A059233 -id:A059233 - cp's OEIS frontend

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

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

Offset: 1

nonn

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

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).

Aart Blokhuis, Andries Brouwer, and Benne de Weger, Binomial collisions and near collisions, INTEGERS, Volume 17, Article A64, 2017 (also available as arXiv:1707.06893 [math.NT]).
Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture, Integers (2021) Vol. 21, #A34.
B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Econometric Institute Research Papers, No. EI 9536-/B, 1995.
B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 373-386.
Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video (2020).
R. K. Guy and V. Klee, Monthly research problems, 1969-1971, Amer. Math. Monthly, 78 (1971), 1113-1122.
D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6 (3) (1968), 86-93.
Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle, arXiv:2106.03335 [math.NT], 2021.
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 69.
David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298.
Eric Weisstein's World of Mathematics, Pascal's Triangle
Tomohiro Yamada, Necessary conditions for binomial collisions, arXiv:2002.07043 [math.NT], 2020.

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

A003016 Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).

Original entry on oeis.org

0, 3, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane

Or, number of occurrences of n as a binomial coefficient. [Except for 1 which occurs infinitely many times. This is the only reason for the restriction "row <= n" in the definition. Any other number can only appear in rows <= n. - M. F. Hasler, Feb 16 2023]
Sequence A138496 gives record values and where they occur. - Reinhard Zumkeller, Mar 20 2008
Singmaster conjectures that this sequence is bounded. - Michael J. Hardy, Jun 09 2025

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
H. L. Abbott, P. Erdős and D. Hanson, On the numbers of times an integer occurs as a binomial coefficient, Amer. Math. Monthly, (1974), 256-261.
Daniel Kane, New Bounds on the Number of Representations of t as a Binomial Coefficient, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper A7, 2004.
Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle, arXiv:2106.03335 [math.NT], 2021.
D. Singmaster, How often does an integer occur as a binomial coefficient?, Amer. Math. Monthly, 78 (1971), 385-386.
Eric Weisstein's World of Mathematics, Pascal's Triangle
Index entries for triangles and arrays related to Pascal's triangle

Cf. A003015, A059233, A138496, A180058.

a003016 n = sum $ map (fromEnum . (== n)) $
                      concat $ take (fromInteger n + 1) a007318_tabl
-- Reinhard Zumkeller, Apr 12 2012

a[0] = 0; t = {{1}}; a[n_] := Count[ AppendTo[t, Table[ Binomial[n, k], {k, 0, n}]], n, {2}]; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Feb 20 2012 *)

{A003016(n)=if(n<4, [0,3,1,2][n+1], my(c=2, k=2, r=sqrtint(2*n)+1, C=r*(r-1)/2); until(, while(C= r\2 && break; C *= r-k; C \= r; r -= 1); c)} \\ M. F. Hasler, Feb 16 2023

from math import isqrt # requires python3.8 or higher
def A003016(n):
    if n < 4: return[0,3,1,2][n]
    cnt = k = 2; r = isqrt(2*n)+1; C = r*(r-1)//2
    while True:
       while C < n and k < r//2:
          C *= r-k; k += 1; C //= k
       if C == n: cnt += 2 - (r == 2*k)
       if k >= r//2: return cnt
       C *= r-k; C //= r; r -= 1 # M. F. Hasler, Feb 16 2023

More terms from Erich Friedman

Edited by N. J. A. Sloane, Nov 18 2007, at the suggestion of Max Alekseyev

A098565 Numbers that appear as binomial coefficients exactly 6 times.

