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 1-10 of 11 results. Next

A000583 Fourth powers: a(n) = n^4.

Original entry on oeis.org

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921
Offset: 0

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Author

Keywords

Comments

Figurate numbers based on 4-dimensional regular convex polytope called the 4-measure polytope, 4-hypercube or tesseract with Schlaefli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
Totally multiplicative sequence with a(p) = p^4 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A058649. The inverse binomial transforms yields the (finite) 0, 1, 14, 36, 24, the 4th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Generate Pythagorean triangles with parameters a and b to get sides of lengths x = b^2-a^2, y = 2*a*b, and z = a^2 + b^2. In particular use a=n-1 and b=n for a triangle with sides (x1,y1,z1) and a=n and b=n+1 for another triangle with sides (x2,y2,z2). Then x1*x2 + y1*y2 + z1*z2 = 8*a(n). - J. M. Bergot, Jul 22 2013
For n > 0, a(n) is the largest integer k such that k^4 + n is a multiple of k + n. Also, for n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n^2. - Derek Orr, Sep 04 2014
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
a(n+2)/2 is the area of a trapezoid with vertices at (T(n), T(n+1)), (T(n+1), T(n)), (T(n+1), T(n+2)), and (T(n+2), T(n+1)) with T(n)=A000292(n) for n >= 0. - J. M. Bergot, Feb 16 2018

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
  • Dov Juzuk, Curiosa 56: An interesting observation, Scripta Mathematica 6 (1939), 218.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 47.

Crossrefs

Programs

Formula

a(n) = A123865(n)+1 = A002523(n)-1.
Multiplicative with a(p^e) = p^(4e). - David W. Wilson, Aug 01 2001
G.f.: x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. More generally, g.f. for n^m is Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292).
Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: (x + 7*x^2 + 6*x^3 + x^4)*e^x. More generally, the general form for the e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial of order n. - Franklin T. Adams-Watters, Sep 11 2005
Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - Jaume Oliver Lafont, Sep 20 2009
a(n) = C(n+3,4) + 11*C(n+2,4) + 11*C(n+1,4) + C(n,4). [Worpitzky's identity for powers of 4. See, e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - Bruno Berselli, May 07 2010
a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - Charlie Marion, Jun 13 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - Ant King, Sep 23 2013
From Amiram Eldar, Jan 20 2021: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).
Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)

A000389 Binomial coefficients C(n,5).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757, 658008, 749398
Offset: 0

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Author

Keywords

Comments

a(n+4) is the number of inequivalent ways of coloring the vertices of a regular 4-dimensional simplex with n colors, under the full symmetric group S_5 of order 120, with cycle index (x1^5 + 10*x1^3*x2 + 20*x1^2*x3 + 15*x1*x2^2 + 30*x1*x4 + 20*x2*x3 + 24*x5)/120.
Figurate numbers based on 5-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 10 of these 5-simplex(n) numbers (compared with g=3 for triangular numbers, g=5 for tetrahedral numbers and g=8 for pentatope numbers). - Jonathan Vos Post, Nov 28 2004
The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which are the convolution of the nonnegative integers with themselves (making appropriate allowances for offsets of all sequences). - Graeme McRae, Jun 07 2006
a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6)^n. - Sergio Falcon, Feb 12 2007
Product of five consecutive numbers divided by 120. - Artur Jasinski, Dec 02 2007
Equals binomial transform of [1, 5, 10, 10, 5, 1, 0, 0, 0, ...]. - Gary W. Adamson, Feb 02 2009
Equals INVERTi transform of A099242 (1, 7, 34, 153, 686, 3088, ...). - Gary W. Adamson, Feb 02 2009
For a team with n basketball players (n>=5), this sequence is the number of possible starting lineups of 5 players, without regard to the positions (center, forward, guard) of the players. - Mohammad K. Azarian, Sep 10 2009
a(n) is the number of different patterns, regardless of order, when throwing (n-5) 6-sided dice. For example, one die can display the 6 numbers 1, 2, ..., 6; two dice can display the 21 digit-pairs 11, 12, ..., 56, 66. - Ian Duff, Nov 16 2009
Sum of the first n pentatope numbers (1, 5, 15, 35, 70, 126, 210, ...), see A000332. - Paul Muljadi, Dec 16 2009
Sum_{n>=0} a(n)/n! = e/120. Sum_{n>=4} a(n)/(n-4)! = 501*e/120. See A067764 regarding the second ratio. - Richard R. Forberg, Dec 26 2013
For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 4 elements, which is 3*C(n+1,5) (for n>=4), hence a(n) = 3*C(n+1,5) = 3*A000389(n+1). - Serhat Bulut, Mar 11 2015
a(n) = fallfac(n,5)/5! is also the number of independent components of an antisymmetric tensor of rank 5 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+1 into exactly 6 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-5 into exactly 6 parts. - Juergen Will, Jan 02 2016
a(n+3) could be the general number of all geodetic graphs of diameter n>=2 homeomorphic to the Petersen Graph. - Carlos Enrique Frasser, May 24 2018
From Robert A. Russell, Dec 24 2020: (Start)
a(n) is the number of chiral pairs of colorings of the 5 tetrahedral facets (or vertices) of the regular 4-D simplex (5-cell, pentachoron, Schläfli symbol {3,3,3}) using subsets of a set of n colors. Each member of a chiral pair is a reflection but not a rotation of the other.
a(n+4) is the number of unoriented colorings of the 5 tetrahedral facets of the regular 4-D simplex (5-cell, pentachoron) using subsets of a set of n colors. Each chiral pair is counted as one when enumerating unoriented arrangements. (End)
For integer m and positive integer r >= 4, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (4 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

