cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 3, 1, 6, 6, 1, 10, 21, 12, 1, 14, 36, 24, 1, 22, 93, 132, 60, 1, 30, 150, 240, 120, 1, 46, 345, 900, 960, 360, 1, 62, 540, 1560, 1800, 720, 1, 94, 1173, 4980, 9300, 7920, 2520, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 190, 3801, 24612, 71400, 103320, 73080, 20160, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320
Offset: 1

Views

Author

Robert A. Russell, Jun 04 2018

Keywords

Comments

The number of achiral necklaces is equivalent to the number of achiral bracelets.

Examples

			The triangle begins with T(1,1):
1;
1,   1;
1,   2;
1,   4,     3;
1,   6,     6;
1,  10,    21,     12;
1,  14,    36,     24;
1,  22,    93,    132,     60;
1,  30,   150,    240,    120;
1,  46,   345,    900,    960,     360;
1,  62,   540,   1560,   1800,     720;
1,  94,  1173,   4980,   9300,    7920,    2520;
1, 126,  1806,   8400,  16800,   15120,    5040;
1, 190,  3801,  24612,  71400,  103320,   73080,   20160;
1, 254,  5796,  40824, 126000,  191520,  141120,   40320;
1, 382, 11973, 113652, 480060, 1048320, 1234800,  745920, 181440;
1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;
For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB.

Crossrefs

Odd rows are A019538.
Even rows are A172106.
Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492.

Programs

  • Mathematica
    Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] + StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 15}, {k, 1, Ceiling[(n + 1)/2]}] // Flatten
  • PARI
    T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2));
tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 02 2018

Formula

T(n,k) = (k!/2) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), where S2(n,k) is the Stirling subset number A008277.
T(n,k) = 2*A273891(n,k) - A087854(n,k).
G.f. for column k>1: (k!/2) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2). - Robert A. Russell, Sep 26 2018

A056343 Number of bracelets of length n using exactly three different colored beads.

Original entry on oeis.org

0, 0, 1, 6, 18, 56, 147, 411, 1084, 2979, 8043, 22244, 61278, 171030, 477929, 1345236, 3795750, 10758902, 30572427, 87149124, 248991822, 713096352, 2046303339, 5883433409, 16944543810, 48879769575
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over will not create a new bracelet.

Examples

			For a(4)=6, the arrangements are ABAC, ABCB, ACBC, AABC, ABBC, and ABCC. Only the last three are chiral, their reverses being AACB, ACBB, and ACCB respectively. - _Robert A. Russell_, Sep 26 2018

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

Crossrefs

Column 3 of A273891.
Equals (A056283 + A056489) / 2 = A056283 - A305542 = A305542 + A056489.

Programs

  • Mathematica
    t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
a[n_] := T[n, 3];
Array[a, 26] (* Jean-François Alcover, Nov 05 2017, after Andrew Howroyd *)
k=3; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

Formula

a(n) = A027671(n) - 3*A000029(n) + 3.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=3 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=3 is the number of colors. (End)

A305542 Number of chiral pairs of color loops of length n with exactly 3 different colors.

Original entry on oeis.org

0, 0, 1, 3, 12, 35, 111, 318, 934, 2634, 7503, 21071, 59472, 167229, 472133, 1333263, 3777600, 10721837, 30516447, 87035631, 248820816, 712751271, 2045784183, 5882388956, 16942974060, 48876617790, 141204945463, 408495109005, 1183247473872, 3431451145390, 9962348798055, 28953196894668
Offset: 1

Views

Author

Robert A. Russell, Jun 04 2018

Keywords

Examples

			For a(4)=3, the chiral pairs of color loops are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.

Crossrefs

Third column of A305541.

Programs

  • Mathematica
    k=3; Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 40}]
  • PARI
    a(n) = my(k=3); -(k!/4)*(stirling(floor((n+1)/2),k,2) + stirling(ceil((n+1)/2),k,2)) + (k!/(2*n))*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Jun 06 2018

Formula

a(n) = -(k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2n))*Sum_{d|n} phi(d)*S2(n/d,k), with k=3 different colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A052823(n) - A056489(n)) / 2.
a(n) = A305541(n,3).
G.f.: -(3/2) * x^4 * (1+x)^2 / Product_{j=1..3} (1-j*x^2) - Sum_{d>0} (phi(d)/(2d)) * (log(1-3x^d) - 3*log(1-2x^d) + 3*log(1-x^d)).

A056499 Number of primitive (period n) periodic palindromes using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 3, 6, 21, 36, 90, 150, 339, 540, 1149, 1806, 3765, 5790, 11880, 18150, 36894, 55980, 113145, 170970, 344541, 519156, 1043190, 1569744, 3149979, 4733670, 9488409, 14250606, 28544205, 42850116, 85786560, 128746410, 257672355, 386634018, 773623116, 1160688606
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

Crossrefs

Column 3 of A327878.
Cf. A056464, A056489.

Formula

a(n) = Sum_{d|n} mu(d)*A056489(n/d).

Extensions

Terms a(32) and beyond from Andrew Howroyd, Sep 28 2019
Showing 1-4 of 4 results.

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