A010677 -id:A010677 - cp's OEIS frontend

A271832 Period 12 zigzag sequence: repeat [0,1,2,3,4,5,6,5,4,3,2,1].

0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1

Offset: 0

Wesley Ivan Hurt, Apr 15 2016

a(n)/36 is the probability that the sum shown after rolling a pair of standard dice is 1+(n mod 12). - Mathew Englander, Jul 11 2022

Decimal expansion of 37037/3000003. - Elmo R. Oliveira, Mar 03 2024

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

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), this sequence (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

Cf. A010677, A011655, A110551.

&cat[[0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]: n in [0..10]];

A271832:=n->[0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1][(n mod 12)+1]: seq(A271832(n), n=0..300);

CoefficientList[Series[x*(1 + x + x^2 + x^3 + x^4 + x^5)/(1 - x + x^6 - x^7), {x, 0, 100}], x]

lista(nn) = for(n=0, nn, print1(abs(n-12*round(n/12)), ", ")); \\ Altug Alkan, Apr 15 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5)/(1 - x + x^6 - x^7).
a(n) = a(n-1) - a(n-6) + a(n-7) for n>6.
a(n) = abs(n - 12*round(n/12)).
a(n) = Sum_{i=1..n} (-1)^floor((i-1)/6).
a(2n) = a(10n) = 2*A260686(n), a(2n+1) = A110551(n).
a(3n) = 3*A007877(n), a(4n) = a(8n) = 4*A011655(n).
a(6n) = A010677(n) = 6*A000035(n).
a(n) = a(n-12) for n >= 12. - Wesley Ivan Hurt, Sep 07 2022

A309315 Number of 5-colorings of an n-wheel graph.

Original entry on oeis.org

60, 120, 420, 1200, 3660, 10920, 32820, 98400, 295260, 885720, 2657220, 7971600, 23914860, 71744520, 215233620, 645700800, 1937102460, 5811307320, 17433922020, 52301766000, 156905298060, 470715894120, 1412147682420, 4236443047200, 12709329141660

Offset: 3

Aalok Sathe, Jul 23 2019

Cf. A010677 (for 3-colorings), A090860 (for 4-colorings).

Colin Barker, Table of n, a(n) for n = 3..1000
Prateek Bhakta, Benjamin Brett Buckner, Lauren Farquhar, Vikram Kamat, Sara Krehbiel, Heather M. Russell, Cut-Colorings in Coloring Graphs, Graphs and Combinatorics, (2019) 35(1), 239-248.
Luis Cereceda, Janvan den Heuvel, Matthew Johnson, Connectedness of the graph of vertex-colourings, Discrete Mathematics, (2008) 308(5-6), 913-919.
Eric Weisstein's World of Mathematics, Wheel Graph
Wikipedia, Chromatic polynomial
Wikipedia, Wheel graph
Index entries for linear recurrences with constant coefficients, signature (2,3).

Cf. A010677, A090860.

Vec(60*x^3 / ((1 + x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Jul 24 2019

a(n) = 5*3^(n-1)-15*(-1)^n.
From Colin Barker, Jul 24 2019: (Start)
G.f.: 60*x^3 / ((1 + x)*(1 - 3*x)).
a(n) = 2*a(n-1) + 3*a(n-2) for n>4.
(End)

A360222 a(n) is the number of permutable pieces in a standard n X n X n Rubik's cube.

Original entry on oeis.org

0, 8, 20, 56, 92, 152, 212, 296, 380, 488, 596, 728, 860, 1016, 1172, 1352, 1532, 1736, 1940, 2168, 2396, 2648, 2900, 3176, 3452, 3752, 4052, 4376, 4700, 5048, 5396, 5768, 6140, 6536, 6932, 7352, 7772, 8216, 8660, 9128, 9596, 10088, 10580, 11096, 11612, 12152

Offset: 1

William Riley Barker, Jan 30 2023

			The 2 X 2 X 2 Rubik's cube consists of 8 corner pieces, so a(2) = 8; the 3 X 3 X 3 cube has 8 corner pieces, 12 edge pieces, and 6 non-permutable center pieces, so a(3) = 8 + 12 = 20.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Rubik cube
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A005897, A010677.

A360222[n_] := If[n == 1, 0, 6*((n-2)*n - Mod[n, 2]) + 8]; Array[A360222, 50] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {0, 8, 20, 56, 92}, 50] (* Paolo Xausa, Oct 04 2024 *)

N = 20
seq = [0]
for n in range(2, N+1):
   seq.append( 8 + 12*(n-2) + 6*((n-2)**2 - (n%2)) )

a(n) = 8 + 12*(n-2) + 6*((n-2)^2 - (n mod 2)) for n > 1, a(1) = 0.
G.f.: 4*x^2*(x^3-4*x^2-x-2)/((x+1)*(x-1)^3).
a(n) = A005897(n-1) - A010677(n) for n>=2.
E.g.f.: 2*(2*(x - 2) + (3*x^2 - 3*x + 4)*cosh(x) + (3*x^2 - 3*x + 1)*sinh(x)). - Stefano Spezia, Feb 02 2023

A021169 Decimal expansion of 1/165.

Original entry on oeis.org

0, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6

Offset: 0

N. J. A. Sloane

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

Cf. A010677 shifted right.

Join[{0,0},RealDigits[1/165,10,120][[1]]] (* or *) PadRight[{0},120,{6,0}] (* Harvey P. Dale, Aug 29 2021 *)

cp's OEIS Frontend

A271832 Period 12 zigzag sequence: repeat [0,1,2,3,4,5,6,5,4,3,2,1].

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A309315 Number of 5-colorings of an n-wheel graph.

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A360222 a(n) is the number of permutable pieces in a standard n X n X n Rubik's cube.

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A021169 Decimal expansion of 1/165.

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