Examples

			G.f. = x^5 + 6*x^6 + 21*x^7 + 56*x^8 + 126*x^9 + 252*x^10 + 462*x^11 + ...
For A={1,2,3,4}, the only subset with 4 elements is {1,2,3,4}; sum of 2 minimum elements of this subset: a(4) = 1+2 = 3 = 3*C(4+1,5).
For A={1,2,3,4,5}, the subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}; sum of 2 smallest elements of each subset: a(5) = (1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 18 = 3*C(5+1,5). - _Serhat Bulut_, Mar 11 2015
a(6) = 6 from the six independent components of an antisymmetric tensor A of rank 5 and dimension 6: A(1,2,3,4,5), A(1,2,3,4,6), A(1,2,3,5,6), A(1,2,4,5,6), A(1,3,4,5,6), A(2,3,4,5,6). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015
		

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
  • Gupta, Hansraj; Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
  • J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A099242. - Gary W. Adamson, Feb 02 2009
Cf. A242023. A104712 (fourth column, k=5).
5-cell colorings: A337895 (oriented), A132366(n-1) (achiral).
Unoriented colorings: A063843 (5-cell edges, faces), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell), A338965 (600-cell vertices, 120-cell facets).
Chiral colorings: A331352 (5-cell edges, faces), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell), A338966 (600-cell vertices, 120-cell facets).

Programs

  • Haskell
    a000389 n = a000389_list !! n
    a000389_list = 0 : 0 : f [] a000217_list where
       f xs (t:ts) = (sum $ zipWith (*) xs a000217_list) : f (t:xs) ts
    -- Reinhard Zumkeller, Mar 03 2015, Apr 13 2012
    
  • Magma
    [Binomial(n, 5): n in [0..40]]; // Vincenzo Librandi, Mar 12 2015
  • Maple
    f:=n->(1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n): seq(f(n), n=0..60);
    ZL := [S, {S=Prod(B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=0..42); # Zerinvary Lajos, Mar 13 2007
    A000389:=1/(z-1)**6; # Simon Plouffe, 1992 dissertation
  • Mathematica
    Table[Binomial[n, 5], {n, 5, 50}] (* Stefan Steinerberger, Apr 02 2006 *)
    CoefficientList[Series[x^5 / (1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 12 2015 *)
    LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,0,0,0,1},50] (* Harvey P. Dale, Jul 17 2016 *)
  • PARI
    (conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w);
    (t(n)=n*(n+1)/2); u=vector(10,i,t(i)); conv(u,u)
    

