A065180 -id:A065180 - cp's OEIS frontend

A065177 Table M(n,b) (columns: n >= 1, rows: b >= 0) gives the number of site swap juggling patterns with exact period n, using exactly b balls, where cyclic shifts are not counted as distinct.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 15, 12, 4, 1, 0, 9, 42, 42, 20, 5, 1, 0, 18, 107, 156, 90, 30, 6, 1, 0, 30, 294, 554, 420, 165, 42, 7, 1, 0, 56, 780, 2028, 1910, 930, 273, 56, 8, 1, 0, 99, 2128, 7350, 8820, 5155, 1806, 420, 72, 9, 1, 0, 186, 5781, 26936

Offset: 0

Antti Karttunen, Oct 19 2001

			Upper left corner starts as:
  1, 0,  0,   0,    0,     0,     0, ...
  1, 1,  2,   3,    6,     9,    18, ...
  1, 2,  6,  15,   42,   107,   294, ...
  1, 3, 12,  42,  156,   554,  2028, ...
  1, 4, 20,  90,  420,  1910,  8820, ...
  1, 5, 30, 165,  930,  5155, 28830, ...
  1, 6, 42, 273, 1806, 11809, 77658, ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
Juggling Information Service, Site Swap FAQs

Row 1: A059966, row 2: A065178, row 3: A065179, row 4: A065180.
Column 1: A002378, column 2: A059270.
Main diagonal gives A306173.
Cf. also A065167. trinv given at A054425.

[seq(DistSS_table(j),j=0..119)]; DistSS_table := (n) -> DistSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2)));
with(numtheory); DistSS := proc(n,b) local d,s; s := 0; for d in divisors(n) do s := s+mobius(n/d)*((b+1)^d - b^d); od; RETURN(s/n); end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
DistSS[n_, b_] := DivisorSum[n, MoebiusMu[n/#]*((b + 1)^# - b^#)&] /n;
a[n_] := DistSS[(((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1, (n - ((trinv[n]*(trinv[n] - 1))/2))];
Table[a[n], {n, 0, 119}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

Row n is the inverse Euler transform of j-> n^(j-1). - Alois P. Heinz, Jun 23 2018

A065178 Number of site swap patterns with 2 balls and exact period n.

Original entry on oeis.org

1, 2, 6, 15, 42, 107, 294, 780, 2128, 5781, 15918, 43885, 122010, 340323, 954394, 2685930, 7588770, 21507696, 61144062, 174283887, 498012094, 1426213191, 4092816966, 11767176070, 33890202192, 97761428205, 282424564744

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

When interspersed with 0's, exponents in expansion of A065481 as a product zeta(n)^(-a(n)).

			We have one period 1 (2), two period 2 (31/13 and 40/04) and six period three 2-ball siteswaps (312, 330, 411, 420, 501, 600) (The average of the digits is always 2).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs
M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, eq. (A1).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Row 2 of A065177.

Cf. A008683, A065179, A065180, A065481.

[seq(DistSS(p,2),p=1..60)];
A065178 := proc(n)
    add( mobius(n/d)*(3^d-2^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065178(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#] * (3^#-2^#)&] / n; Array[a, 30] (* Jean-François Alcover, Mar 05 2016, after R. J. Mathar *)

a(n) ~ 3^n/n. - Vaclav Kotesovec, Mar 05 2016
Inverse Euler transform of A133494. - Alois P. Heinz, Jun 23 2018
G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 3*x^k))/k. - Seiichi Manyama, Apr 14 2025

A065179 Number of site swap patterns with 3 balls and exact period n.

Original entry on oeis.org

1, 3, 12, 42, 156, 554, 2028, 7350, 26936, 98874, 365196, 1353520, 5039580, 18831306, 70626140, 265741350, 1002984060, 3796211692, 14406086604, 54801192684, 208932673508, 798218225802, 3055417434732, 11716355452900

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

			We have one period 1 (3, the three-ball cascade), three period two (42/24, 51/15 = three-ball shower and 60/06) and twelve period three 3-ball siteswaps (423, 441, 450, 522, 531, 603, 612, 630, 711, 720, 801, 900) (The average of digits is always 3).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs
M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, equation A1.

