[1, 1] = 0;
T[0, ] = T[, 0] = 1;
T[n_ /; n >= 2, m_] /; m <= n := T[n, m] = Switch[m, 1, 0, 2, n + 2, 3, n + 3, _, Sum[ U[m, k]*U[n, k] - 2*V[m, k]*V[n, k]*(-1)^k, {k, 0, max-3}]];
T[n_, m_] /; m > n := T[m, n];
Table[T[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

From Andrew Howroyd, Feb 26 2017: (Start)
T(n,m) = Sum_{k>=0} U(m,k)*U(n,k) - 2*V(m,k)*V(n,k)*(-1)^k
  where U(r,k)=binomial(r-k+2*floor(k/3), floor(k/3)), V(r,k)=binomial(r-ceiling(k/2)-1, floor(k/2)).
T(n,0)=1 for n>=1, T(n,1)=0 for n>=1, T(n,2)=n+2 for n>=2, T(n,3)=n+3 for n>=2.
T(n,4)=(n-1)*(n+4)/2 for n>=3, T(n,5)=(n-2)*(n+5) for n>=3. (End)

a(66)-a(77) from Andrew Howroyd, Feb 26 2017

A263658 Number of (0, 1)-necklaces with n zeros and n ones without zigzags (see reference for precise definition).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 7, 12, 27, 57, 128, 285, 659, 1518, 3561, 8389, 19936, 47607, 114397, 276018, 669035, 1627491, 3973106, 9728991, 23892779, 58828866, 145201423, 359182693, 890350290, 2211257973, 5501701981, 13711368630, 34225162345, 85555609119, 214166692430, 536810116905

Offset: 0

Felix Fröhlich, Oct 23 2015

nonn

See page 16 in the reference.

A zigzag is a substring which is either 010 or 101. The necklaces 01 and 10 are considered to be with a zigzag. Necklaces do not allow turnover.

			For n=2 the necklace is 0011.
For n=3 the necklace is 000111.
For n=4 the necklaces are 00001111, 00110011.
For n=5 the necklaces are 0000011111, 0001110011, 0001100111.

Andrew Howroyd, Table of n, a(n) for n = 0..100
E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Main diagonal of A263657.

Cf. A007039, A263655, A263656, A263659.

(* b = A263656 *)
b[n_ /; n < 6] := {0, 0, 4, 6, 12, 30}[[n + 1]];
b[n_] := b[n] = (1/n)*(3*(n - 1)*b[n - 1] - 4*(n - 4)*b[n - 2] + (7*n - 27)*b[n - 3] - 6*b[n - 4] + (7*n - 37)*b[n - 5] - 3*(n - 6)*b[n - 6]);
a[0] = 0;
a[n_] := (1/n)*DivisorSum[n, EulerPhi[n/#] * b[#]/2&];
Array[a, 36, 0] (* Jean-François Alcover, Sep 11 2017, after Andrew Howroyd *)

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A263656(d) / 2. - Andrew Howroyd, Feb 26 2017

Offset corrected and a(21)-a(35) from Andrew Howroyd, Feb 26 2017

A263659 Number of (0, 1)-necklaces of length n without zigzags (see reference for precise definition).

Original entry on oeis.org

0, 2, 2, 2, 3, 4, 5, 6, 8, 10, 15, 20, 31, 42, 64, 94, 143, 212, 329, 494, 766, 1170, 1811, 2788, 4341, 6714, 10462, 16274, 25415, 39652, 62075, 97110, 152288, 238838, 375167, 589528, 927555, 1459962, 2300348, 3626242, 5721045

Offset: 0

Felix Fröhlich, Oct 23 2015

nonn

See page 16 in the reference.

A zigzag is a substring which is either 010 or 101. The necklaces 01 and 10 are considered to be with a zigzag. Necklaces do not allow turnover.

			For n=5 the necklaces are 00000, 11111, 00011, 00111 so a(5)=4.

Andrew Howroyd, Table of n, a(n) for n = 0..200
E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Antidiagonal sums of A263657.

Cf. A007039, A263655, A263656, A263658.

(* b = A007039 *) b[n_ /; n<4] = 2; b[4] = 6; b[n_] := b[n] = 2*b[n-1] - b[n-2] + b[n-4];
a[0] = 0; a[n_] := (1/n) * DivisorSum[n, EulerPhi[n/#] * b[#]&];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A007039(d). - Andrew Howroyd, Feb 26 2017

a(25)-a(40) from Andrew Howroyd, Feb 26 2017

cp's OEIS Frontend

A263656 Number of length-2n central circular binary strings without zigzags (see reference for precise definition).

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A263655 Table T(m, n) of number of circular binary strings with m ones and n zeros without zigzags, read by antidiagonals (see reference for precise definition).

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A263658 Number of (0, 1)-necklaces with n zeros and n ones without zigzags (see reference for precise definition).

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A263659 Number of (0, 1)-necklaces of length n without zigzags (see reference for precise definition).

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