A208727 -id:A208727 - cp's OEIS frontend

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A208671 T(n,k) = number of 2n-bead necklaces labeled with numbers 1..k allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 4, 1, 0, 5, 7, 8, 6, 1, 0, 6, 9, 12, 14, 8, 1, 0, 7, 11, 16, 23, 24, 13, 1, 0, 8, 13, 20, 32, 44, 47, 18, 1, 0, 9, 15, 24, 41, 65, 97, 89, 30, 1, 0, 10, 17, 28, 50, 86, 152, 212, 187, 46, 1, 0, 11, 19, 32, 59, 107, 208, 360, 512, 396, 78, 1, 0, 12, 21, 36

Offset: 1

R. H. Hardin, Feb 29 2012

Table starts
.0.1..2...3...4...5....6....7....8....9...10..11..12..13.14.15.16
.0.1..3...5...7...9...11...13...15...17...19..21..23..25.27.29
.0.1..4...8..12..16...20...24...28...32...36..40..44..48.52
.0.1..6..14..23..32...41...50...59...68...77..86..95.104
.0.1..8..24..44..65...86..107..128..149..170.191.212
.0.1.13..47..97.152..208..264..320..376..432.488
.0.1.18..89.212.360..514..669..824..979.1134
.0.1.30.187.512.937.1398.1866.2335.2804

			All solutions for n=4, k=3:
..1....1....1....1....1....2
..2....2....2....2....2....3
..3....1....1....1....3....2
..2....2....2....2....2....3
..1....3....1....1....3....2
..2....2....2....2....2....3
..3....3....3....1....3....2
..2....2....2....2....2....3

R. H. Hardin, Table of n, a(n) for n = 1..148

Column 3 is A000029.

Cf. A208727, A220062.

T(n,k) = (2*A208727(n) + A220062(n+1,k))/4. - Andrew Howroyd, Mar 19 2017

A208724 Number of 2n-bead necklaces labeled with numbers 1..5 not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

4, 7, 12, 25, 52, 131, 316, 835, 2196, 5935, 16108, 44369, 122644, 341803, 956636, 2690845, 7596484, 21524543, 61171660, 174342217, 498112276, 1426419859, 4093181692, 11767920119, 33891544420, 97764131647, 282429537948, 817028472961, 2366564736724, 6863038218843

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=3:
..4....1....3....2....1....2....3....1....2....3....1....2
..5....2....4....3....2....3....4....2....3....4....2....3
..4....1....3....2....3....4....3....3....2....5....1....4
..5....2....4....3....2....3....4....4....3....4....2....5
..4....3....3....2....3....4....5....3....4....5....1....4
..5....2....4....3....2....3....4....2....3....4....2....3

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 5 of A208727.

a[n_] := (1/n)*DivisorSum[n, EulerPhi[n/#] * (2*3^# + 2) &] / 2;
Array[a, 30] (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A198635(d) / 2. - Andrew Howroyd, Mar 18 2017

a(15)-a(30) from Andrew Howroyd, Mar 18 2017

A208722 Number of 2n-bead necklaces labeled with numbers 1..n not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

0, 1, 4, 15, 52, 210, 796, 3175, 12468, 49288, 194140, 766110, 3019224, 11905330, 46938192, 185111095, 730148332, 2880799554, 11369141308, 44881708072, 177229210656, 700047776214, 2765947680124, 10931565973950, 43215517211240, 170888480306500

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=4:
..1....1....3....2....1....2....2....2....2....1....1....1....1....1....1
..2....2....4....3....2....3....3....3....3....2....2....2....2....2....2
..1....3....3....2....3....2....4....4....2....3....1....1....3....3....1
..2....4....4....3....2....3....3....3....3....2....2....2....4....2....2
..3....3....3....2....1....2....4....2....4....3....1....3....3....3....1
..2....4....4....3....2....3....3....3....3....4....2....4....2....2....2
..3....3....3....2....3....4....4....4....4....3....1....3....3....3....3
..2....2....4....3....2....3....3....3....3....2....2....2....2....2....2

