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.
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).
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).
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved