A220062 -id:A220062 - cp's OEIS frontend

A208666 Number of 2n-bead necklaces labeled with numbers 1..n allowing reversal, with neighbors differing by exactly 1.

0, 1, 4, 14, 44, 152, 514, 1866, 6884, 26137, 100442, 390592, 1526272, 5989223, 23548688, 92727898, 365445200, 1441195226, 5686268314, 22444465311, 88622259788, 350040069245, 1383007946774, 5465854718664, 21607909105528, 85444555330132, 337962745845558, 1337094537703089

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

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

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

Diagonal of A208671.

a(n) = (2*A208722(n) + A220062(n+1,n))/4. - Andrew Howroyd, Mar 19 2017

a(11)-a(28) from Andrew Howroyd, Mar 19 2017

A153361 Number of zig-zag paths from top to bottom of a rectangle of width 12 with n rows.

Original entry on oeis.org

12, 22, 42, 80, 154, 296, 572, 1104, 2138, 4136, 8020, 15536, 30148, 58450, 113472, 220110, 427410, 829352, 1610628, 3125954, 6071028, 11784514, 22887536, 44431506, 86293452, 167532792, 325373382, 631721620, 1226878704, 2382108386

Offset: 1

Joseph Myers, Dec 24 2008

Number of words of length n using a 12-symbol alphabet where neighboring letters are neighbors in the alphabet. - Andrew Howroyd, Apr 17 2017

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-6,3,1).

Column 12 of A220062.

Twice A129638.

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[i < k, b[n - 1, i + 1, k], 0]]]; a[n_] := b[n, 0, 12]; Array[a, 30] (* Jean-François Alcover, Oct 10 2017, after Alois P. Heinz *)

G.f.: -2*x*(3*x^5 + 12*x^4 - 12*x^3 - 20*x^2 + 5*x + 6)/(x^6 + 3*x^5 - 6*x^4 - 4*x^3 + 5*x^2 + x - 1). - Colin Barker, Sep 02 2012

A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 3, 9, 11, 10, 24, 21, 30, 50, 43, 75, 93, 96, 161, 170, 215, 312, 323, 456, 574, 639, 906, 1046, 1276, 1710, 1935, 2501, 3135, 3642, 4760, 5699, 6893, 8823, 10401, 12952, 16079, 19104, 24002, 29097, 35165, 43865, 52628, 64503, 79363, 95329

Offset: 0

John Tyler Rascoe, Apr 29 2025

A rise is any pair of parts (p_{i-1},p_i) with p_{i-1} < p_i.

By reversal a(n) is also the number of descents in all compositions of n of this kind.

			For n = 6 the following compositions have 5 rises: (1,2,1,2), (1,2,3), (2,1,2,1), (3,2,1).

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,2,-1,0,-2,0,-1).

Cf. A011782, A076118, A124760, A151842, A173258, A214247, A220062, A238343.

A_x(N) = {my(x='x+O('x^N)); concat([0,0,0], Vec(x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2))}
A_x(40)

G.f.: x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2.

cp's OEIS Frontend

A208666 Number of 2n-bead necklaces labeled with numbers 1..n allowing reversal, with neighbors differing by exactly 1.

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A153361 Number of zig-zag paths from top to bottom of a rectangle of width 12 with n rows.

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A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

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