Andrew Howroyd, Table of n, a(n) for n = 1..100

Diagonal of A208727.

a(13)-a(26) from Andrew Howroyd, Mar 18 2017

A208723 Number of 2n-bead necklaces labeled with numbers 1..4 not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

3, 5, 8, 15, 27, 60, 123, 285, 648, 1529, 3603, 8680, 20883, 50825, 124056, 304575, 750123, 1855100, 4600203, 11442087, 28527448, 71292605, 178526883, 447919420, 1125750147, 2833906685, 7144450568, 18036423975, 45591631803, 115381823348

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=4:
..1....1....3....2....1....2....2....2....2....1....1....1....1....1....1
..2....2....4....3....2....3....3....3....3....2....2....2....2....2....2
..1....3....3....2....3....2....4....4....2....3....1....1....3....3....1
..2....4....4....3....2....3....3....3....3....2....2....2....4....2....2
..3....3....3....2....1....2....4....2....4....3....1....3....3....3....1
..2....4....4....3....2....3....3....3....3....4....2....4....2....2....2
..3....3....3....2....3....4....4....4....4....3....1....3....3....3....3
..2....2....4....3....2....3....3....3....3....2....2....2....2....2....2

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 4 of A208727.

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A005248(d). - Andrew Howroyd, Mar 18 2017

a(26)-a(30) from Andrew Howroyd, Mar 18 2017

A208725 Number of 2n-bead necklaces labeled with numbers 1..6 not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

5, 9, 16, 35, 78, 210, 551, 1569, 4475, 13078, 38465, 114584, 343026, 1034471, 3134135, 9540969, 29154478, 89407073, 275016292, 848329872, 2623322133, 8130714643, 25252366057, 78577560856, 244933963301, 764707458720, 2391026407058, 7486342546939

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=3:
..3....2....1....3....5....2....4....1....2....3....4....4....1....2....1....3
..4....3....2....4....6....3....5....2....3....4....5....5....2....3....2....4
..5....4....3....5....5....2....6....1....4....3....4....4....3....2....1....3
..4....3....4....6....6....3....5....2....5....4....5....5....2....3....2....4
..5....4....3....5....5....2....6....1....4....5....4....6....3....4....3....3
..4....3....2....4....6....3....5....2....3....4....5....5....2....3....2....4

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 6 of A208727.

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A198636(d). - Andrew Howroyd, Mar 18 2017

a(17)-a(28) from Andrew Howroyd, Mar 18 2017

A208726 Number of 2n-bead necklaces labeled with numbers 1..7 not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

6, 11, 20, 45, 104, 290, 796, 2366, 7076, 21684, 66996, 209594, 659560, 2090590, 6659120, 21312716, 68476512, 220798322, 714144612, 2316303916, 7531662344, 24545750474, 80160395804, 262280793024, 859662037240, 2822180991508, 9278652591844, 30547892144882

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=2:
..5....1....2....3....5....4....2....4....6....1....3
..6....2....3....4....6....5....3....5....7....2....4
..5....1....2....5....7....4....4....6....6....3....3
..6....2....3....4....6....5....3....5....7....2....4

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 7 of A208727.

a(13)-a(28) from Andrew Howroyd, Mar 18 2017

cp's OEIS Frontend

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A208671 T(n,k) = number of 2n-bead necklaces labeled with numbers 1..k allowing reversal, with neighbors differing by exactly 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A208724 Number of 2n-bead necklaces labeled with numbers 1..5 not allowing reversal, with neighbors differing by exactly 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A208722 Number of 2n-bead necklaces labeled with numbers 1..n not allowing reversal, with neighbors differing by exactly 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A208723 Number of 2n-bead necklaces labeled with numbers 1..4 not allowing reversal, with neighbors differing by exactly 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A208725 Number of 2n-bead necklaces labeled with numbers 1..6 not allowing reversal, with neighbors differing by exactly 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A208726 Number of 2n-bead necklaces labeled with numbers 1..7 not allowing reversal, with neighbors differing by exactly 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions