A034931 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 4.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 0, 3, 2, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 1, 2, 1, 0, 2, 0, 2, 0, 1, 2, 1, 1, 3, 3, 1, 2, 2, 2, 2, 1, 3, 3, 1, 1, 0, 2, 0, 3, 0, 0, 0, 3, 0, 2, 0, 1, 1, 1, 2, 2, 3, 3, 0, 0, 3, 3, 2, 2, 1, 1

Offset

0,5

Comments

The number of 3's in row n is given by 2^(A000120(n)-1) if A014081(n) is nonzero, else by 0 [Davis & Webb]. - R. J. Mathar, Jul 28 2017

Links

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Kenneth S. Davis and William A. Webb, Lucas' theorem for prime powers, European Journal of Combinatorics 11:3 (1990), pp. 229-233.

Kenneth S. Davis and William A. Webb, Pascal's triangle modulo 4, Fib. Quart., 29 (1991), 79-83.

Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 8), European Journal of Combinatorics 19:1 (1998), pp. 45-62.

Ivan Korec, Definability of Pascal's Triangles Modulo 4 and 6 and Some Other Binary Operations from Their Associated Equivalence Relations, Acta Univ. M. Belii Ser. Math. 4 (1996), pp. 53-66.

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(n+1,k) = (T(n,k) + T(n,k-1)) mod 4. - Reinhard Zumkeller, Mar 14 2015

Example

Triangle begins:
                 1
                1 1
               1 2 1
              1 3 3 1
             1 0 2 0 1
            1 1 2 2 1 1
           1 2 3 0 3 2 1
          1 3 1 3 3 1 3 1
         1 0 0 0 2 0 0 0 1
        1 1 0 0 2 2 0 0 1 1
       1 2 1 0 2 0 2 0 1 2 1
      1 3 3 1 2 2 2 2 1 3 3 1
  ...

Maple

A034931 := proc(n, k)
    modp(binomial(n, k), 4) ;
end proc:
seq(seq(A034931(n, k), k=0..n), n=0..10); # R. J. Mathar, Jul 28 2017

Mathematica

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 4] (* Robert G. Wilson v, May 26 2004 *)

Prog

(Haskell)
a034931 n k = a034931_tabl !! n !! k
a034931_row n = a034931_tabl !! n
a034931_tabl = iterate
   (\ws -> zipWith ((flip mod 4 .) . (+)) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Mar 14 2015
(PARI) C(n, k)=binomial(n, k)%4 \\ Charles R Greathouse IV, Aug 09 2016
(PARI) f(n, k)=2*(bitand(n-k, k)==0);
T(n, j)=if(j==0, return(1)); my(k=logint(n, 2), K=2^k, K1=K/2, L=n-K); if(L<K1, if(j<=L, T(L, j), j<K1, 0, j<=K1+L, f(L, j-K1), j<K, 0, T(L, j-K)), if(j<K1, T(L, j), j<=L, bitxor(T(L, j), f(L, j-K1)), j<K, f(L, j-K1), j<=L+K, bitxor(T(L, j-K), f(L, j-K1)), T(L, j-K))); \\ See Davis & Webb 1991. - Charles R Greathouse IV, Aug 11 2016

Crossrefs

Cf. A007318, A047999, A083093, A034930, A008975, A034932, A163000 (# 2's), A270438 (# 1's), A249732 (# 0's).

Cf. A000120, A014081.

Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (m = 3), (this sequence) (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

Sequence in context: A186332 A129571 A180180 * A248473 A307116 A212626

Adjacent sequences: A034928 A034929 A034930 * A034932 A034933 A034934

Keyword

nonn,tabl

Author

N. J. A. Sloane

Status

approved