A007997 -id:A007997 - cp's OEIS frontend

A007998 Number of primitive solutions to x + y + z = 0 (mod n).

1, 2, 3, 3, 4, 5, 5, 7, 8, 9, 9, 15, 11, 17, 17, 18, 18, 31, 21, 33, 30, 33, 30, 53, 34, 48, 46, 53, 43, 87, 47, 73, 69, 79, 66, 110, 65, 100, 94, 111, 80, 159, 86, 135, 130, 135, 108, 202, 116, 181, 153, 175, 133, 255, 150, 212

Offset: 1

N. J. A. Sloane

I can't reconstruct this. Possibly an erroneous version of A003050 or A048259. - N. J. A. Sloane

Cf. A007997.

A032195 Number of necklaces with 10 black beads and n-10 white beads.

Original entry on oeis.org

1, 1, 6, 22, 73, 201, 504, 1144, 2438, 4862, 9252, 16796, 29414, 49742, 81752, 130752, 204347, 312455, 468754, 690690, 1001603, 1430715, 2016144, 2804880, 3856892, 5245128, 7060984, 9414328, 12440668, 16301164

Offset: 10

Christian G. Bower

nonn

The g.f. is Z(C_10,x)/x^10, the 10-variate cycle index polynomial for the cyclic group C_10, with substitution x[i]->1/(1-x^i), i=1,...,10. By Polya enumeration, a(n+10) is the number of cyclically inequivalent 10-necklaces whose 10 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_10,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,9,-32,10,29,-29,-9,28,-7,-5,-5,-7,28,-9,-29,29,10,-32,9,17,-12,-2,4,-1).

Column k=10 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194.

k = 10; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 10 ]" (necklace, indistinct, unlabeled, 10 parts) transform of 1, 1, 1, 1...
G.f.: (x^10)*(1-3*x+4*x^2+12*x^3-8*x^4-x^5+31*x^6-4*x^8+16*x^9 +11*x^10 +3*x^11+8*x^12+4*x^13+4*x^14+x^15+x^16) /((1-x)^4*(1-x^2)^4 *(1-x^5)*(1-x^10)).
G.f.: (1/10)*x^10*(1/(1 - x)^10 + 1/(1 - x^2)^5 + 4/(1 - x^5)^2 + 4/(1 - x^10)^1). - Herbert Kociemba, Oct 22 2016

A032196 Number of necklaces with 11 black beads and n-11 white beads.

Original entry on oeis.org

1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32066, 58786, 104006, 178296, 297160, 482885, 766935, 1193010, 1820910, 2731365, 4032015, 5864750, 8414640, 11920740, 16689036, 23107896, 31666376, 42975796

Offset: 11

Christian G. Bower

nonn

The g.f. is Z(C_11,x)/x^11, the 11-variate cycle index polynomial for the cyclic group C_11, with substitution x[i]->1/(1-x^i), i=1..11. By Polya enumeration, a(n+11) is the number of cyclically inequivalent 11-necklaces whose 11 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_11,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1,1,-10,45,-120,210,-252,210,-120,45,-10,1).

Column k=11 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194, A032195.

k = 11; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
DeleteCases[CoefficientList[Series[(x^11) (1 - 9 x + 41 x^2 - 109 x^3 + 191 x^4 - 229 x^5 + 191 x^6 - 109 x^7 + 41 x^8 - 9 x^9 + x^10)/((1 - x)^10 (1 - x^11)), {x, 0, 39}], x], 0] (* Michael De Vlieger, Oct 10 2016 *)

"CIK[ 11 ]" (necklace, indistinct, unlabeled, 11 parts) transform of 1, 1, 1, 1...
G.f.: (x^11) * (1 - 9*x + 41*x^2 - 109*x^3 + 191*x^4 - 229*x^5 + 191*x^6 - 109*x^7 + 41*x^8 - 9*x^9 + x^10) / ((1-x)^10 * (1-x^11)).
a(n) = ceiling(binomial(n, 11)/n) (conjecture Wolfdieter Lang).
From Herbert Kociemba, Oct 11 2016: (Start)
This conjecture indeed is true.
Sketch of proof:
There are binomial(n,11) ways to place the 11 black beads in the necklace with n beads. If n is not divisible by 11 there are no necklaces with a rotational symmetry. So exactly n necklaces are equivalent up to rotation and there are binomial(n,11)/n = ceiling(binomial(n,11)/n) equivalence classes.
If n is divisible by 11 the only way to get a necklace with rotational symmetry is to space out the 11 black beads evenly. There are n/11 ways to do this and all ways are equivalent up to rotation. So there are  binomial(n,11) - n/11 unsymmetric necklaces which give binomial(n,11)/n - 1/11 equivalence classes. If we add the single symmetric equivalence class we get Binomial(n,11)/n - 1/11 + 1 which also is ceiling(binomial(n,11)/n). (End)
G.f.: (10/(1 - x^11) + 1/(1 - x)^11)*x^11/11. - Herbert Kociemba, Oct 16 2016

A032197 Number of necklaces with 12 black beads and n-12 white beads.

