A182237 -id:A182237 - 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).

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

Fabian Rothelius, Jan 20 2001

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

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

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

T. D. Noe, Table of n, a(n) for n=2..10000
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
Wikipedia, Singmaster's conjecture
Index entries for triangles and arrays related to Pascal's triangle

Cf. A003016, A003015.

a059233 n = length $ filter (n `elem`) $
                            take (fromInteger n) $ tail a007318_tabl
a059233_list = map a059233 [2..]
-- Reinhard Zumkeller, Dec 24 2012

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 *)

A059233(n)=A003016(n)\/2 \\ M. F. Hasler, Mar 01 2023

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

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

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)

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

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

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A098565 Numbers that appear as binomial coefficients exactly 6 times.

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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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