Row 3 of A065177.

Cf. A008683, A065178, A065180.

[seq(DistSS(p,3),p=1..60)];
A065179 := proc(n)
    add( mobius(n/d)*(4^d-3^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065179(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#]*(4^# - 3^#)&]/n; Array[a, 25] (* Jean-François Alcover, Mar 06 2016 *)

G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 4*x^k))/k. - Seiichi Manyama, Apr 14 2025

A255253 Complete list of siteswaps (indecomposable ground-state in concatenated decimal notation organized first by sum of digits and then by magnitude).

Original entry on oeis.org

0, 1, 2, 3, 4, 31, 40, 5, 6, 42, 51, 60, 312, 330, 411, 420, 501, 600, 7, 8, 53, 62, 71, 3122, 3302, 4013, 4112, 4130, 4202, 4400, 5111, 5120, 5201, 5300, 6011, 6020, 7001, 8000, 9, 423, 441, 450, 522, 531, 603, 612, 630

Offset: 1

Gordon Hamilton, Feb 18 2015

Siteswaping is worthy of exploration in the elementary school classroom. In my experience (Gordon Hamilton) students across a full spectrum of ability find the subject matter intriguing and the mathematics engaging.
By "indecomposable" we mean that the juggling state sequence associated to each loop should not return to the ground state 7 (xxx) until after the last throw.
By "ground state" we mean that the permutation is chosen that is as large as possible. Example: 3302 is the same as 3023 and 0233 and 2330. Only the 3302 is in the list because it is the largest number.
The list breaks down at term 57, which requires a digit for "10." In the classroom this can be solved by writing "10" vertically or using commas.

			There are 13 siteswap sequences that have a digit-sum of 9. In order, these are 9, 423, 441, 450, 522, 531, 603, 612, 630, 711, 720, 801, 900.

Cf. A065178, A084509, A065180.

A383042 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Euler transform of j-> k^(j-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 15, 6, 0, 1, 5, 20, 42, 42, 9, 0, 1, 6, 30, 90, 156, 107, 18, 0, 1, 7, 42, 165, 420, 554, 294, 30, 0, 1, 8, 56, 273, 930, 1910, 2028, 780, 56, 0, 1, 9, 72, 420, 1806, 5155, 8820, 7350, 2128, 99, 0

Offset: 1

Seiichi Manyama, Apr 13 2025

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  2,   6,   12,   20,    30,    42, ...
  0,  3,  15,   42,   90,   165,   273, ...
  0,  6,  42,  156,  420,   930,  1806, ...
  0,  9, 107,  554, 1910,  5155, 11809, ...
  0, 18, 294, 2028, 8820, 28830, 77658, ...
  ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=1..5 give A000007, A059966, A065178, A065179, A065180.
Main diagonal gives A306173.
Cf. A065177 (another version).
Cf. A008683, A074650, A383033.

a(n, k) = sumdiv(n, d, moebius(n/d)*(k^d-(k-1)^d))/n;

A(n,k) = (1/n) * Sum_{d|n} mu(n/d) * (k^d - (k-1)^d).

A(n,k) = (1/n) * (k^n - (k-1)^n - Sum_{d

A(n,k) = A074650(n,k) - A074650(n,k-1).

Product_{n>=1} 1/(1 - x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + x^j/(1 - k*x^j)) / j.

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A065177 Table M(n,b) (columns: n >= 1, rows: b >= 0) gives the number of site swap juggling patterns with exact period n, using exactly b balls, where cyclic shifts are not counted as distinct.

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A065178 Number of site swap patterns with 2 balls and exact period n.

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A065179 Number of site swap patterns with 3 balls and exact period n.

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A255253 Complete list of siteswaps (indecomposable ground-state in concatenated decimal notation organized first by sum of digits and then by magnitude).

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A383042 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Euler transform of j-> k^(j-1).

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