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 1-6 of 6 results.

A056452 a(n) = 6^floor((n+1)/2).

Original entry on oeis.org

1, 6, 6, 36, 36, 216, 216, 1296, 1296, 7776, 7776, 46656, 46656, 279936, 279936, 1679616, 1679616, 10077696, 10077696, 60466176, 60466176, 362797056, 362797056, 2176782336, 2176782336, 13060694016, 13060694016, 78364164096
Offset: 0

Views

Author

Marks R. Nester

Keywords

Comments

Number of achiral rows of length n using up to six different colors. For a(3) = 36, the rows are AAA, ABA, ACA, ADA, AEA, AFA, BAB, BBB, BCB, BDB, BEB, BFB, CAC, CBC, CCC, CDC, CEC, CFC, DAD, DBD, DCD, DDD, DED, DFD, EAE, EBE, ECE, EDE, EEE, EFE, FAF, FBF, FCF, FDF, FEF, and FFF. - Robert A. Russell, Nov 08 2018
Also: a(n) is the number of palindromes with n digits using a maximum of six different symbols. - David A. Corneth, Nov 09 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 k=6 of A321391.
Cf. A016116.
Cf. A000400 (oriented), A056308 (unoriented), A320524 (chiral).

Programs

  • Magma
    [6^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011
  • Maple
    A056452:=n->6^floor((n+1)/2);
  • Mathematica
    Riffle[6^Range[0, 20], 6^Range[20]] (* Harvey P. Dale, Jun 18 2017 *)
Table[6^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 6}, {1, 6}, 40] (* Robert A. Russell, Nov 08 2018 *)

Formula

a(n) = 6^floor((n+1)/2).
a(n) = 6*a(n-2). - Colin Barker, May 06 2012
G.f.: (1+6*x) / (1-6*x^2). - Colin Barker, May 06 2012 [Adapted to offset 0 by Robert A. Russell, Nov 08 2018]
a(n) = C(6,0)*A000007(n) + C(6,1)*A057427(n) + C(6,2)*A056453(n) + C(6,3)*A056454(n) + C(6,4)*A056455(n) + C(6,5)*A056456(n) + C(6,6)*A056457(n). - Robert A. Russell, Nov 08 2018

Extensions

a(0)=1 prepended by Robert A. Russell, Nov 08 2018
Name corrected by David A. Corneth, Nov 08 2018

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

A056313 Number of reversible strings with n beads using exactly six different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64615680, 476515080, 3355679880, 22837101840, 151449674040, 984573656640, 6302070915840, 39847411326600, 249509384858160, 1550188410555960, 9570844671224760
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent.

Examples

			For n=6, the 360 rows are 360 permutations of ABCDEF that do not include any mutual reversals.  Each of the 360 chiral pairs, such as ABCDEF-FEDCBA, is then counted just once. - _Robert A. Russell_, Sep 25 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

Cf. A056308, A056322.
Column 6 of A305621.
Equals (A000920 + A056457) / 2 = A000920 - A305626 = A305626 + A056457.

Programs

  • Mathematica
    k=6; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)
  • PARI
    a(n) = my(k=6); k!/2*(stirling(n, k, 2) + stirling(ceil(n/2), k, 2)); \\ Altug Alkan, Sep 27 2018

Formula

a(n) = A056308(n) - 6*A032122(n) + 15*A032121(n) - 20*A032120(n) + 15*A005418(n+1) - 6.
G.f.: 360*x^6*(8*x^2 - x - 1)*(90*x^7 - 9*x^6 - 29*x^5 - 34*x^4 + 15*x^3 + 9*x^2 - x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)*(6*x^2 - 1)). - Colin Barker, Sep 03 2012
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=6 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A056467 Number of primitive (aperiodic) palindromes using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 720, 720, 15120, 15120, 191520, 191520, 1905120, 1905120, 16435440, 16435440, 129230640, 129229920, 953029440, 953028720, 6711344640, 6711329520, 45674188560, 45674173440, 302899156560, 302898965040, 1969147121760, 1969146930240
Offset: 1

Views

Author

Marks R. Nester

Keywords

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 6 of A327873.
Cf. A056457, A056462.

Formula

a(n) = Sum_{d|n} mu(d)*A056457(n/d).

Extensions

Terms a(28) and beyond from Andrew Howroyd, Sep 29 2019

A305626 Number of chiral pairs of rows of n colors with exactly 6 different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64614960, 476514360, 3355664760, 22837086720, 151449482520, 984573465120, 6302069010720, 39847409421480, 249509368422720, 1550188394120520, 9570844541994120, 58789922099665680, 359629148397511080, 2192484972513916080, 13329510116645202480, 80854267307329446840, 489528474458978944080, 2959252601445086408280, 17866194139995100525080, 107751636988750077294240, 649286502010403671101240
Offset: 1

Views

Author

Robert A. Russell, Jun 06 2018

Keywords

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			For a(6) = 360, the chiral pairs are the 6! = 720 permutations of ABCDEF, each paired with its reverse.

Crossrefs

Sixth column of A305622.
A056457(n) is number of achiral rows of n colors with exactly 6 different colors.
Cf. A000920, A056313.

Programs

  • Mathematica
    k=6; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]
  • PARI
    a(n) = 360*(stirling(n, 6, 2) - stirling(ceil(n/2), 6, 2)); \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=6 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000920(n) - A056457(n)) / 2.
a(n) = A000920(n) - A056313(n) = A056313(n) - A056457(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=6 colors used.

A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 6, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 126, 1806, 8400, 16800, 15120, 5040
Offset: 0

Views

Author

Robert A. Russell, Nov 09 2018

Keywords

Comments

Each zero in the data is the beginning of a new row.
Same as A131689, with rows (except for the first) repeated. - Joerg Arndt, Sep 08 2019

Examples

			The triangle begins with T(0,0):
1
0 1
0 1
0 1   2
0 1   2
0 1   6     6
0 1   6     6
0 1  14    36     24
0 1  14    36     24
0 1  30   150    240    120
0 1  30   150    240    120
0 1  62   540   1560   1800    720
0 1  62   540   1560   1800    720
0 1 126  1806   8400  16800   15120    5040
0 1 126  1806   8400  16800   15120    5040
0 1 254  5796  40824 126000  191520  141120   40320
0 1 254  5796  40824 126000  191520  141120   40320
0 1 510 18150 186480 834120 1905120 2328480 1451520 362880
For T(7,2)=14, the rows are AAABAAA, AABABAA, AABBBAA, ABAAABA, ABABABA, ABBABBA, ABBBBBA, BAAAAAB, BAABAAB, BABABAB, BABBBAB, BBAAABB, BBABABB, and BBBABBB.

Crossrefs

Columns 0-6 are A000007, A057427, A056453, A056454, A056455, A056456, A056457.
Cf. A019538 (oriented), A305621 (unoriented), A305622 (chiral).
Cf. A131689.

Programs

  • Mathematica
    Table[k! StirlingS2[Ceiling[n/2], k], {n, 0, 18}, {k, 0, (n+1)/2}] // Flatten

Formula

T(n,k) = k!*S2(ceiling(n/2),k), where S2 is the Stirling subset number A008277.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.