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

Author

Fabian Rothelius, Jan 20 2001

Keywords

• easy
• nice
• nonn

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

• 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

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