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.

A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.

Original entry on oeis.org

0, 1, 8, 15, 24, 16, 32, 102, 288, 640, 960, 744, 80, 370, 1584, 5920, 18240, 43080, 69120, 56256, 160, 975, 5664, 30080, 141120, 564120, 1835520, 4542336, 7580160, 6385920, 280, 2121, 15624, 108080, 684480, 3876600, 19138560, 79805376
Offset: 2

Views

Author

Andrew Weimholt, Nov 16 2009

Keywords

Comments

Row n contains 2n-2 elements.
The n-orthoplex is the dual polytope of the n-cube.
The orthoplex is also known as the cross-polytope.
Also the triangle of coefficients of the cocktail party graph cycle polynomials ordered from smallest to largest exponent starting with x^3. - Eric W. Weisstein

Examples

			T(3,3) = 8, because in dimension n=3, the cross-polytope is the octahedron, which has 8 3-cycles in its graph.
Triangle starts
   0,   1;
   8,  15,   24,   16;
  32, 102,  288,  640,   960,   744;
  80, 370, 1584, 5920, 18240, 43080, 69120, 56256;
  ...
In terms of cycle polynomials:
   0*x^3 +    1*x^4;
   8*x^3 +  15*x^4 +  24*x^5 +  16*x^6;
  32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8;
  ...

Links

Crossrefs

Cf. A167987 (row sums).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (k-cycles on (n-1)-simplex for k>3).

Programs

  • Magma
    b:= func< n,k,j | (-1)^j*Binomial(n,j)*Binomial(2*(n-j),k-2*j)*2^(j-1)*Factorial(k-j-1) >;
A167986:= func< n,k | (&+[b(n,k,j): j in [0..Floor(k/2)]]) >;
[A167986(n,k): k in [3..2*n], n in [2..10]]; // G. C. Greubel, Jan 17 2023
  • Mathematica
    T[n_, k_]:= Sum[(-1)^j*Binomial[n, j]*Binomial[2*(n-j), k-2*j]*2^j*(k - j - 1)!/2, {j, 0, Floor[k/2]}];
Table[T[n, k], {n,2,7}, {k,3,2*n}]//Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Binomial[2n, k]*Gamma[k]*HypergeometricPFQ[{(1-k)/2, -k/2}, {1 - k, 1/2 -n}, -2]/2, {n,7}, {k,3,2n}]//Flatten (* Eric W. Weisstein, Mar 25 2020 *)
  • PARI
    a(n,k)=sum(j=0,k\2, (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!)/2;
for (n=2,6,for (k=3,2*n, print1(a(n,k), ","));print); \\ Andrew Howroyd, May 09 2017
  • SageMath
    def A167986(n,k): return simplify(binomial(2*n, k)*gamma(k)*hypergeometric([(1-k)/2, -k/2], [1-k, 1/2-n], -2)/2)
flatten([[A167986(n,k) for k in range(3,2*n+1)] for n in range(2,11)]) # G. C. Greubel, Jan 17 2023

Formula

T(n,k) = Sum_{j=0..floor(k/2)} (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!/2. - Andrew Howroyd, May 09 2017

A167981 Number of 2n-cycles on the graph of the tesseract, 2 <= n <= 8.

Original entry on oeis.org

24, 128, 696, 2112, 5024, 5736, 1344
Offset: 2

Views

Author

Andrew Weimholt, Nov 16 2009

Keywords

Comments

Row n=4 of the triangle in A085452
The graph of any n-cube (n>1) contains only even length cycles.
The tesseract is the 4 dimensional cube, and is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the tesseract is {4,3,3}.

Examples

			a(2) = 24 because there are 24 4-cycles on the graph of the tesseract.
The cycle polynomial is  24*x^4 + 128*x^6 + 696*x^8 + 2112*x^10 + 5024*x^12 + 5376*x^14 + 1344*x^16.

Links

Crossrefs

Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

A167982 Number of n-cycles on the graph of the regular 16-cell, 3 <= n <= 8.

Original entry on oeis.org

32, 102, 288, 640, 960, 744
Offset: 3

Views

Author

Andrew Weimholt, Nov 16 2009

Keywords

Comments

Row n=3 of the triangle in A167986
The 16-cell is the dual polytope of the tesseract, and is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the 16-cell is {3,3,4}.

Examples

			a(3) = 32, because there are 32 3-cycles on the graph of the 16-cell.
Cycle polynomial is 32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8.

Links

Crossrefs

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

A167983 Number of n-cycles on the graph of the regular 24-cell, 3 <= n <= 24.

Original entry on oeis.org

96, 360, 1440, 7120, 37728, 196488, 974592, 4536000, 19934208, 82689264, 322437312, 1171745280, 3924079104, 11964375936, 32761139328, 79244294016, 165800420352, 291640320576, 413774810112, 443415854592, 318534709248, 114869295744
Offset: 3

Views

Author

Andrew Weimholt, Nov 16 2009

Keywords

Comments

The 24-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol of the 24-cell is {3,4,3}.

