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A227428 Number of twos in row n of triangle A083093.

%I A227428 #63 Jul 24 2025 20:34:33
%S A227428 0,0,1,0,0,2,1,2,4,0,0,2,0,0,4,2,4,8,1,2,4,2,4,8,4,8,13,0,0,2,0,0,4,2,
%T A227428 4,8,0,0,4,0,0,8,4,8,16,2,4,8,4,8,16,8,16,26,1,2,4,2,4,8,4,8,13,2,4,8,
%U A227428 4,8,16,8,16,26,4,8,13,8,16,26,13,26,40
%N A227428 Number of twos in row n of triangle A083093.
%C A227428 "The number of entries with value r in the n-th row of Pascal's triangle modulo k is found to be 2^{#_r^k (n)}, where now #_r^k (n) gives the number of occurrences of the digit r in the base-k representation of the integer n." [Wolfram] - _R. J. Mathar_, Jul 26 2017 [This is not correct: there are entries in the sequence that are not powers of 2. - _Antti Karttunen_, Jul 26 2017]
%H A227428 Reinhard Zumkeller (terms 0..1000) & Antti Karttunen, <a href="/A227428/b227428.txt">Table of n, a(n) for n = 0..19683</a>
%H A227428 R. Garfield and H. S. Wilf, <a href="https://dx.doi.org/10.1016/0022-314X(92)90078-4">The distribution of the binomial coefficients modulo p</a>, J. Numb. Theory 41 (1) (1992) 1-5.
%H A227428 Marcus Jaiclin, et al. <a href="https://web.archive.org/web/20170823000349/http://pyrrho.wsc.ma.edu/math/faculty/jaiclin/writings/research/pascals_triangle/">Pascal's Triangle, Mod 2,3,5</a>
%H A227428 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.
%H A227428 Avery Wilson, <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol47_No2.pdf">Pascal's Triangle Modulo 3</a>, Mathematics Spectrum, 47-2 - January 2015, pp. 72-75.
%H A227428 S. Wolfram, <a href="http://dx.doi.org/10.2307/2323743">Geometry of binomial coefficients</a>, Am. Math. Monthly 91 (9) (1984) 566-571.
%F A227428 a(n) = A006047(n) - A206424(n) = n + 1 - A062296(n) - A206424(n).
%F A227428 a(n) = 2^(N_1-1)*(3^N_2-1) where N_1 = A062756(n), N_2 = A081603(n). [Wilson, Theorem 2, Wells] - _R. J. Mathar_, Jul 26 2017
%F A227428 a(n) = A206424(n) * ((3^A081603(n))-1) / ((3^A081603(n))+1). - _Antti Karttunen_, Jul 27 2017
%F A227428 a(n) = (1/2)*Sum_{k = 0..n} mod(C(n,k)^2 - C(n,k), 3). - _Peter Bala_, Dec 17 2020
%e A227428 Example of Wilson's formula: a(26) = 13 = 2^(0-1)*(3^3-1) = 26/2, where A062756(26)=0, A081603(26)=3, 26=(222)_3. - _R. J. Mathar_, Jul 26 2017
%p A227428 A227428 := proc(n)
%p A227428     local a;
%p A227428     a := 0 ;
%p A227428     for k from 0 to n do
%p A227428         if A083093(n,k) = 2 then
%p A227428             a := a+1 ;
%p A227428         end if;
%p A227428     end do:
%p A227428     a ;
%p A227428 end proc:
%p A227428 seq(A227428(n),n=0..20) ; # _R. J. Mathar_, Jul 26 2017
%t A227428 Table[Count[Mod[Binomial[n, Range[0, n]], 3], 2], {n, 0, 99}] (* _Alonso del Arte_, Feb 07 2012 *)
%o A227428 (Haskell)
%o A227428 a227428 = sum . map (flip div 2) . a083093_row
%o A227428 (PARI) A227428(n) = sum(k=0,n,2==(binomial(n,k)%3)); \\ (Naive implementation, from the description) _Antti Karttunen_, Jul 26 2017
%o A227428 (Python)
%o A227428 from sympy import binomial
%o A227428 def a(n):
%o A227428     return sum(1 for k in range(n + 1) if binomial(n, k) % 3 == 2)
%o A227428 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jul 26 2017
%o A227428 (Python)
%o A227428 from sympy.ntheory import digits
%o A227428 def A227428(n):
%o A227428     s = digits(n,3)[1:]
%o A227428     return 3**s.count(2)-1<<s.count(1)>>1 # _Chai Wah Wu_, Jul 24 2025
%o A227428 (Scheme) (define (A227428 n) (* (A000079 (- (A062756 n) 1)) (+ -1 (A000244 (A081603 n))))) ;; After Wilson's direct formula, _Antti Karttunen_, Jul 26 2017
%Y A227428 Cf. A006047, A062296, A062756, A083093, A081603, A206424, A206428.
%K A227428 nonn
%O A227428 0,6
%A A227428 _Reinhard Zumkeller_, Jul 11 2013