Original entry on oeis.org

1, 1, 7, 31, 116, 364, 1038, 2652, 6310, 14000, 29414, 58786, 112720, 208012, 371516, 643856, 1086601, 1789515, 2883289, 4552275, 7056280, 10752060, 16128424, 23841480, 34769374, 50067108, 71250060, 100276894, 139672312

Offset: 12

Christian G. Bower

nonn

The g.f. is Z(C_12,x)/x^12, the 12-variate cycle index polynomial for the cyclic group C_12, with substitution x[i]->1/(1-x^i), i=1,...,12. Therefore by Polya enumeration a(n+12) is the number of cyclically inequivalent 12-necklaces whose 12 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_12,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,-4,12,-12,2,2,-12,24,-18,4,4,-6,15,-20,0,10,-4,10,0,-20,15,-6,4,4,-18,24,-12,2,2,-12,12,-4,4,-2,-4,4,-1).

Column k=12 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192-A032196.

k = 12; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 12 ]" (necklace, indistinct, unlabeled, 12 parts) transform of 1, 1, 1, 1...
G.f.: (x^12)*(1-3*x+7*x^2+9*x^3+18*x^4+38*x^5+72*x^6+92*x^7+168*x^8+160*x^9+238*x^10+230*x^11+296*x^12+234*x^13+330*x^14+248*x^15+284*x^16+238*x^17+230*x^18+166*x^19+172*x^20+78*x^21+80*x^22+38*x^23+21*x^24+7*x^25+3*x^26+x^27) /((1+x)*(1-x)*(1-x^2)*(1-x^3)*(1-x)^5*(1+x+x^2)*(1-x^4)^2*(1-x^6)*(1-x^12)). - Wolfdieter Lang, Feb 15 2005 (see comment)
G.f.: 1/12 x^12 ((1 - x)^-12 + (1 - x^2)^-6 + 2 (1 - x^3)^-4 + 2 (1 - x^4)^-3 + 2 (1 - x^6)^-2 + 4 (1 - x^12)^-1). - Herbert Kociemba, Oct 22 2016

A058936 Decomposition of Stirling's S(n,2) based on associated numeric partitions.

Original entry on oeis.org

0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260, 45360, 25920, 20160, 18144, 403200, 226800, 172800, 151200, 72576, 3991680, 2217600, 1663200, 1425600, 1330560, 43545600, 23950080, 17740800, 14968800, 13685760, 6652800, 518918400

Offset: 1

Alford Arnold, Jan 11 2001

These values also appear in a wider context when counting elements of finite groups by cycle structure. For example, the alternating group on four symbols has 12 elements; eight associated with the partition 3+1, three associated with 2+2 and the identity associated with 1+1+1+1. The cross-referenced sequences are all associated with similar numeric partitions and "M2" weights.

			Triangle begins:
  0;
  1;
  3;
  8, 3;
  30, 20;
  144, 90, 40;
  840, 504, 420;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000012, A000035, A000027, A004526, A022003, A008619, A000217, A007997, A001399, A011765 A008620, A027656, A002620, A000292, A008627.

From Sean A. Irvine, Sep 05 2022: (Start)
T(1,1) = 0.
T(n,k) = n! / (k * (n-k)) for 1 <= k < n/2.
T(2n,n) = (2*n)! / (2*n^2).
(End)

More terms from Sean A. Irvine, Sep 05 2022

A195798 Number of triangular n X n X n 0..1 arrays with all rows and diagonals having the same length having the same sum, with corners zero.

Original entry on oeis.org

1, 1, 2, 8, 16, 64, 1184, 5300, 130324, 14748808, 421963232, 54990266540

Offset: 1

R. H. Hardin, Sep 23 2011

From Pontus von Brömssen, Sep 08 2022: (Start)
For n <= 6, all solutions are rotationally symmetric, which implies that a(n) = 2^(A007997(n+4)-1). But for n >= 7 there exist asymmetric solutions like
        0
       0 0
      0 0 1
     0 1 0 0
    1 0 0 0 0
   0 0 0 0 1 0
  0 0 0 1 0 0 0.
(End)

			Some solutions for n=5
......0..........0..........0..........0..........0..........0..........0
.....0.0........0.0........0.0........0.1........0.1........0.1........0.1
....0.1.0......1.0.1......1.1.1......0.0.0......0.1.0......1.0.1......1.1.1
...0.1.1.0....0.0.0.0....0.1.1.0....1.0.0.0....1.1.1.0....1.0.0.0....1.1.1.0
..0.0.0.0.0..0.0.1.0.0..0.0.1.0.0..0.0.0.1.0..0.0.0.1.0..0.0.1.1.0..0.0.1.1.0

Column 1 of A195805.

