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 1-7 of 7 results.

A003015 Numbers that occur 5 or more times in Pascal's triangle.

Original entry on oeis.org

1, 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The subject of a recent thread on sci.math. Apparently it has been known for many years that there are infinitely many numbers that occur at least 6 times in Pascal's triangle, namely the solutions to binomial(n,m-1) = binomial(n-1,m) given by n = F_{2k}*F_{2k+1}; m = F_{2k-1}*F_{2k} where F_i is the i-th Fibonacci number. The first of these outside the range of the existing database entry is {104 choose 39} = {103 choose 40} = 61218182743304701891431482520. - Christopher E. Thompson, Mar 09 2001
It may be that there are no terms that appear exactly 5 times in Pascal's triangle, in which case the title could be changed to "Numbers that occur 6 or more times in Pascal's triangle". - N. J. A. Sloane, Nov 24 2004
No other terms below 33*10^16 (David W. Wilson).
61218182743304701891431482520 really is the next term. Weger shows this and I checked it. - T. D. Noe, Nov 15 2004
Blokhuis et al. show that there are no other solutions less than 10^60, nor within the first 10^6 rows of Pascal's triangle other than those given by the parametric solution mentioned above. - Christopher E. Thompson, Jan 19 2018
See the b-file of A090162 for the explicit numbers produced by the parametric formula. - Jeppe Stig Nielsen, Aug 23 2020

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003016, A059233.
Cf. A182237, A098565 (subsequence).
Cf. A090162 (easy subsequence).

A059233 Number of rows in which n appears in Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 2

Views

Author

Fabian Rothelius, Jan 20 2001

Keywords

Comments

Central binomial coefficients c = A000984(n) > 1 appear once in the middle column C(2n, n), and thereafter in one or more later rows to the left as C(r,k) and to the right as C(r, r-k), k < r/2; the last time in row r = c = C(c,1) = C(c,c-1). For these, a(n) = (A003016(n)+1)/2. For all other numbers n > 1, a(n) = A003016(n)/2. - M. F. Hasler, Mar 01 2023

Examples

			6 appears in both row 4 and row 6 in Pascal's triangle, therefore a(6) = 2.

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
  • C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 96.

Links

Crossrefs

Cf. A003016, A003015.

Programs

  • Haskell
    a059233 n = length $ filter (n `elem`) $
                            take (fromInteger n) $ tail a007318_tabl
a059233_list = map a059233 [2..]
-- Reinhard Zumkeller, Dec 24 2012
  • Mathematica
    nmax = 101; A007318 = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, n}]; a[n_] := Position[A007318, n][[All, 1]] // Union // Length; Table[a[n], {n, 2, nmax}] (* Jean-François Alcover, Sep 09 2013 *)
  • PARI
    A059233(n)=A003016(n)\/2 \\ M. F. Hasler, Mar 01 2023

Formula

a(A180058(n)) = n and a(m) < n for m < A180058(n); a(A182237(n)) = 2; a(A098565(n)) = 3. - Reinhard Zumkeller, Dec 24 2012
a(n) = ceiling(A003016(n)/2). - M. F. Hasler, Mar 01 2023

A182237 Numbers occurring exactly in 2 rows of Pascal's triangle.

Original entry on oeis.org

6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105, 126, 136, 153, 165, 171, 190, 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
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 24 2012

Keywords

Comments

A059233(a(n)) = 2.

Links

Crossrefs

Cf. A098565.

Programs

  • Haskell
    import Data.List (elemIndices)
a182237 n = a182237_list !! (n-1)
a182237_list = map (+ 2 ) $ elemIndices 2 a059233_list
  • Mathematica
    nn = 1000; t = Table[s = {}; k = 1; While[k++; b = Binomial[n, k]; k <= n/2 && b <= nn, AppendTo[s, b]]; s, {n, nn}]; t2 = Select[t, Length[#] > 0 &]; Transpose[Select[Tally[Sort[Flatten[t2]]], #[[2]] == 1 &]][[1]] (* T. D. Noe, Mar 13 2013 *)

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

A180058 Smallest number occurring in exactly n rows of Pascal's triangle.

Original entry on oeis.org

2, 6, 120, 3003
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 24 2012

Keywords

Comments

A059233(a(n)) = n and A059233(m) < n for m < a(n).

Examples

			.  n  A180058  referred equal binomial coefficients (A007318)  A059233
.  -  -------  ----------------------------------------------  -------
.  1        2   C (2, 1)                                             1
.  2        6   C (4, 2)   C (6, 1)                                  2
.  3      120   C (10, 3)  C (16, 2)  C (120, 1)                     3
.  4     3003   C (14, 6)  C (15, 5)  C (78, 2)   C (3003, 1)        4 .

Links

Crossrefs

Cf. A185024, A182237, A098565, A003015, A003016.

Programs

  • Haskell
    import Data.List (elemIndex)
import Data.Maybe (fromJust)
a180058 = (+ 2) . fromJust . (`elemIndex` a059233_list)

A138496 Where record values occur in A003016.

Original entry on oeis.org

0, 1, 10, 120, 3003
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 20 2008

Keywords

Comments

It appears that the record values are 0, 3, 4, 6, 8, 10, 12, ...
From M. F. Hasler, Feb 16 2023: (Start)
The numbers that appear 3 times in Pascal's triangle are the central binomial coefficients (A000984), except for the number 2 that is the only number to appear only once. For n = 1 there would be an infinite number of occurrences, but sequence A003016 counts only the occurrences of n in rows <= n so that n = 1 also gives 3.
All C(n,k) with 1 < k < n/2 (in particular triangular numbers A000217) appear at least 4 times; see A098564 for those appearing exactly 4 times.
Numbers that appear 5 or more times are quite rare, they are listed in A003015 with subsequence A098565 of those appearing exactly 6 times.
They are mostly C(n,k) with 2 < k < n/2 which are also triangular numbers, but some are also of the form C(n+1,k) = C(n,k+1) with 3 < k < n/2, and a subsequence of these has n and k given in terms of Fibonacci numbers. (End)

Programs

Showing 1-7 of 7 results.

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

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