A208721 -id:A208721 - cp's OEIS frontend

A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less.

1, 1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 35, 22, 13, 6, 1, 64, 83, 54, 29, 16, 7, 1, 128, 199, 134, 73, 36, 19, 8, 1, 256, 479, 340, 185, 92, 43, 22, 9, 1, 512, 1155, 872, 481, 236, 111, 50, 25, 10, 1, 1024, 2787, 2254, 1265, 622, 287, 130, 57, 28, 11

Offset: 1

Andrew Howroyd, Apr 15 2017

All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.

			Array starts:
   1  1  1   1   1    1    1    1     1     1 ...
   2  4  8  16  32   64  128  256   512  1024 ...
   3  7 15  35  83  199  479 1155  2787  6727 ...
   4 10 22  54 134  340  872 2254  5854 15250 ...
   5 13 29  73 185  481 1265 3361  8993 24193 ...
   6 16 36  92 236  622 1658 4468 12132 33146 ...
   7 19 43 111 287  763 2051 5575 15271 42099 ...
   8 22 50 130 338  904 2444 6682 18410 51052 ...
   9 25 57 149 389 1045 2837 7789 21549 60005 ...
  10 28 64 168 440 1186 3230 8896 24688 68958 ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

Rows 3-32 are A124696-A124719, A124726, A124783, A124784, A124799, A124803, A124804.

Cf. A188866, A220062, A285280, A285281, A208777, A208721.

T[m_, n_] := Sum[(1 + 2*Cos[j*Pi/(m+1)])^n, {j, 1, m}] // FullSimplify;
Table[T[m-n+1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)

\\ from Knopfmacher et al.
ChebyshevU(n,x) = sum(i=0, n/2, 2*poltchebi(n-2*i,x)) + (n%2-1);
RowGf(k,x) = 1 + (k*x*(1+3*x) - 2*(k+1)*x*subst(ChebyshevU(k-1,z)/ChebyshevU(k,z),z,(1-x)/(2*x)))/((1+x)*(1-3*x));
a(m,n)=Vec(RowGf(m,x)+O(x^(n+1)))[n+1];
for(m=1, 10, print(RowGf(m,x)));
for(m=1, 10, for(n=1, 9, print1( a(m,n), ", ") ); print(); );

T(m, n) = Sum_{j=1..m} (1 + 2*cos(j*pi/(m+1)))^n. - Andrew Howroyd, Apr 15 2017

A208715 Number of n-bead necklaces labeled with numbers 1..n allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

1, 3, 7, 18, 38, 100, 224, 600, 1498, 4124, 11054, 31347, 87887, 253251, 727867, 2117551, 6159994, 18022052, 52757222, 154874172, 454989753, 1338737083, 3941860229, 11618101775, 34264747423, 101127761931, 298632667837, 882370316215, 2608423280534, 7714578529582

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

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

Diagonal of A208721.

a(17)-a(30) from Andrew Howroyd, Mar 03 2017

A208716 Number of n-bead necklaces labeled with numbers 1..3 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

3, 5, 7, 12, 18, 34, 56, 111, 207, 427, 859, 1851, 3930, 8672, 19092, 42845, 96243, 218567, 497183, 1138084, 2610226, 6009662, 13861968, 32057868, 74260243, 172351415, 400589343, 932486879, 2173368730, 5071877864, 11849063220, 27711739481

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..100
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], 2024. See p. 22.

Column 3 of A208721.

Cf. A208772, A078057.

a(2n+1) = (1/2) * (A208772(2n+1) + A078057(n+1)). - Andrew Howroyd, Mar 03 2017

a(2n) = (1/2) * A208772(2n) + (1/4) * (A078057(n) + A078057(n+1)). - Andrew Howroyd, Mar 03 2017

A208717 Number of n-bead necklaces labeled with numbers 1..4 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

4, 7, 10, 18, 28, 56, 98, 208, 418, 933, 2044, 4777, 11072, 26548, 63672, 155248, 379348, 935278, 2311294, 5741228, 14292966, 35699049, 89339860, 224097602, 563074848, 1417313897, 3572747650, 9019154944, 22797181996, 57693378135, 146168107034, 370712004868

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

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

Column 4 of A208721.

