cp's OEIS Frontend

A062296 a(n) = number of entries in n-th row of Pascal's triangle divisible by 3.

%I A062296 #36 Jul 24 2025 20:48:41
%S A062296 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,
%T A062296 24,27,24,21,28,23,18,33,30,27,32,25,18,31,20,9,40,35,30,37,26,15,34,
%U A062296 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
%N A062296 a(n) = number of entries in n-th row of Pascal's triangle divisible by 3.
%C A062296 Number of zeros in row n of triangle A083093. - _Reinhard Zumkeller_, Jul 11 2013
%H A062296 Reinhard Zumkeller, <a href="/A062296/b062296.txt">Table of n, a(n) for n = 0..1000</a>
%H A062296 D. L. Wells, <a href="http://dx.doi.org/10.1007/978-94-009-0223-7_42">Residue counts modulo three for the fibonacci triangle</a>, Appl. Fib. Numbers, Proc. 6th Int Conf Fib. Numbers, Pullman, 1994 (1996) 521-536.
%F A062296 a(n) = n + 1 - A006047(n).
%F A062296 a(n) = n + 1 - A206424(n) - A227428(n). - _Reinhard Zumkeller_, Jul 11 2013
%F A062296 a(n) = n + 1 - 2^A062756(n)*3^A081603(n). - _Shenghui Yang_, Jan 08 2025
%e A062296 When n=3 the row is 1,3,3,1 so a(3) = 2.
%p A062296 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_
%t A062296 a[n_] := Count[(Binomial[n, #] & )[Range[0, n]], _?(Divisible[#, 3] & )];
%t A062296 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jan 26 2018 *)
%t A062296 Table[n + 1 - 2^(DigitCount[n, 3, 1])*3^(DigitCount[n, 3, 2]), {n, 0, 76}] (* _Shenghui Yang_, Jan 08 2025 *)
%o A062296 (Haskell)
%o A062296 a062296 = sum . map ((1 -) . signum) . a083093_row
%o A062296 -- _Reinhard Zumkeller_, Jul 11 2013
%o A062296 (Python)
%o A062296 from sympy.ntheory import digits
%o A062296 def A062296(n):
%o A062296     s = digits(n,3)[1:]
%o A062296     return n+1-(3**s.count(2)<<s.count(1)) # _Chai Wah Wu_, Jul 24 2025
%Y A062296 Cf. A006047, A083093, A206424, A227428.
%Y A062296 Cf. A062756, A081603.
%K A062296 nonn,look
%O A062296 0,4
%A A062296 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001
%E A062296 More terms from _Emeric Deutsch_, Feb 03 2005