Examples

			a(3) = 96, because there are 96 3-cycles on the graph of the 24-cell.
Cycle polynomial is 96*x^3 + 360*x^4 + 1440*x^5 + 7120*x^6 + 37728*x^7 + 196488*x^8 + 974592*x^9 + 4536000*x^10 + 19934208*x^11 + 82689264*x^12 + 322437312*x^13 + 1171745280*x^14 + 3924079104*x^15 + 11964375936*x^16 + 32761139328*x^17 + 79244294016*x^18 + 165800420352*x^19 + 291640320576*x^20 + 413774810112*x^21 + 443415854592*x^22 + 318534709248*x^23 + 114869295744*x^24.

Links

Crossrefs

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

Extensions

a(16)-a(24) and "full" keyword from Max Alekseyev, Nov 18 2009

A167984 Number of n-cycles on the graph of the regular 120-cell, 3 <= n <= 600.

Original entry on oeis.org

0, 0, 720, 0, 0, 3600, 2400, 4320, 28800, 35400, 64800, 284400, 540000, 1139400, 3708000, 8557200, 19677600, 55725120, 140359200, 346456800, 935942400, 2442469200, 6282571680
Offset: 3

Views

Author

Andrew Weimholt, Nov 16 2009

Keywords

Comments

The 120-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol of the 120-cell is {5,3,3}.

Examples

			a(5) = 720, because there are 720 5-cycles on the graph of the 120-cell.
Cycle polynomial is 720*x^5 + 3600*x^8 + 2400*x^9 + 4320*x^10 + 28800*x^11 + 35400*x^12 + 64800*x^13 +  284400*x^14 + 540000*x^15 + 1139400*x^16 + 3708000*x^17 + 8557200*x^18 + 19677600*x^19 + 55725120*x^20 + 140359200*x^21 + 346456800*x^22 + 935942400*x^23 + ...

Links

Crossrefs

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).
Cf. A108997 (number of vertices n-steps from a given vertex on graph of 120-cell).

Extensions

a(24) from Eric W. Weisstein, Feb 21 2014
a(25) from Eric W. Weisstein, Mar 11 2014

A167985 Number of n-cycles on the graph of the regular 600-cell, 3 <= n <= 120.

Original entry on oeis.org

1200, 5400, 29520, 187200, 1310400, 9813600, 77193600, 630538632, 5307656400
Offset: 3

Views

Author

Andrew Weimholt, Nov 16 2009

Keywords

Comments

The 600-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the 600-cell is {3,3,5}.

Examples

			a(3) = 1200, because there are 1200 3-cycles on the graph of the 600-cell.
Cycle polynomial is 1200*x^3 + 5400*x^4 + 29520*x^5 + 187200*x^6 + 1310400*x^7 + 9813600*x^8 + 77193600*x^9 + 630538632*x^10 + ...

Links

Crossrefs

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).
Cf. A118785 (number of vertices n-steps from a given vertex on graph of the 600-cell).

Extensions

a(11) from Eric W. Weisstein, Feb 09 2014

A085451 Numbers n such that n and prime[n] together use only distinct digits.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 35, 39, 40, 45, 53, 57, 58, 60, 61, 69, 70, 72, 79, 85, 89, 90, 91, 93, 96, 98, 104, 108, 120, 124, 145, 146, 147, 150, 162, 236, 237, 253, 254, 259, 315, 316, 359, 380, 384, 390, 405, 406, 460, 461, 518
Offset: 1

Views

Author

Zak Seidov, Jul 01 2003

Keywords

Comments

There are exactly 101 such numbers in the sequence. Numbers with distinct digits in A010784. Primes with distinct digits in A029743. The case n and n^2 (exactly 22 numbers) in A059930.
A178788(A045532(a(n))) = 1. [From Reinhard Zumkeller, Jun 30 2010]

Examples

			3106 is in the sequence (and the last term) because it and prime[3106]=28549 together use all 10 distinct digits.

Links

Crossrefs

Cf. A010784 A029743 A059930 A085452.

Programs

  • Mathematica
    bb = {}; Do[idpn = IntegerDigits[Prime[n]]; idn = IntegerDigits[n]; If[Length[idn] + Length[idpn] == Length[Union[idn, idpn]], bb = {bb, n}], {n, 1, 100000}]; Flatten[bb]

A085408 Total number of cycles in the binary n-cube.

Original entry on oeis.org

0, 1, 28, 14704, 51109385408
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 12 2003

Keywords

References

  • Computed by Daniele Degiorgi (danieled(AT)INF.ETHZ.CH).

Links

Crossrefs

Cf. A066037, A001788. Row sums of A085452.

Programs

  • Mathematica
    Table[Total[Table[Length[FindCycle[HypercubeGraph[n], {k}, All]], {k, 4, 2^n, 2}]], {n, 4}] (* Eric W. Weisstein, Mar 24 2020 *)
Showing 1-8 of 8 results.

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