Cf. A208773, A126358.

a(2n+1) = (1/2) * (A208773(2n+1) + A126358(n+1)). - Andrew Howroyd, Mar 03 2017

a(2n) = (1/2) * A208773(2n) + (1/4) * (A126358(n) + A126358(n+1)). - Andrew Howroyd, Mar 03 2017

a(29)-a(32) from Andrew Howroyd, Mar 03 2017

A208718 Number of n-bead necklaces labeled with numbers 1..5 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

5, 9, 13, 24, 38, 78, 140, 306, 634, 1464, 3326, 8066, 19454, 48534, 121294, 308154, 785222, 2018548, 5203634, 13482426, 35019010, 91251438, 238278314, 623629333, 1635062126, 4294493670, 11296419934, 29757590061, 78489973742, 207281830814

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

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

Column 5 of A208721.

Cf. A208774, A057960.

a(2n+1) = (1/2) * (A208774(2n+1) + r(n+1)) where r(n) = A057960(n+1). - Andrew Howroyd, Mar 03 2017

a(2n) = (1/2) * A208774(2n) + (1/4) * (r(n) + r(n+1)) where r(n) = A057960(n+1). - Andrew Howroyd, Mar 03 2017

a(25)-a(30) from Andrew Howroyd, Mar 03 2017

A208719 Number of n-bead necklaces labeled with numbers 1..6 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

6, 11, 16, 30, 48, 100, 182, 404, 850, 1996, 4614, 11391, 28002, 71236, 181710, 471437, 1227712, 3226816, 8509930, 22564205, 60002088, 160113626, 428273964, 1148410792, 3085474406, 8305718893, 22394409228, 60474491712, 163531569910, 442782412569

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

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

Column 6 of A208721.

Cf. A208775, A126360.

a(2n+1) = (1/2) * (A208775(2n+1) + A126360(n+1)). - Andrew Howroyd, Mar 03 2017

a(2n) = (1/2) * A208775(2n) + (1/4) * (A126360(n) + A126360(n+1)). - Andrew Howroyd, Mar 03 2017

a(26)-a(30) from Andrew Howroyd, Mar 03 2017

A208720 Number of n-bead necklaces labeled with numbers 1..7 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

7, 13, 19, 36, 58, 122, 224, 502, 1066, 2528, 5902, 14717, 36557, 93987, 242387, 635990, 1675743, 4457771, 11903835, 31970464, 86144382, 233001926, 631920076, 1718567054, 4684178054, 12794677456, 35012475762, 95976563897, 263496604025, 724445459431

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

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

Column 7 of A208721.

Cf. A208776, A002714.

a(2n+1) = (1/2) * (A208776(2n+1) + A002714(n+1)). - Andrew Howroyd, Mar 03 2017

a(2n) = (1/2) * A208776(2n) + (1/4) * (A002714(n) + A002714(n+1)). - Andrew Howroyd, Mar 03 2017

a(22)-a(30) from Andrew Howroyd, Mar 03 2017

cp's OEIS Frontend

A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less.

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A208715 Number of n-bead necklaces labeled with numbers 1..n allowing reversal, with no adjacent beads differing by more than 1.

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A208716 Number of n-bead necklaces labeled with numbers 1..3 allowing reversal, with no adjacent beads differing by more than 1.

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A208717 Number of n-bead necklaces labeled with numbers 1..4 allowing reversal, with no adjacent beads differing by more than 1.

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A208718 Number of n-bead necklaces labeled with numbers 1..5 allowing reversal, with no adjacent beads differing by more than 1.

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A208719 Number of n-bead necklaces labeled with numbers 1..6 allowing reversal, with no adjacent beads differing by more than 1.

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A208720 Number of n-bead necklaces labeled with numbers 1..7 allowing reversal, with no adjacent beads differing by more than 1.

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