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 1-4 of 4 results.

A006987 Binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

Original entry on oeis.org

6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105, 120, 126, 136, 153, 165, 171, 190, 210, 220, 231, 252, 253, 276, 286, 300, 325, 330, 351, 364, 378, 406, 435, 455, 462, 465, 495, 496, 528, 560, 561, 595, 630, 666, 680, 703, 715, 741, 780, 792, 816, 820
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

Complement of A137905; a(n) > A058084(a(n)). - Reinhard Zumkeller, Mar 20 2009
Or numbers l which, for the first time, appear in m-th row of the Pascal triangle for m < l. - Vladimir Shevelev, Apr 28 2010
Appears to be the set of simplex numbers of order > 2 and dimension > 1. - Dylan Hamilton, Nov 05 2010
This is correct (assuming the notational choice of giving the first n-simplicial number index 1), as the n-th diagonal or antidiagonal of Pascal's triangle gives the n-simplicial numbers. - Thomas Anton, Dec 04 2018

Examples

			Pascal's triangle (A007318) with the outer two layers removed:
             6
          10  10
        15  20  15
      21  35  35  21
    28  56  70  56  28
  36  84 126 126  84  36
  ...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002808, A007318.

Programs

  • Mathematica
    Take[ Union[ Flatten[ Table[ Binomial[n, k], {n, 2, 45}, {k, 2, n - 2}]]], 58] (* Robert G. Wilson v, May 25 2004 *)
  • PARI
    list(lim)=my(v=List(), t); for(n=4, sqrtint(2*lim)+1, for(k=2, n\2, t=binomial(n, k); if(t>lim, break, listput(v, t)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Apr 03 2012

Extensions

More terms from David W. Wilson
Spelling corrected by Jason G. Wurtzel, Aug 22 2010

A058084 Smallest m such that binomial(m,k) = n for some k.

Original entry on oeis.org

0, 2, 3, 4, 5, 4, 7, 8, 9, 5, 11, 12, 13, 14, 6, 16, 17, 18, 19, 6, 7, 22, 23, 24, 25, 26, 27, 8, 29, 30, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 42, 43, 44, 10, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 8, 57, 58, 59, 60, 61, 62, 63, 64, 65, 12, 67, 68, 69, 8, 71, 72, 73, 74, 75, 76
Offset: 1

Views

Author

Fabian Rothelius, Nov 25 2000

Keywords

Comments

Index of first row of Pascal's triangle (A007318) containing n.

Examples

			a(28)=8 because 28 is first found in row 8 of Pascal's triangle (where the first row is counted as 0).

Links

Crossrefs

Cf. A007318, A006987, A137905, A357327.

Programs

  • Haskell
    import Data.List (findIndex); import Data.Maybe (fromJust)
a058084 n = fromJust $ findIndex (elem n) a007318_tabl
-- Reinhard Zumkeller, Nov 09 2011
  • Maple
    with(combinat): for n from 2 to 150 do flag := 0: for m from 1 to 150 do for k from 1 to m do if binomial(m,k) = n then printf(`%d,`,m); flag := 1; break fi: od: if flag=1 then break fi; od: od:
  • Mathematica
    nmax = 76; t = Table[Binomial[m, k], {m, 0, nmax}, {k, 0, m}]; a[n_] := Position[t, n, 2, 1][[1, 1]]-1; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Nov 30 2011 *)
  • PARI
    a(n) = my(k=0); while (!vecsearch(vector((k+2)\2, i, binomial(k, i-1)), n), k++); k; \\ Michel Marcus, Dec 07 2021
  • Python
    def A058084(n):
    if n == 1: return 0
    c = [1]
    for m in range(n):
        d = [1]
        for i in range(m):
            a = c[i]+c[i+1]
            if a == n:
                return m+1
            d.append(a)
        c = d+[1] # Chai Wah Wu, Aug 23 2025

Formula

a(A006987(n)) < A006987(n); a(A137905(n)) = A137905(n). - Reinhard Zumkeller, Mar 20 2009
A007318(a(n), A357327(n)) = n. - Pontus von Brömssen, Sep 24 2022

Extensions

More terms from James Sellers, Nov 27 2000

A098564 Numbers that appear as binomial coefficients exactly 4 times.

Original entry on oeis.org

10, 15, 21, 28, 35, 36, 45, 55, 56, 66, 78, 84, 91, 105, 126, 136, 153, 165, 171, 190, 220, 231, 253, 276, 286, 300, 325, 330, 351, 364, 378, 406, 435, 455, 462, 465, 495, 496, 528, 560, 561, 595, 630, 666, 680, 703, 715, 741, 780, 792, 816, 820
Offset: 1

Views

Author

Paul D. Hanna, Oct 27 2004

Keywords

Comments

Let f(k) be the sequence of numbers that appear as binomial coefficients exactly k times:
f(1) = {2}.
f(2) = A137905.
f(3) appears to be A000984 \ {1, 2}: central binomial coefficients greater than 2.
f(4) = this sequence.
f(5) appears to be empty.
f(6) = A098565.
f(7) appears to be empty.
f(8) begins with 3003.

Links

Crossrefs

Cf. A000984, A098565, A137905.

Programs

  • Mathematica
    binmax = 10^5; dm = 100; Clear[f]; f[m_] := f[m] = (Join[Table[Binomial[n, k], {n, 1, m}, {k, 1, n-1}], Table[Table[{Binomial[n, 1], Binomial[n, 2]}, {2}], {n, m+1, binmax}]] // Flatten // Tally // Select[#, #[[1]] <= binmax && #[[2]] == 4&]&)[[All, 1]] // Sort; f[dm]; f[m = 2*dm]; While[f[m] != f[m-dm], Print[m]; m = m+dm]; f[m] (* Jean-François Alcover, Mar 10 2014 *)

A185024 Numbers occurring in just one row of Pascal's triangle.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 24 2012

Keywords

Links

Crossrefs

Cf. A137905, A098565, A182237, A059233.

Programs

  • Haskell
    import Data.List (elemIndices)
a185024 n = a185024_list !! (n-1)
a185024_list = map (+ 2 ) $ elemIndices 1 a059233_list

Formula

A059233(a(n)) = 1.
a(n) = A137905(n-1), n >= 2. - Elijah Beregovsky, May 14 2019
Showing 1-4 of 4 results.

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

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