Original entry on oeis.org

120, 210, 1540, 7140, 11628, 24310, 61218182743304701891431482520

Offset: 1

Paul D. Hanna, Oct 27 2004

Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture, arXiv:2107.09107 [math.NT], 2021.
Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video, 2020.
David Singmaster, How Often Does An Integer Occur As A Binomial Coefficient?, American Mathematical Monthly, 78(4), 1971, pp. 385-386; also on Fermat's Library.
Wikipedia, Singmaster's conjecture
Index entries for triangles and arrays related to Pascal's triangle

See A098564 for more information.
Cf. A185024, A182237. Subsequence of A003015.
Cf. A059233.

import Data.List (elemIndices)
a098565 n = a098565_list !! (n-1)
a098565_list = map (+ 2 ) $ elemIndices 3 a059233_list
-- Reinhard Zumkeller, Dec 24 2012

A059233(a(n)) = 3. - Reinhard Zumkeller, Dec 24 2012

a(7) from T. D. Noe, Jul 13 2005

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

Reinhard Zumkeller, Dec 24 2012

nonn

A059233(a(n)) = 2.

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)
Wikipedia, Singmaster's conjecture
Index entries for triangles and arrays related to Pascal's triangle

Cf. A098565.

import Data.List (elemIndices)
a182237 n = a182237_list !! (n-1)
a182237_list = map (+ 2 ) $ elemIndices 2 a059233_list

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 *)

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

Reinhard Zumkeller, Dec 24 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..2500
Wikipedia, Singmaster's conjecture
Index entries for triangles and arrays related to Pascal's triangle

Cf. A137905, A098565, A182237, A059233.

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

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

Reinhard Zumkeller, Dec 24 2012

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

			.  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 .

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

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

A062527 Smallest number (>1) which appears at least n times in Pascal's triangle.

Original entry on oeis.org

2, 3, 6, 10, 120, 120, 3003, 3003

Offset: 1

Henry Bottomley, Jul 10 2001

Singmaster's conjecture is that this sequence is finite.

			a(8)=3003 since 3003 =C(3003,1) =C(3003,3002) =C(78,2) =C(78,76) =C(15,5) =C(15,10) =C(14,6) = C(14,8).

H. L. Abbott, P. Erdos and D. Hanson, On the numbers of times an integer occurs as a binomial coefficient, Amer. Math. Monthly, (1974), 256-261.
FTL Magazine, One Thousand and One Coincidences
D. Singmaster, How often does an integer occur as a binomial coefficient?, Amer. Math. Monthly, 78 (1971), 385-386.
David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298.

Cf. A003015, A003016, A006987, A007318, A059233.

(* Computation lasts a few minutes *) max = 4000; Clear[cnt]; cnt[] = 0; Do[b = Binomial[n, k]; If[b <= max, cnt[b] += 1], {n, 0, max}, {k, 1, n - 1}]; sel = Select[Table[{b, cnt[b]}, {b, 1, max }], #[[2]] >= 1&]; a[n] := Select[sel, #[[2]] >= n&][[1, 1]]; Array[a, 8] (* Jean-François Alcover, Oct 05 2015 *)

A128373 Irregular triangle read by rows: row n (n>=2) lists positions in the sequence A007318 where n appears.

Original entry on oeis.org

4, 7, 8, 11, 13, 16, 19, 12, 22, 26, 29, 34, 37, 43, 46, 53, 17, 18, 56, 64, 67, 76, 79, 89, 92, 103, 106, 118, 23, 25, 121, 134, 137, 151, 154, 169, 172, 188, 191, 208, 24, 211, 229, 30, 33, 232, 251, 254, 274, 277, 298, 301, 323, 326, 349, 352, 376, 379, 404, 38

Offset: 2

Nick Hobson, Mar 01 2007

			In A007318, the number 2 appears in position 4, so the first row is 4. The number 3 appears in positions 7 and 8 in A007318, so the second row is 7, 8.
The irregular triangle begins:
   4
   7,  8
  11, 13
  16, 19
  12, 22, 26
  29, 34
  37, 43
  46, 53
  ...

Cf. A003015, A003016, A007318, A059233.

cp's OEIS Frontend

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

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A003016 Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).

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A098565 Numbers that appear as binomial coefficients exactly 6 times.

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

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

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

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A062527 Smallest number (>1) which appears at least n times in Pascal's triangle.

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A128373 Irregular triangle read by rows: row n (n>=2) lists positions in the sequence A007318 where n appears.

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