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.

Previous Showing 21-30 of 50 results. Next

A079430 Marks on lexicographically earliest 11-mark optimal Golomb ruler.

Original entry on oeis.org

0, 1, 4, 13, 28, 33, 47, 54, 64, 70, 72
Offset: 1

Views

Author

Daniel Smith (dsmith(AT)globalnet.co.uk), Feb 16 2003

Keywords

Links

Crossrefs

Cf. A003022.

A241094 Triangle read by rows: T(n,i) = number of gracefully labeled graphs with n edges that do not use the label i, 1 <= i <= n-1, n > 1.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 18, 24, 24, 18, 96, 144, 144, 96, 600, 960, 1080, 1080, 960, 600, 4320, 7200, 8460, 8460, 8460, 7200, 4320, 35280, 60840, 75600, 80640, 80640, 75600, 60480, 35280, 322560, 564480, 725760, 806400, 806400, 806400, 725760, 564480, 322560
Offset: 2

Views

Author

Christian Barrientos and Sarah Minion, Apr 15 2014

Keywords

Comments

A graph with n edges is graceful if its vertices can be labeled with distinct integers in the range 0,1,...,n in such a way that when the edges are labeled with the absolute differences between the labels of their end-vertices, the n edges have the distinct labels 1,2,...,n.

Examples

			For n=7 and i=3, g(7,3) = 1080.
For n=7 and i=5, g(7,5) = 960.
Triangle begins:
[n\i]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]     0;
[3]     1,      1;
[4]     4,      4,      4;
[5]    18,     24,     24,     18;
[6]    96,    144,    144,    144,     96;
[7]   600,    960,   1080,   1080,    960,    600;
[8]  4320,   7200,   8640,   8640,   8640,   7200,   4320;
[9] 35280,  60480,  75600,  80640,  80640,  75600,  60480,  35280;
...
- _Bruno Berselli_, Apr 23 2014

Links

Crossrefs

Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661.

Programs

  • Magma
    /* As triangle: */ [[i le Floor(n/2) select Factorial(n-2)*(n-1-i)*i else Factorial(n-2)*(n-i)*(i-1): i in [1..n-1]]: n in [2..10]]; // Bruno Berselli, Apr 23 2014
  • Maple
    Labeled:=(i,n) piecewise(n<2 or i<1, -infinity, 1 <= i <= floor(n/2), GAMMA(n-1)*(n-1-i)*i, ceil((n+1)/2) <= i <= n-1, GAMMA(n-1)*(n-i)*(i-1), infinity):
  • Mathematica
    n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *)
For[i = 1, i <= Floor[n/2], i++, g[n_,i_]:=(n-2)!*(n-1-i)*i; Print["g(",n,",",i,")=", g[n,i]]]
For[i = Ceiling[(n+1)/2], i <= (n-1), i++, g[n_,i_]:=(n-2)!*(n-i)*(i-1); Print["g(",n,",",i,")=",g[n,i]]]

Formula

For n >=2, if 1 <= i <= floor(n/2), g(n,i) = (n-2)!*(n-1-i)*i; if ceiling((n+1)/2) <= i <= n-1, g(n,i) = (n-2)!*(n-i)*(i-1).
# alternative
A241094 := proc(n,i)
if n <2 or i<1 or i >= n then
0;
elif i <= floor(n/2) then
GAMMA(n-1)*(n-1-i)*i;
else
GAMMA(n-1)*(n-i)*(i-1) ;
fi ;
end proc:
seq(seq(A241094(n,i),i=1..n-1),n=2..12); # R. J. Mathar, Jul 30 2024

A351700 T(n,k) is the maximum number of points that can be chosen from a rectangle of n X k lattice points such that their mutual distances are distinct, where T(n,k) is a triangle read by rows, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 3, 4, 5, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 4, 5, 5, 6, 7, 7, 7, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 4, 6, 6, 7, 7, 8, 8, 8, 9, 9, 4, 6, 6, 7, 8, 8, 8, 9, 9, 9, 10, 5, 6, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 5, 6, 7, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11
Offset: 1

