A124698 -id:A124698 - cp's OEIS frontend

A208774 Number of n-bead necklaces labeled with numbers 1..5 not allowing reversal, with no adjacent beads differing by more than 1.

5, 9, 13, 24, 41, 91, 185, 435, 1009, 2445, 5945, 14813, 36977, 93465, 237313, 606465, 1556033, 4010205, 10367897, 26891385, 69930457, 182302161, 476262761, 1246710303, 3269321393, 8587489185, 22590646417, 59511087087, 156973954865, 414552479249, 1096017973385, 2900753690865, 7684758676201, 20377462520193

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..100
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

Column 5 of A208777.

Cf. A215337 (cyclically smooth Lyndon words with 5 colors).

sn[n_, k_] := 1/n*Sum[DivisorSum[n, EulerPhi[#]*(1 + 2*Cos[i*Pi/(k + 1)])^(n/#)&], {i, 1, k}]; Table[sn[n, 5], {n, 1, 34}] // Simplify (* Jean-François Alcover, Oct 31 2017, after Joerg Arndt *)

/* from the Knopfmacher et al. reference */
default(realprecision,99); /* using floats */
sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
vector(66,n, round(sn(n,5)) )
/* Joerg Arndt, Aug 09 2012 */

a(n) = Sum_{ d | n } A215337(d). - Joerg Arndt, Aug 13 2012

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124698(n). - Andrew Howroyd, Mar 18 2017

A136493 Triangle of coefficients of characteristic polynomials of symmetrical pentadiagonal matrices of the type (1,-1,1,-1,1).

Original entry on oeis.org

1, -1, 1, 1, -2, 0, -1, 3, 0, 0, 1, -4, 1, 2, 0, -1, 5, -3, -5, 1, 1, 1, -6, 6, 8, -5, -2, 1, -1, 7, -10, -10, 14, 4, -4, 0, 1, -8, 15, 10, -29, -4, 12, 0, 0, -1, 9, -21, -7, 50, -4, -30, 4, 4, 0, 1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1

Offset: 0

Roger L. Bagula, Mar 21 2008

From Georg Fischer, Mar 29 2021: (Start)
The pentadiagonal matrices have 1 in the main diagonal, -1 in the first lower and upper diagonal, 1 in the second lower and upper diagonal, and 0 otherwise.
The linear recurrences that yield A124805, A124806, A124807 and similar can be derived from the rows of this triangle (the first element of a row must be removed and multiplied onto the remaining elements).
This observation extends to other sequences. For example the linear recurrence signature (5,-6,2,4,0) of A124698 "Number of base 5 circular n-digit numbers with adjacent digits differing by 1 or less" can be derived from the coefficients of the characteristic polynomial of a tridiagonal (type -1,1,-1) 5 X 5 matrix.
(End)

			Triangle begins:
   1;
  -1,   1;
   1,  -2,   0;
  -1,   3,   0,   0;
   1,  -4,   1,   2,   0;
  -1,   5,  -3,  -5,   1,  1;
   1,  -6,   6,   8,  -5, -2,   1;
  -1,   7, -10, -10,  14,  4,  -4,   0;
   1,  -8,  15,  10, -29, -4,  12,   0,   0;
  -1,   9, -21,  -7,  50, -4, -30,   4,   4,  0;
   1, -10,  28,   0, -76, 28,  61, -20, -15,  2,  1;

Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A011658, A087509, A124805 ff., A124696 ff., A124999 ff., A161680, A181983.

T[n_, m_]:= Piecewise[{{-1, 1+m==n || m==1+n}, {1, 2+m==n || m==n || m==2+n}}];
MO[d_]:= Table[T[n, m], {n,d}, {m,d}];
CL[n_]:= CoefficientList[CharacteristicPolynomial[MO[n], x], x];
Join[{{1}}, Table[Reverse[CL[n]], {n,10}]]//Flatten
(* For the signature of A124698 added by Georg Fischer, Mar 29 2021 : *)
Reverse[CoefficientList[CharacteristicPolynomial[{{1,-1,0,0,0}, {-1, 1,-1,0,0}, {0,-1,1,-1,0}, {0,0,-1,1,-1}, {0,0,0,-1,1}}, x], x]]

Sum_{k=1..n} T(n, k) = (-1)^(n mod 3) * A087509(n+1) + [n=1].
From G. C. Greubel, Aug 01 2023: (Start)
T(n, n) = A011658(n+2).
T(n, 1) = (-1)^(n-1).
T(n, 2) = A181983(n-1).
T(n, 3) = (-1)^(n-3)*A161680(n-3). (End)

Edited by Georg Fischer, Mar 29 2021

cp's OEIS Frontend

A208774 Number of n-bead necklaces labeled with numbers 1..5 not allowing reversal, with no adjacent beads differing by more than 1.

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