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A249733
Number of (not necessarily distinct) multiples of 9 on row n of Pascal's triangle.
8
0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 4, 2, 0, 2, 1, 0, 12, 6, 0, 8, 4, 0, 4, 2, 0, 24, 21, 18, 19, 14, 9, 14, 7, 0, 28, 20, 12, 20, 13, 6, 12, 6, 0, 32, 19, 6, 21, 12, 3, 10, 5, 0, 48, 42, 36, 38, 28, 18, 28, 14, 0, 50, 37, 24, 36, 24, 12, 22, 11, 0, 52, 32, 12, 34, 20, 6, 16, 8, 0
OFFSET
0,10
COMMENTS
Number of zeros on row n of A095143 (Pascal's triangle reduced modulo 9).
This should have a formula. See for example A062296, A006047 and A048967.
LINKS
FORMULA
For all n >= 0, the following holds:
a(n) <= A048277(n).
a(n) <= A062296(n).
a(2*A249719(n)) > 0 and a((2*A249719(n))-1) > 0.
a(n) is odd if and only if n is one of the terms of A249720.
EXAMPLE
Row 9 of Pascal's triangle is {1, 9, 36, 84, 126, 126, 84, 36, 9, 1}. The terms 9, 36, and 126 are the only multiples of nine, and each of them occurs two times on that row, thus a(9) = 2*3 = 6.
Row 10 of Pascal's triangle is {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1}. The terms 45 (= 9*5) and 252 (= 9*28) are the only multiples of nine, and the former occurs twice, while the latter is alone at the center, thus a(10) = 2+1 = 3.
MATHEMATICA
Total/@Table[If[Mod[Binomial[n, k], 9]==0, 1, 0], {n, 0, 80}, {k, 0, n}] (* Harvey P. Dale, Feb 12 2020 *)
PROG
(PARI)
A249733(n) = { my(c=0); for(k=0, n\2, if(!(binomial(n, k)%9), c += (if(k<(n/2), 2, 1)))); return(c); } \\ Unoptimized.
for(n=0, 6561, write("b249733.txt", n, " ", A249733(n)));
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 04 2014
STATUS
approved