Views

Author

Hugo Pfoertner, Mar 05 2022

Keywords

Examples

			The triangle begins:
  1
  2  2
  2  3  3
  3  4  4  4
  3  4  4  5  5
  3  4  5  5  5  6
  4  5  5  6  6  6  7
  4  5  5  6  7  7  7  7
  4  5  6  6  7  7  8  8  8
  4  6  6  7  7  8  8  8  9  9
  4  6  6  7  8  8  8  9  9  9 10
  5  6  7  7  8  9  9  9  9 10 10 10
  5  6  7  8  8  9  9 10 10 10 10 11 11

Links

Crossrefs

First occurrence of n in first column: A227590.
Main diagonal: A271490.
Cf. A003022, A193838, A351699.

Extensions

T(13,2)=a(80) and T(13,8)=a(86) corrected by Fausto A. C. Cariboni, Jul 10 2022

A039953 Triangle of "canonical" optimal Golomb rulers.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 3, 5, 2, 1, 3, 6, 2, 5, 1, 3, 6, 8, 5, 2, 1, 3, 5, 6, 7, 10, 2, 1, 4, 7, 13, 2, 8, 6, 3, 1, 5, 4, 13, 3, 8, 7, 12, 2, 1, 3, 9, 15, 5, 14, 7, 10, 6, 2, 2, 4, 18, 5, 11, 3, 12, 13, 7, 1, 9, 2, 3, 20, 12, 6, 16, 11, 15, 4, 9, 1, 7
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

n-th row gives first differences of the (n+1)-th row of A106683. - Andrey Zabolotskiy, Aug 22 2017

Examples

			Triangle begins:
  1;
  1, 2;
  1, 3, 2;
  1, 3, 5, 2;
  1, 3, 6, 2, 5;
  1, 3, 6, 8, 5, 2;
  ...

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 315.
  • A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 5-13.

Links

Crossrefs

Cf. A003022, A036501.
These all start at 1. For rulers starting at 0, see A079283, A079287, A079423, A079425, A079426, A079430, A079433, A079434, A079435, A079454 and A079467, A079604, A079605, A079606, A079607, A079608, A079625, A079634.

Extensions

Corrected by Philip Newton, Feb 06 2002
Corrected by Andrey Zabolotskiy, Aug 22 2017

A079423 Marks on lexicographically earliest 8-mark optimal Golomb ruler.

Original entry on oeis.org

0, 1, 4, 9, 15, 22, 32, 34
Offset: 1

Views

Author

Daniel Smith (dsmith(AT)globalnet.co.uk), Feb 16 2003

Keywords

Links

Crossrefs

Cf. A003022.

A079425 Marks on lexicographically earliest 9-mark optimal Golomb ruler.

Original entry on oeis.org

0, 1, 5, 12, 25, 27, 35, 41, 44
Offset: 1

Views

Author

Daniel Smith (dsmith(AT)globalnet.co.uk), Feb 16 2003

Keywords

Links

Crossrefs

Cf. A003022.

A079426 Marks on lexicographically earliest 10-mark optimal Golomb ruler.

Original entry on oeis.org

0, 1, 6, 10, 23, 26, 34, 41, 53, 55
Offset: 1

Views

Author

Daniel Smith (dsmith(AT)globalnet.co.uk), Feb 16 2003

Keywords

Links

Crossrefs

Cf. A003022. Equals A031873 - 1.

A079433 Marks on lexicographically earliest 12-mark optimal Golomb ruler.

Original entry on oeis.org

0, 2, 6, 24, 29, 40, 43, 55, 68, 75, 76, 85
Offset: 1

Views

Author

Daniel Smith (dsmith(AT)globalnet.co.uk), Feb 16 2003

Keywords

Links

Crossrefs

Cf. A003022.

