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 1-3 of 3 results.

A273891 Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 18, 24, 12, 1, 11, 56, 136, 150, 60, 1, 16, 147, 612, 1200, 1080, 360, 1, 28, 411, 2619, 7905, 11970, 8820, 2520, 1, 44, 1084, 10480, 46400, 105840, 129360, 80640, 20160, 1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440
Offset: 1

Views

Author

Marko Riedel, Jun 02 2016

Keywords

Comments

For bracelets, chiral pairs are counted as one.

Examples

			Triangle begins with T(1,1):
1;
1,  1;
1,  2,    1;
1,  4,    6,     3;
1,  6,   18,    24,     12;
1, 11,   56,   136,    150,     60;
1, 16,  147,   612,   1200,   1080,     360;
1, 28,  411,  2619,   7905,  11970,    8820,    2520;
1, 44, 1084, 10480,  46400, 105840,  129360,   80640,  20160;
1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440;
For T(4,2)=4, the arrangements are AAAB, AABB, ABAB, and ABBB, all achiral.
For T(4,4)=3, the arrangements are ABCD, ABDC, and ACBD, whose chiral partners are ADCB, ACDB, and ADBC respectively. - _Robert A. Russell_, Sep 26 2018

Links

Crossrefs

Columns 1-6: A057427, A056342, A056343, A056344, A056345, A056346.
Row sums give A019537.
Cf. A087854 (oriented), A305540 (achiral), A305541 (chiral).

Programs

  • Mathematica
    (* t = A081720 *) 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}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)
Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,10}, {k,1,n}] // Flatten (* Robert A. Russell, Sep 26 2018 *)

Formula

T(n,k) = Sum_{i=0..k-1} (-1)^i * binomial(k,i) * A081720(n,k-i). - Andrew Howroyd, Mar 25 2017
From Robert A. Russell, Sep 26 2018: (Start)
T(n,k) = (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 S2 is the Stirling subset number A008277.
G.f. for column k>1: (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).
T(n,k) = (A087854(n,k) + A305540(n,k)) / 2 = A087854(n,k) - A305541(n,k) = A305541(n,k) + A305540(n,k).
(End)

A056492 Number of periodic palindromes using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 360, 720, 7920, 15120, 103320, 191520, 1048320, 1905120, 9170280, 16435440, 72833040, 129230640, 541130040, 953029440, 3832187040, 6711344640, 26192766600, 45674188560, 174286672560, 302899156560, 1136023139160, 1969147121760
Offset: 1

Views

Author

Marks R. Nester

Keywords

Examples

			For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
There are 720 permutations of the six letters used in ABACDEFEDC.  These 720 arrangements can be paired up with a half turn (e.g., ABACDEFEDC-EFEDCABACD) to arrive at the 360 arrangements for n=10.

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

Cf. A056286, A056346, A056457.
Column 6 of A305540.

Programs

  • GAP
    a:=[0,0,0,0,0,0,0,0,0,360,720];; for n in [12..35] do a[n]:=a[n-1] +20*a[n-2]-20*a[n-3]-155*a[n-4]+155*a[n-5]+580*a[n-6] -580*a[n-7] -1044*a[n-8]+1044*a[n-9]+720*a[n-10]-720*a[n-11]; od; a; # Muniru A Asiru, Sep 26 2018
  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!(360*x^10*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)*(6*x^2-1)))); // G. C. Greubel, Oct 13 2018
  • Maple
    with(combinat):  a:=n->(factorial(6)/2)*(Stirling2(floor((n+1)/2),6)+Stirling2(ceil((n+1)/2),6)): seq(a(n),n=1..35); # Muniru A Asiru, Sep 26 2018
  • Mathematica
    k = 6; Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] + StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 40}] (* Robert A. Russell, Jun 05 2018 *)
LinearRecurrence[{1,20,-20,-155,155,580,-580,-1044,1044,720,-720}, Join[Table[0,{9}],{360,720}],40] (* Robert A. Russell, Sep 29 2018 *)
  • PARI
    a(n) = my(k=6); (k!/2)*(stirling(floor((n+1)/2), k, 2) + stirling(ceil((n+1)/2), k, 2)); \\ Michel Marcus, Jun 05 2018

Formula

a(n) = 2*A056346(n) - A056286(n).
G.f.: 360*x^10*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)*(6*x^2-1)). - Colin Barker, Jul 08 2012
a(n) = (k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), with k=6 different colors used and where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, Jun 05 2018
a(n) = a(n-1) + 20*a(n-2) - 20*a(n-3) - 155*a(n-4) + 155*a(n-5) + 580*a(n-6) - 580*a(n-7) - 1044*a(n-8) + 1044*a(n-9) + 720*a(n-10) - 720*a(n-11). - Muniru A Asiru, Sep 26 2018

A056352 Number of primitive (period n) bracelets using exactly six different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 60, 1080, 11970, 105840, 821952, 5874480, 39713490, 258136200, 1631272140, 10096734312, 61536365730, 370710950400, 2213749552980, 13132080672480, 77509456122366, 455754569691600
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Turning over will not create a new bracelet.

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]

Crossrefs

Cf. A056291.

Formula

Sum mu(d)*A056346(n/d) where d|n.
Showing 1-3 of 3 results.

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