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A034931
Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 4.
23
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
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.
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.
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).
Sequence in context: A186332 A129571 A180180 * A248473 A307116 A212626
KEYWORD
nonn,tabl
STATUS
approved