Formula

G.f.: x^5/(1-x)^6.
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/120.
a(n) = (n^5-10*n^4+35*n^3-50*n^2+24*n)/120. (Replace all x_i's in the cycle index with n.)
a(n+2) = Sum_{i+j+k=n} i*j*k. - Benoit Cloitre, Nov 01 2002
Convolution of triangular numbers (A000217) with themselves.
Partial sums of A000332. - Alexander Adamchuk, Dec 19 2004
a(n) = -A110555(n+1,5). - Reinhard Zumkeller, Jul 27 2005
a(n+3) = (1/2!)*(d^2/dx^2)S(n,x)|A049310.%20-%20_Wolfdieter%20Lang">{x=2}, n>=2, one half of second derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - _Wolfdieter Lang, Apr 04 2007
a(n) = A052787(n+5)/120. - Zerinvary Lajos, Apr 26 2007
Sum_{n>=5} 1/a(n) = 5/4. - R. J. Mathar, Jan 27 2009
For n>4, a(n) = 1/(Integral_{x=0..Pi/2} 10*(sin(x))^(2*n-9)*(cos(x))^9). - Francesco Daddi, Aug 02 2011
Sum_{n>=5} (-1)^(n + 1)/a(n) = 80*log(2) - 655/12 = 0.8684411114... - Richard R. Forberg, Aug 11 2014
a(n) = -a(4-n) for all n in Z. - Michael Somos, Oct 07 2014
0 = a(n)*(+a(n+1) + 4*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Oct 07 2014
a(n) = 3*C(n+1, 5) = 3*A000389(n+1). - Serhat Bulut, Mar 11 2015
From Ilya Gutkovskiy, Jul 23 2016: (Start)
E.g.f.: x^5*exp(x)/120.
Inverse binomial transform of A054849. (End)
From Robert A. Russell, Dec 24 2020: (Start)
a(n) = A337895(n) - a(n+4) = (A337895(n) - A132366(n-1)) / 2 = a(n+4) - A132366(n-1).
a(n+4) = A337895(n) - a(n) = (A337895(n) + A132366(n-1)) / 2 = a(n) + A132366(n-1).
a(n+4) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 1*C(n,5), where the coefficient of C(n,k) is the number of unoriented pentachoron colorings using exactly k colors. (End)

Extensions

Corrected formulas that had been based on other offsets. - R. J. Mathar, Jun 16 2009
I changed the offset to 0. This will require some further adjustments to the formulas. - N. J. A. Sloane, Aug 01 2010

A338967 Number of achiral colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.

Original entry on oeis.org

1, 314843647550280564736, 5068890957390271123224826359979956, 11893730816857265534982913331475052373213184, 220581496716947452381892465686737251285705566406250
Offset: 1

Views

Author

Robert A. Russell, Dec 04 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. There are 7200 elements in the automorphism group of the 120-cell that are not in its rotation group. They divide into 9 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Odd Cycle Indices Count Odd Cycle Indices
60 x_1^30x_2^45 1200 x_1^2x_2^2x_6^19
60 x_1^2x_2^59 720+720 x_2^5x_5^6x_10^8
1800 x_2^2x_4^29 720+720 x_1^2x_2^4x_10^11
1200 x_2^3x_3^10x_6^14
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the cycle indices are:
Count Odd Cycle Indices Count Odd Cycle Indices
60 x_1^60x_2^270 1200 x_2^6x_6^98
60 x_2^300 720+720 x_5^12x_10^54
1800 x_1^2x_2^1x_4^149 720+720 x_10^60
1200 x_2^6x_3^20x_6^88
The formula is (24*n^60 + 24*n^66 + 20*n^104 + 20*n^114 + 30*n^152 + n^300 + n^330) / 120.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:
Count Odd Cycle Indices Count Odd Cycle Indices
60 x_1^72x_2^324 1200 x_6^120
60 x_2^360 720+720 x_1^2x_2^4x_5^14x_10^64
1800 x_2^4x_4^178 720+720 x_2^5x_10^71
1200 x_3^24x_6^108
The formula is (24*n^76 + 24*n^84 + 20*n^120 + 20*n^132 + 30*n^182 + n^360 + n^396) / 120.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:
Count Odd Cycle Indices Count Odd Cycle Indices
60 x_1^80x_2^560 1200 x_2^3x_6^199
60 x_2^600 720+720 x_5^16x_10^112
1800 x_2^4x_4^298 720+720 x_10^120
1200 x_1^2x_2^2x_3^26x_6^186
The formula is (24*n^120 + 24*n^128 + 20*n^202 + 20*n^216 + 30*n^302 + n^600 + n^640) / 120.

