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

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

2,5

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

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

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

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

Example

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

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 *)

Prog

(Haskell)
a059233 n = length $ filter (n `elem`) $
                            take (fromInteger n) $ tail a007318_tabl
a059233_list = map a059233 [2..]
-- Reinhard Zumkeller, Dec 24 2012
(PARI) A059233(n)=A003016(n)\/2 \\ M. F. Hasler, Mar 01 2023

Crossrefs

Cf. A003016, A003015.

Sequence in context: A256554 A321649 A003650 * A357327 A327924 A354057

Adjacent sequences: A059230 A059231 A059232 * A059234 A059235 A059236

Keyword

easy,nice,nonn

Author

Fabian Rothelius, Jan 20 2001

Status

approved