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 1-8 of 8 results.

A056308 Number of reversible strings with n beads using a maximum of six different colors.

Original entry on oeis.org

1, 6, 21, 126, 666, 3996, 23436, 140616, 840456, 5042736, 30236976, 181421856, 1088414496, 6530486976, 39182222016, 235093332096, 1410555793536, 8463334761216, 50779983373056, 304679900238336, 1828079250264576, 10968475501587456, 65810852102532096
Offset: 0

Views

Author

Marks R. Nester

Keywords

Comments

A string and its reverse are considered to be equivalent. Thus aabc and cbaa are considered to be identical, but abca is a different string.

Examples

			For a(2)=21, the six achiral strings are AA, BB, CC, DD, EE, and FF; the 15 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, AF-FA, BC-CB, BD-DB, BE-EB, BF-FB, CD-DC, CE-EC, CF-FC, DE-ED, DF-FD, and EF-FE.

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 A277504.
Cf. A000400 (oriented), A320524 (chiral), A056452 (achiral).

Programs

  • Magma
    I:=[1,6,21]; [n le 3 select I[n] else 6*Self(n-1) +6*Self(n-2) - 36*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 10 2018
  • Mathematica
    k=6; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (* Robert A. Russell, Nov 25 2017 *)
a[ n_] := (6^n + 6^Quotient[n + 1, 2]) / 2; (* Michael Somos, Jul 10 2018 *)
LinearRecurrence[{6, 6, -36}, {1, 6, 21}, 31] (* Robert A. Russell, Nov 10 2018 *)
  • PARI
    Vec((1-21*x^2) / ((1 - 6*x)*(1 - 6*x^2)) + O(x^30)) \\ Colin Barker, Mar 20 2017 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
  • PARI
    {a(n) = (6^n + 6^((n+1)\2)) / 2}; \\ Michael Somos, Jul 10 2018

Formula

a(n) = (6^n + 6^floor((n+1)/2))/2.
G.f.: (1-21*x^2) / ((1-6*x)*(1-6*x^2)). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
a(n) = 6*a(n-1) + 6*a(n-2) - 36*a(n-3) for n > 3. - Colin Barker, Mar 20 2017
a(n) = (A000400(n) + A056452(n)) / 2. - Robert A. Russell, Jun 19 2018
a(n) = 6^(n + floor((n-1)/2)) * a(1-n) for all n in Z. - Michael Somos, Jul 10 2018

Extensions

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

A321391 Array read by antidiagonals: T(n,k) is the number of achiral rows of n colors using up to k colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 4, 1, 0, 1, 6, 5, 16, 9, 8, 1, 0, 1, 7, 6, 25, 16, 27, 8, 1, 0, 1, 8, 7, 36, 25, 64, 27, 16, 1, 0, 1, 9, 8, 49, 36, 125, 64, 81, 16, 1, 0, 1, 10, 9, 64, 49, 216, 125, 256, 81, 32, 1, 0
Offset: 0

Views

Author

Robert A. Russell, Nov 08 2018

Keywords

Comments

The antidiagonals go from top-right to bottom-left.

Examples

			The array begins with T(0,0):
1 1  1   1    1     1     1      1      1      1       1       1 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 ...
For T(3,3)=9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC.

Crossrefs

Columns 0-6 are A000007, A000012, A060546, A056449, A056450, A056451, A056452.
Cf. A003992 (oriented), A277504 (unoriented), A293500 (chiral).

Programs

  • Mathematica
    Table[If[n>0, (n-k)^Ceiling[k/2], 1], {n, 0, 12}, {k, 0, n}] // Flatten

Formula

T(n,k) = [n==0] + [n>0] * k^ceiling(n/2).
The generating function for column k is (1+k*x) / (1-k*x^2).

A056457 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, 129230640, 953029440, 953029440, 6711344640, 6711344640, 45674188560
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

Cf. A056452, A000920.

Programs

  • Mathematica
    k=6; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}]
  • PARI
    a(n) = 6!*stirling((n+1)\2, 6, 2); \\ Altug Alkan, Sep 25 2018

Formula

a(n) = 6! * Stirling2( [(n+1)/2], 6).
G.f.: 720*x^11/((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, Sep 03 2012
G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-ix^2), where k=6 is the number of symbols. - Robert A. Russell, Sep 25 2018

A056471 Number of palindromic structures using a maximum of six different symbols.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 15, 15, 52, 52, 203, 203, 876, 876, 4111, 4111, 20648, 20648, 109299, 109299, 601492, 601492, 3403127, 3403127, 19628064, 19628064, 114700315, 114700315, 676207628, 676207628, 4010090463
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Permuting the symbols will not change the structure.

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. A056452.

Formula

Sum stirling2( [(n+1)/2], k) for k from 1 to 6.
G.f.: -x*(91*x^8-135*x^6+68*x^4-14*x^2+1) / ((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(6*x^2-1)). [Colin Barker, Dec 05 2012]

A056488 Number of periodic palindromes using a maximum of six different symbols.

Original entry on oeis.org

6, 21, 36, 126, 216, 756, 1296, 4536, 7776, 27216, 46656, 163296, 279936, 979776, 1679616, 5878656, 10077696, 35271936, 60466176, 211631616, 362797056, 1269789696, 2176782336, 7618738176, 13060694016, 45712429056, 78364164096, 274274574336, 470184984576
Offset: 1

Views

Author

Marks R. Nester

Keywords

Comments

Also number of necklaces with n beads and 6 colors that are the same when turned over and hence have reflection symmetry. - Herbert Kociemba, Nov 24 2016

Examples

			G.f. = 6*x + 21*x^2 + 36*x^3 + 126*x^4 + 216*x^5 + 756*x^6 + 1296*x^7 + ...
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 6 of A284855.
Cf. A029744, A038754, A056452.