Crossrefs

Cf. A338964 (oriented), A338965 (unoriented), A338966 (chiral), A338983 (exactly n colors), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958(16-cell vertices, 8-cell facets), A338951 (24-cell).

Programs

  • Mathematica
    Table[(24n^17+24n^19+20n^23+20n^27+30n^31+n^61+n^75)/120,{n,10}]
  • PARI
    a(n)=(24*n^17+24*n^19+20*n^23+20*n^27+30*n^31+n^61+n^75)/120 \\ Charles R Greathouse IV, Jul 05 2024

Formula

a(n) = (24*n^17 + 24*n^19 + 20*n^23 + 20*n^27 + 30*n^31 + n^61 + n^75) / 120.
a(n) = Sum_{j=1..Min(n,75)} A338983(n) * binomial(n,j).
a(n) = 2*A338965(n) - A338964(n) =(A338964(n) - 2*A338966(n)) / 2 = A338965(n) - A338966(n).

A325001 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 5, 1, 5, 16, 15, 6, 1, 6, 25, 34, 21, 7, 1, 7, 36, 65, 56, 28, 8, 1, 8, 49, 111, 125, 84, 36, 9, 1, 9, 64, 175, 246, 210, 120, 45, 10, 1, 10, 81, 260, 441, 461, 330, 165, 55, 11, 1, 11, 100, 369, 736, 917, 792, 495, 220, 66, 12, 1
Offset: 1

Views

Author

Robert A. Russell, Mar 23 2019

Keywords

Comments

For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four triangular faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. An achiral coloring is the same as its reflection.

Examples

			The array begins with A(1,1):
  1  2  3   4   5    6    7     8     9    10    11     12     13 ...
  1  4  9  16  25   36   49    64    81   100   121    144    169 ...
  1  5 15  34  65  111  175   260   369   505   671    870   1105 ...
  1  6 21  56 125  246  441   736  1161  1750  2541   3576   4901 ...
  1  7 28  84 210  461  917  1688  2919  4795  7546  11452  16848 ...
  1  8 36 120 330  792 1715  3424  6399 11320 19118  31032  48672 ...
  1  9 45 165 495 1287 3003  6434 12861 24265 43593  75087 124683 ...
  1 10 55 220 715 2002 5005 11440 24309 48610 92323 167740 293215 ...
  ...
For A(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
		

Crossrefs

Cf. A324999 (oriented), A325000 (unoriented), A325000(n,k-n) (chiral), A325003 (exactly k colors), A327086 (edges, ridges), A337886 (faces, peaks), A325007 (orthotope facets, orthoplex vertices), A325015 (orthoplex facets, orthotope vertices).
Rows 1-4 are A000027, A000290, A006003, A132366(n-1).
Column 2 is A162880.

Programs

  • Mathematica
    Table[Binomial[d+1,n+1] - Binomial[d+1-n,n+1], {d,1,15}, {n,1,d}] // Flatten

Formula

A(n,k) = binomial(n+k,n+1) - binomial(k,n+1).
A(n,k) = Sum_{j=1..n} A325003(n,j) * binomial(k,j).
A(n,k) = 2*A325000(n,k) - A324999(n,k) = A324999(n,k) - 2*A325000(n,k-n) = A325000(n,k) - A325000(n,k-n).
G.f. for row n: (x - x^(n+1)) / (1-x)^(n+2).
Linear recurrence for row n: A(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * A(n,k-j).
G.f. for column k: (1 - (1-x^2)^k) / (x*(1-x)^k).

A338951 Number of achiral colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 6504, 8416440, 1455789440, 80139247500, 2125945744776, 34026498820524, 376045864704000, 3131319814422255, 20854395850585000, 115919421344402676, 554976171149122944, 2343894146343268610, 8896568181794053320
Offset: 1

Views

Author

Robert A. Russell, Nov 17 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Odd Cycle Indices Count Odd Cycle Indices
12 x_1^12x_2^6 72 x_2^2x_4^5
12 x_1^6x_2^9 96 x_1^2x_2^2x_6^3
12 x_1^2x_2^11 96 x_2^3x_3^2x_6^2
12 x_2^12 96 x_3^4x_6^2
72 x_1^2x_2^1x_4^5 96 x_6^4

Crossrefs

Cf. A338948 (oriented), A338949 (unoriented), A338950 (chiral), A338955 (edges, faces), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338967 (120-cell, 600-cell).