A212661 a(n) = smallest number of edges that must be removed from K_n to obtain a graceful graph.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 9, 12, 16, 20, 23, 26, 30, 35, 41, 48, 52, 57, 63, 70, 78, 87, 93
Offset: 3

Views

Author

N. J. A. Sloane, Jun 05 2012

Keywords

References

  • G. S. Bloom and S. W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications. Theory and Applications of Graphs, Lecture Notes in Math. 642, (1978), 53-65.
  • Miller, J. C. P. Difference bases. Three problems in additive number theory. Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pp. 299--322. Academic Press, London,1971. MR0316269 (47 #4817) - From N. J. A. Sloane, Jun 05 2012
  • Rhys Price Jones, Gracelessness, Proc. 10th S.-E. Conf. Combin., Graph Theory and Computing, 1979, pp. 547-552. - From N. J. A. Sloane, Jun 05 2012

Links

Crossrefs

Cf. A003022, A078106, A004137.

Formula

n(n-1)/2 = a(n)+A004137(n). - Kellen Myers, Jun 06 2016

Extensions

a(26) using extension of A004137 from Hugo Pfoertner, Feb 28 2021

A347498 Least k such that there exists an n-element subset S of {1,2,...,k} with the property that all products i * j are distinct for i <= j.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 20, 23, 25, 28, 29, 31, 33, 37, 40, 41, 42, 43, 47, 51, 53, 55, 57, 59, 61, 67, 69, 71, 73, 75, 79, 83
Offset: 1

Views

Author

Peter Kagey, Sep 03 2021

Keywords

Comments

a(n) <= A066720(n) and a(n+1) >= a(n) + 1

Examples

			   n | example set
-----+-------------------------------------------------------
   1 | {1}
   2 | {1, 2}
   3 | {1, 2, 3}
   4 | {1, 2, 3, 5}
   5 | {1, 3, 4, 5,  6}
   6 | {1, 3, 4, 5,  6,  7}
   7 | {1, 2, 5, 6,  7,  8,  9}
   8 | {1, 2, 5, 6,  7,  8,  9, 11}
   9 | {1, 2, 5, 6,  7,  8,  9, 11, 13}
  10 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15}
  11 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15, 17}
  12 | {1, 2, 5, 7,  8,  9, 11, 12, 13, 15, 17, 19}
  13 | {1, 5, 6, 7,  9, 11, 13, 14, 15, 16, 17, 19, 20}
  14 | {1, 2, 5, 7, 11, 12, 13, 16, 17, 18, 19, 20, 21, 23}
For n = 4, the set {1,2,3,4} does not have distinct products because 2*2 = 1*4. However, the set {1,2,3,5} does have distinct products because 1*1, 1*2, 1*3, 1*5, 2*2, 2*3, 2*5, 3*3, 3*5, and 5*5 are all distinct.

Crossrefs

Analogous for sums: A003022 and A227590.
Cf. A066720, A338006, A347499.

Programs

  • Mathematica
    Table[k=1;While[!Or@@(Length[s=Union[Sort/@Tuples[#,{2}]]]==Length@Union[Times@@@s]&/@Subsets[Range@k,{n}]),k++];k,{n,12}] (* Giorgos Kalogeropoulos, Sep 08 2021 *)
  • Python
    from itertools import combinations, combinations_with_replacement
def a(n):
    k = n
    while True:
        for Srest in combinations(range(1, k), n-1):
            S = Srest + (k, )
            allprods = set()
            for i, j in combinations_with_replacement(S, 2):
                if i*j in allprods: break
                else: allprods.add(i*j)
            else: return k
        k += 1
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Sep 08 2021

Formula

a(n) = min {k >= 1; A338006(k) = n}. - Pontus von Brömssen, Sep 09 2021

Extensions

a(15)-a(20) from Michael S. Branicky, Sep 08 2021
a(21)-a(38) (based on the terms in A338006) from Pontus von Brömssen, Sep 09 2021
Previous Showing 21-30 of 50 results. Next

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

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