Cf. A007997.

A332050 Number of ways to arrange Palago tiles in a triangle of side length n, up to rotation, reflection, and swapping colors.

Original entry on oeis.org

1, 1, 7, 129, 9882, 2391930, 1743402771, 3812799008214, 25015772571200361, 492385451093553791610, 29074868501520453489499806, 5150525730438768829942800034449, 2737200544710109691113626131721984885, 4363981784043856212945753449232929426200329

Offset: 0

Peter Kagey, Feb 06 2020

nonn

A Palago tile is a hexagonal tile with four regions of alternating colors. See links for illustrations.

Peter Kagey, Table of n, a(n) for n = 0..64
Code Golf Stack Exchange, Counting creatures on a hexagonal tiling
Peter Kagey, Example of for n = 2.

Cf. A000217, A002620, A007997.

Cf. A325936.

a[n_] = (3^Binomial[n + 1, 2] +
    3*3^((Binomial[n + 1, 2] - Ceiling[n/2])/2) +
    If[Mod[n, 3] == 1, 0, 2*3^(Binomial[n + 1, 2]/3)])/6

a(n) = (3^A000217(n) + 3*3^A002620(n) + 2*3^A007997(n+4))/6 if n = 1 (mod 3), and

a(n) = (3^A000217(n) + 3*3^A002620(n))/6 otherwise.

A349216 Number of ternary triples (u,v,w) with 1 <= u < v < w <= n.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 13, 20, 30, 40, 53, 70, 88, 110, 137, 166, 200, 240, 281, 328, 382, 438, 501, 572, 646, 728, 819, 910, 1010, 1120, 1233, 1356, 1490, 1628, 1777, 1938, 2100, 2274, 2461, 2652, 2856, 3074, 3297, 3534, 3786, 4040, 4309, 4594, 4884, 5190, 5513, 5842, 6188, 6552, 6917

Offset: 1

Peter J. Dukes, Nov 10 2021

A triple of integers (u,v,w) is a ternary triple if in the ternary expansions of u,v,w, all three disagree at the least significant position at which any two disagree.

Equivalently, (u,v,w) is a ternary triple if the highest power of three dividing 2w-u-v is greater than the highest power of three dividing gcd(w-u,w-v).

			For n = 7 the 13 ternary triples are (1, 2, 3), (2, 3, 4), (1, 3, 5), (3, 4, 5), (1, 2, 6), (2, 4, 6), (1, 5, 6), (4, 5, 6), (2, 3, 7), (1, 4, 7), (3, 5, 7), (2, 6, 7), (5, 6, 7).

Coen del Valle and Peter J. Dukes, Balancing permuted copies of multigraphs and integer matrices, arXiv:2201.00897 [math.CO], 2022.

Cf. A061866, A005704, A038500, A007089, A007997.

Array[Sum[Sum[Sum[Boole[IntegerExponent[w + w - u - v, 3] > IntegerExponent[GCD[w - u, w - v], 3]], {u, (v - 1)}], {v, 2, (w - 1)}], {w, 3, #}] &, 55] (* Michael De Vlieger, Feb 15 2022 *)

A349216(n) = sum(w=3,n,sum(v=2,(w-1),sum(u=1,(v-1),valuation(w+w-u-v,3) > valuation(gcd(w-u,w-v),3)))); \\ Antti Karttunen, Nov 13 2021

def a(n):
    t=3^ceil(log(n,3))
    counter=0
    for w in range(n):
        for v in range(w):
            for u in range(v):
                if min(gcd(w-u,3^t),gcd(w-v,3^t))

cp's OEIS Frontend

A007998 Number of primitive solutions to x + y + z = 0 (mod n).

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A032195 Number of necklaces with 10 black beads and n-10 white beads.

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A032196 Number of necklaces with 11 black beads and n-11 white beads.

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A032197 Number of necklaces with 12 black beads and n-12 white beads.

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A058936 Decomposition of Stirling's S(n,2) based on associated numeric partitions.

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A195798 Number of triangular n X n X n 0..1 arrays with all rows and diagonals having the same length having the same sum, with corners zero.

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Comments

Examples

Crossrefs

A332050 Number of ways to arrange Palago tiles in a triangle of side length n, up to rotation, reflection, and swapping colors.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A349216 Number of ternary triples (u,v,w) with 1 <= u < v < w <= n.

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