Programs

  • Mathematica
    Table[(8n^4+8n^6+22n^7+6n^8+n^12+n^13+n^15+n^18)/48,{n,15}]

Formula

a(n) = (8*n^4 + 8*n^6 + 22*n^7 + 6*n^8 + n^12 + n^13 + n^15 + n^18) / 48.
a(n) = 1*C(n,1) + 6502*C(n,2) + 8396931*C(n,3) + 1422162700*C(n,4) + 72944399665*C(n,5) + 1666778870130*C(n,6) + 20777144613015*C(n,7) + 158973991255800*C(n,8) + 803196369526320*C(n,9) + 2806639981714800*C(n,10) + 6979192091902800*C(n,11) + 12538220293368000*C(n,12) + 16327662245294400*C(n,13) + 15272334392515200*C(n,14) + 10003736158416000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A338949(n) - A338948(n) = A338948(n) - 2*A338950(n) = A338949(n) - A338950(n).

A337895 Number of oriented colorings of the tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors.

Original entry on oeis.org

1, 6, 21, 56, 127, 258, 483, 848, 1413, 2254, 3465, 5160, 7475, 10570, 14631, 19872, 26537, 34902, 45277, 58008, 73479, 92114, 114379, 140784, 171885, 208286, 250641, 299656, 356091, 420762, 494543, 578368, 673233
Offset: 1

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Author

Robert A. Russell, Sep 28 2020

Keywords

Comments

Each chiral pair is counted as two when enumerating oriented arrangements. Also called a 5-cell or pentachoron. The Schläfli symbol is {3,3,3}, and it has 5 tetrahedral facets (vertices).
There are 60 elements in the rotation group of the 4-dimensional simplex. Each is an even permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Partition Count Even Cycle Indices
5 24 x_5^1
311 20 x_1^2x_3^1
221 15 x_1^1x_2^2
11111 1 x_1^5

Examples

			For a(2)=6, the colors are AAAAA, AAAAB, AAABB, AABBB, ABBBB, and BBBBB.
		

Crossrefs

Cf. A000389(n+4) (unoriented), A000389(chiral), A132366(n-1) (achiral), A331350 (edges, faces), A337952 (8-cell vertices, 16-cell facets), A337956(16-cell vertices, 8-cell facets), A338948 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A324999 (oriented colorings of facets or vertices of an n-simplex).

Programs

  • Mathematica
    Table[n (24 + 35 n^2 + n^4)/60, {n, 40}]

Formula

a(n) = n * (24 + 35*n^2 + n^4) / 60.
a(n) = binomial[4+n,5] + binomial[n,5].
a(n) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 2*C(n,5), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A000389(n+4) + A000389(n) = 2*A000389(n+4) - A132366(n-1) = 2*A000389(n) + A132366(n-1).

A337955 Number of achiral colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

Original entry on oeis.org

1, 308, 34128, 1056576, 15303750, 136236276, 865711763, 4296782848, 17656466751, 62510672500, 196174554026, 557301826368, 1456216515468, 3543525156276, 8109415963125, 17592637669376, 36414622551373
Offset: 1

Views

Author

Robert A. Russell, Oct 03 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual. There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each involves a permutation of the axes that can be associated with a partition of 4 based on the conjugacy class of the permutation. This table shows the hyperoctahedron facet (tesseract vertex) cycle indices for each member of such a class. The first formula is obtained by averaging these cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Partition Count Odd Cycle Indices
4 6 8x_1^2x_2^1x_4^3
31 8 8x_2^2x_6^2
22 3 8x_4^4
211 6 2x_1^8x_2^4 + 2x_2^8 + 4x_4^4
1111 1 8x_2^8

Crossrefs

Cf. A337952 (oriented), A128767 (unoriented), A337954 (chiral).
Other elements: A331361 (tesseract edges, hyperoctahedron faces), A331357 (tesseract faces, hyperoctahedron edges), A337958 (tesseract facets, hyperoctahedron vertices).
Other polychora: A132366(n-1) (4-simplex facets/vertices), A338951 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A325015 (orthoplex facets, orthotope vertices).