Programs

  • Magma
    [IsEven(n) select 6^(n div 2)*7/2 else 6^((n+1) div 2): n in [1..30]]; // Vincenzo Librandi, Sep 22 2018
  • Mathematica
    LinearRecurrence[{0,6},{6,21},30] (* Harvey P. Dale, Feb 02 2015 *)
k = 6; Table[(k^Floor[(n + 1)/2] + k^Ceiling[(n + 1)/2]) / 2, {n, 30}] (* Robert A. Russell, Sep 21 2018 *)
If[EvenQ[#], 6^(# / 2) 7/2, 6^((# + 1) / 2)]&/@Range[30] (* Vincenzo Librandi, Sep 22 2018 *)
  • PARI
    a(n) = if(n%2, 6^((n+1)/2), 7*6^(n/2)/2); \\ Altug Alkan, Sep 21 2018

Formula

a(n) = 6^((n+1)/2) for n odd, a(n) = 6^(n/2)*7/2 for n even.
From Colin Barker, Jul 08 2012: (Start)
a(n) = 6*a(n-2).
G.f.: 3*x*(2+7*x)/(1-6*x^2). (End)
a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 2, where k = 6 is the number of possible colors. - Robert A. Russell, Sep 22 2018

Extensions

More terms from Vincenzo Librandi, Sep 22 2018

A056462 Number of primitive (aperiodic) palindromes using a maximum of six different symbols.

Original entry on oeis.org

6, 0, 30, 30, 210, 180, 1290, 1260, 7740, 7560, 46650, 46410, 279930, 278640, 1679370, 1678320, 10077690, 10069740, 60466170, 60458370, 362795730, 362750400, 2176782330, 2176734420, 13060693800, 13060414080, 78364156320, 78363884130, 470184984570, 470183297220
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]

Crossrefs

Cf. A056391, A056452.
Column 6 of A284823.

Programs

  • PARI
    a(n) = sumdiv(n, d, moebius(d)*6^((1 + n/d)\2));
for(n=1, 32, print1(a(n), ", ")); \\ Petros Hadjicostas, Apr 24 2020

Formula

a(n) = Sum_{d|n} mu(d)*A056452(n/d).

Extensions

More terms from Petros Hadjicostas, Apr 24 2020

A154237 a(n) = ( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 12, 114, 1008, 8676, 73872, 626184, 5298048, 44791056, 378551232, 3198883104, 27030060288, 228394230336, 1929828955392, 16306120554624, 137778577993728, 1164159319286016, 9836554491620352, 83113874320863744, 702269857101754368
Offset: 1

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

Comments

Fifth binomial transform of A002533 without initial term 1. Sixth binomial transform of 1 followed by A056452.
Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(6) = 8.4494897427....

Links

Crossrefs

Cf. A010464 (decimal expansion of square root of 6), A002533, A056452.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
  • Magma
    I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-30*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Sep 07 2016
  • Mathematica
    Join[{a=1,b=12},Table[c=12*b-30*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011*)
LinearRecurrence[{12, -30}, {1, 12}, 25] (* or *) Table[( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)), {n,1,25}] (* G. C. Greubel, Sep 07 2016 *)
  • Sage
    [lucas_number1(n,12,30) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

Formula

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 12*a(n-1) - 30*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 12*x + 30*x^2). (End)

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009
Edited by Klaus Brockhaus, Oct 06 2009

A320524 Number of chiral pairs of a row of n colors using 6 or fewer colors.

Original entry on oeis.org

0, 15, 90, 630, 3780, 23220, 139320, 839160, 5034960, 30229200, 181375200, 1088367840, 6530207040, 39181942080, 235091652480, 1410554113920, 8463324683520, 50779973295360, 304679839772160, 1828079189798400, 10968475138790400, 65810851739735040, 394865110438410240, 2369190668072417280, 14215144008434503680, 85290864083258757120
Offset: 1

Views

Author

Robert A. Russell, Oct 14 2018

Keywords

Comments

A chiral row is different from its reverse.

Examples

			For a(2)=15, the chiral pairs are the fifteen combinations of six colors taken two at a time, e.g., AB-BA.

Links

Crossrefs

Column 6 of A293500.
Cf. A000400 (oriented), A056308 (unoriented), A056452 (achiral).

Programs

  • Magma
    [(6^n - 6^Ceiling(n / 2)) / 2: n in [1..25]]; // Vincenzo Librandi, Oct 15 2018
  • Mathematica
    k = 6; Table[(k^n - k^Ceiling[n/2])/2, {n, 1, 30}]
LinearRecurrence[{6, 6, -36}, {0, 15, 90}, 30]
  • PARI
    m=40; v=concat([0,15,90], vector(m-3)); for(n=4, m, v[n] = 6*v[n-1] +6*v[n-2] -36*v[n-3]); v \\ G. C. Greubel, Oct 17 2018

Formula

a(n) = (k^n - k^ceiling(n/2)) / 2, where k=6 is maximum number of colors.
G.f.: k*x^2*(k-1) / (2*(1-k*x)*(1-k*x^2)), where k=6.
a(n) = (A000400(n) - A056452(n)) / 2 = A000400(n) - A056308(n) = A056308(n) - A056452(n).
Showing 1-8 of 8 results.

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