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.

User: Stewart Gordon

Stewart Gordon's wiki page.

Stewart Gordon has authored 6 sequences.

A231091 Number of distinct (modulo rotation) unicursal star polygons (not necessarily regular, no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 27, 175, 1533, 14361, 151575, 1735869, 21594863, 289365383, 4158887007, 63822480809, 1041820050629, 18027531255745, 329658402237171, 6352776451924233, 128686951765990343, 2733851297673484765, 60781108703102022027, 1411481990523638719737
Offset: 1

Views

Author

Stewart Gordon, Nov 03 2013

Keywords

Comments

For polygons in general see A000939 and A000949, and especially the Golomb-Welch reference. - N. J. A. Sloane, Nov 21 2013

Examples

			For n=5, only solution is the regular pentagram.
For n=6, only solution is the unicursal hexagram (see Wikipedia link).
For n=7, two regular heptagrams and three irregular forms are possible.

Links

Crossrefs

Cf. A000939 (if edges may join adjacent vertices), A000940, A002816 (rotations and reflections counted separately), A326411, A370459 (up to rotations and reflections), A370068 (directed edges).
Cf. A283184.

Programs

  • PARI
    \\ Requires a370068 from A370068, b(n) is A283184.
b(n)={subst(serlaplace(polcoef((1 - x)/(1 + (1 - 2*y)*x + 2*y*x^2) + O(x*x^n), n)), y, 1)}
a(n)={(if(n%2==0 && n > 2, b(n/2-1)/2) + a370068(n))/2} \\ Andrew Howroyd, Mar 01 2024

Formula

a(n) = (A370068(n) + A283184(n/2-1)/2)/2 for even n >= 4; a(n) = A370068(n)/2 for odd n. - Andrew Howroyd, Feb 24 2024

Extensions

a(15) onwards from Andrew Howroyd, Feb 23 2024

A127702 Least value of x*y giving Max ( sigma(x*y) : 1<=x<=n, 1<=y<=n ).

Original entry on oeis.org

1, 4, 9, 16, 20, 36, 42, 64, 72, 90, 90, 144, 144, 168, 210, 240, 240, 324, 324, 360, 420, 420, 420, 576, 600, 600, 600, 756, 756, 840, 840, 960, 960, 960, 960, 1260, 1260, 1260, 1260, 1560, 1560, 1680, 1680, 1680, 1980, 1980, 1980, 2160, 2160, 2400, 2400
Offset: 1

Views

Author

Stewart Gordon, Sep 26 2011

Keywords

Comments

Value is unique by this definition up to a(103) = 9900. To go beyond this number, we could define the sequence in terms of "smallest value of x*y" or "largest value of x*y". We choose smallest here.

Examples

			a(5) = 4 * 5 = 20; sigma(20) = 42

Crossrefs

Cf. A074963, A074964.

A058179 Numbers whose English names include all five vowels at least once.

Original entry on oeis.org

105, 106, 108, 109, 113, 115, 116, 118, 119, 125, 126, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 145, 146, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 175, 176, 178, 179
Offset: 1

Views

Author

Stewart Gordon, Nov 14 2000

Keywords

Comments

Might be called abstemious numbers (since that adjective contains all vowels). - Michael Halm, Aug 24 2002

Examples

			105 = one hundred and five; 106 = one hundred and six; etc.

Crossrefs

Cf. A058180.

A057853 Number of letters in n (in Esperanto).

Original entry on oeis.org

3, 3, 2, 3, 4, 4, 3, 3, 2, 3, 3, 6, 5, 6, 7, 7, 6, 6, 5, 6, 5, 8, 7, 8, 9, 9, 8, 8, 7, 8, 6, 9, 8, 9, 10, 10, 9, 9, 8, 9, 7, 10, 9, 10, 11, 11, 10, 10, 9, 10, 7, 10, 9, 10, 11, 11, 10, 10, 9, 10, 6, 9, 8, 9, 10, 10, 9, 9, 8, 9, 6, 9, 8, 9, 10, 10, 9, 9, 8, 9, 5, 8, 7, 8, 9, 9, 8, 8, 7, 8, 6, 9, 8, 9
Offset: 0

Views

Author

Stewart Gordon, Nov 11 2000

Keywords

Examples

			0 = nul; 1 = unu; 2 = du; 3 = tri; 4 = kvar; 5 = kvin; 6 = ses; 7 = sep; 8 = ok; 9 = naŭ; 10 = dek; ...

Links

A058180 Numbers whose English names include all five vowels exactly once.

Original entry on oeis.org

206, 230, 250, 260, 602, 640, 5000, 8000, 9000, 26000, 80000, 90000
Offset: 1

Views

Author

Stewart Gordon, Nov 14 2000

Keywords

Examples

			206 = two hundred and six 80000 = eighty thousand

References

  • GCHQ, The GCHQ Puzzle Book II, Penguin, 2018. See question 189(b).

Crossrefs

Cf. A058179.

A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

Original entry on oeis.org

0, 3, 3, 2, 2, 2, 3, 1, 1, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 5, 3, 3, 3, 3, 3, 3, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 6, 6, 4, 4, 4, 4, 4, 4, 4, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 6, 6, 6, 6, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7
Offset: 0

Views

Author

Stewart Gordon, Dec 05 2001

Keywords

Examples

			From _Clark Kimberling_, Dec 20 2010: (Start)
T(i,j) for -2<=i<=2 and -2<=j<=2:
  4 1 2 1 4=T(2,2)
  1 2 3 2 1=T(2,1)
  2 3 0 3 2=T(2,0)
  1 2 3 2 1=T(2,-1)
  4 1 2 1 4=T(2,-2)
Corner of the array, T(i,j) for i>=0, j>=0: [Corrected Oct 14 2016]
  0 3 2 3 2 3 4...
  3 2 1 2 3 4 3...
  2 1 4 3 2 3 4...
  3 2 3 2 3 4 2... (End)

Crossrefs

Identical to A049604 except for T(1, 1).
For number of knight's moves to various subsets of the chessboard, see A018837, A183041-A183053.

Formula

From Clark Kimberling, Dec 20 2010: (Start)
T(i,j) is given in cases:
Case 1: row 0
T(0,0)=0, T(1,0)=3, and for m>=1,
T(4m-2,0)=2m, T(4m-1,0)=2m+1, T(4m,0)=2m,
T(4m+1,0)=2m+1.
Case 2: row 1
T(0,1)=3, T(1,1)=2, and for m>=1,
T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1,
T(4m+1,1)=2m+2.
Case 3: columns 0 and 1
(column 0 = row 0); (column 1 = row 1).
Case 4: For i>=2 and j>=2,
T(i,j)=1+min{T(i-2,j-1),T(i-1,j-2)}.
Cases 1-4 determine T in the 1st quadrant;
all other T(i,j) are easily obtained by symmetry. (End)

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

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