Programs

  • Mathematica
    Table[(3n^12+5n^8+12n^6+28n^4)/48,{n,30}]

Formula

a(n) = n^4 * (3*n^8 + 5*n^4 + 12*n^2 + 28) / 48.
a(n) = 1*C(n,1) + 306*C(n,2) + 33207*C(n,3) + 921908*C(n,4) + 10359075*C(n,5) + 59584470*C(n,6) + 197644440*C(n,7) + 400752240*C(n,8) + 505197000*C(n,9) + 386694000*C(n,10) + 164656800*C(n,11) + 29937600*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A128767(n) - A337952(n) = A337952(n) - 2*A337954(n) = A128767(n) - A337954(n).

A337958 Number of achiral colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

Original entry on oeis.org

1, 15, 126, 700, 2850, 9261, 25480, 61776, 135675, 275275, 523446, 943020, 1623076, 2686425, 4298400, 6677056, 10104885, 14942151, 21641950, 30767100, 43008966, 59208325, 80378376, 107730000, 142699375, 186978051, 242545590
Offset: 1

Views

Author

Robert A. Russell, Oct 03 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.

Crossrefs

Cf. A337956 (oriented), A337956 (unoriented), A234249(n+1) (chiral).
Other elements: A331357 (hyperoctahedron edges, tesseract faces), A331361 (hyperoctahedron faces, tesseract edges), A337955 (hyperoctahedron facets, tesseract vertices).
Other polychora: A132366(n-1) (5-cell), A338951 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A325007 (orthotope facets, orthoplex vertices).

Programs

  • Mathematica
    Table[Binomial[Binomial[n+1,2]+3,4] - Binomial[Binomial[n,2],4],{n,30}]

Formula

a(n) = binomial(binomial(n+1,2)+3,4) - binomial(binomial(n,2),4).
a(n) = n^2 * (n+1)^2 * (n+3) * (n^2 -2n +4) / 48.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 282*C(n,4) + 465*C(n,5) + 360*C(n,6) + 105*C(n,7), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A337957(n) - A337956(n) = A337956(n) - 2 * A234249(n+1) = A337957(n) - A234249(n+1).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1 + 7*x + 34*x^2 + 56*x^3 + 8*x^4 - x^5)/(1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n > 8.
(End)

A338983 Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 314843647550280564734, 5068890957389326592282175518285751, 11893730796581701705423717900461048616681772, 220581437248293418784474364671733389683204494492535
Offset: 0

Views

Author

Robert A. Russell, Dec 13 2020

Keywords

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>75, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is bp(60)/5 + bp(66)/5 + bp(104)/6 + bp(114)/6 + bp(152)/4 + bp(300)/120 + bp(330)/120.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(76)/5 + bp(84)/5 + bp(120)/6 + bp(132)/6 + bp(182)/4 + bp(360)/120 + bp(396)/120.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(120)/5 + bp(128)/5 + bp(202)/6 + bp(216)/6 + bp(302)/4 + bp(600)/120 + bp(640)/120.

Crossrefs

Cf. A338980 (oriented), A338981 (unoriented), A338982 (chiral), A338967 (up to n colors), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell).

Programs

  • Mathematica
    bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
    CoefficientList[bp[17]/5+bp[19]/5+bp[23]/6+bp[27]/6+bp[31]/4+bp[61]/120+bp[75]/120,x]

Formula

A338967(n) = Sum_{j=1..Min(n,75)} a(n) * binomial(n,j).
a(n) = 2*A338981(n) - A338980(n) = A338980(n) - 2*A338982(n) = A338981(n) - A338982(n).
G.f.: bp(17)/5 + bp(19)/5 + bp(23)/6 + bp(27)/6 + bp(31)/4 + bp(61)/120 + bp(75)/120, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

A329634 Numbers that are sums of consecutive centered tetrahedral numbers (A005894).

Original entry on oeis.org

1, 5, 6, 15, 20, 21, 35, 50, 55, 56, 69, 104, 119, 121, 124, 125, 190, 195, 225, 240, 245, 246, 295, 316, 385, 420, 425, 435, 440, 441, 490, 589, 611, 680, 715, 720, 730, 735, 736, 791, 915, 1014, 1035, 1036, 1105, 1140, 1155, 1160, 1161, 1309, 1325, 1380, 1504, 1625, 1665, 1694
Offset: 1

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Author

Ilya Gutkovskiy, Nov 18 2019

Keywords

Crossrefs

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