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 A034931 Pascal's triangle read modulo 4. 23

%I

%S 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,

%T 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,

%U 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

%N Pascal's triangle read modulo 4.

%C 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

%H Reinhard Zumkeller, <a href="/A034931/b034931.txt">Rows n = 0..120 of triangle, flattened</a>

%H Kenneth S. Davis and William A. Webb, <a href="http://www.sciencedirect.com/science/article/pii/S0195669813801229">Lucas' theorem for prime powers</a>, European Journal of Combinatorics 11:3 (1990), pp. 229-233.

%H Kenneth S. Davis and William A. Webb, <a href="http://www.fq.math.ca/Scanned/29-1/davis.pdf">Pascal's triangle modulo 4</a>, Fib. Quart., 29 (1991), 79-83.

%H James G. Huard, Blair K. Spearman, and Kenneth S. Williams, <a href="http://www.sciencedirect.com/science/article/pii/S0195669897901463">Pascal's triangle (mod 8)</a>, European Journal of Combinatorics 19:1 (1998), pp. 45-62.

%H Ivan Korec, <a href="http://actamath.savbb.sk/acta0405.shtml">Definability of Pascal's Triangles Modulo 4 and 6 and Some Other Binary Operations from Their Associated Equivalence Relations</a>, Acta Univ. M. Belii Ser. Math. 4 (1996), pp. 53-66.

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n+1,k) = (T(n,k) + T(n,k-1)) mod 4. - _Reinhard Zumkeller_, Mar 14 2015

%e Triangle begins:

%e {1},

%e {1, 1},

%e {1, 2, 1},

%e {1, 3, 3, 1},

%e {1, 0, 2, 0, 1},

%e {1, 1, 2, 2, 1, 1},

%e {1, 2, 3, 0, 3, 2, 1},

%e {1, 3, 1, 3, 3, 1, 3, 1},

%e {1, 0, 0, 0, 2, 0, 0, 0, 1},

%e {1, 1, 0, 0, 2, 2, 0, 0, 1, 1},

%e {1, 2, 1, 0, 2, 0, 2, 0, 1, 2, 1},

%e {1, 3, 3, 1, 2, 2, 2, 2, 1, 3, 3, 1},

%e ...

%p A034931 := proc(n,k)

%p modp(binomial(n,k),4) ;

%p end proc:

%p seq(seq(A034931(n,k),k=0..n),n=0..10); # _R. J. Mathar_, Jul 28 2017

%t Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 4] (* _Robert G. Wilson v_, May 26 2004 *)

%o a034931 n k = a034931_tabl !! n !! k

%o a034931_row n = a034931_tabl !! n

%o a034931_tabl = iterate

%o (\ws -> zipWith ((flip mod 4 .) . (+)) ([0] ++ ws) (ws ++ [0])) [1]

%o -- _Reinhard Zumkeller_, Mar 14 2015

%o (PARI) C(n, k)=binomial(n, k)%4 \\ _Charles R Greathouse IV_, Aug 09 2016

%o (PARI) f(n,k)=2*(bitand(n-k, k)==0);

%o 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

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

%Y Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (m = 3), A034931 (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).

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_

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Last modified January 23 19:12 EST 2020. Contains 331175 sequences. (Running on oeis4.)