A062296 - OEIS

A062296

a(n) = number of entries in n-th row of Pascal's triangle divisible by 3.

0, 0, 0, 2, 1, 0, 4, 2, 0, 8, 7, 6, 9, 6, 3, 10, 5, 0, 16, 14, 12, 16, 11, 6, 16, 8, 0, 26, 25, 24, 27, 24, 21, 28, 23, 18, 33, 30, 27, 32, 25, 18, 31, 20, 9, 40, 35, 30, 37, 26, 15, 34, 17, 0, 52, 50, 48, 52, 47, 42, 52, 44, 36, 58, 53, 48, 55, 44, 33, 52, 35, 18, 64, 56, 48, 58, 41

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OFFSET

0,4

COMMENTS

Number of zeros in row n of triangle A083093. - Reinhard Zumkeller, Jul 11 2013

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

D. L. Wells, Residue counts modulo three for the fibonacci triangle, Appl. Fib. Numbers, Proc. 6th Int Conf Fib. Numbers, Pullman, 1994 (1996) 521-536.

FORMULA

a(n) = n + 1 - A006047(n).

a(n) = n + 1 - A206424(n) - A227428(n). - Reinhard Zumkeller, Jul 11 2013

a(n) = n + 1 - 2^A062756(n)*3^A081603(n). - Shenghui Yang, Jan 08 2025

EXAMPLE

When n=3 the row is 1,3,3,1 so a(3) = 2.

MAPLE

p:=proc(n) local ct, k: ct:=0: for k from 0 to n do if binomial(n, k) mod 3 = 0 then else ct:=ct+1 fi od: end: seq(n+1-p(n), n=0..83); # Emeric Deutsch

MATHEMATICA

a[n_] := Count[(Binomial[n, #] & )[Range[0, n]], _?(Divisible[#, 3] & )];

Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 26 2018 *)

Table[n + 1 - 2^(DigitCount[n, 3, 1])*3^(DigitCount[n, 3, 2]), {n, 0, 76}] (* Shenghui Yang, Jan 08 2025 *)

PROG

(Haskell)

a062296 = sum . map ((1 -) . signum) . a083093_row

-- Reinhard Zumkeller, Jul 11 2013

CROSSREFS

Cf. A006047, A083093, A206424, A227428.

Cf. A062756, A081603.

Sequence in context: A348218 A138002 A261877 * A249343 A369206 A378015