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.
Marc Evanstein, Hearing Pascal's Triangle Mod 4, YouTube video.
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
(Python)
from math import isqrt, comb
def A034931(n):
g = (m:=isqrt(f:=n+1<<1))-(f<=m*(m+1))
k = n-comb(g+1, 2)
if k.bit_count()+(g-k).bit_count()-g.bit_count()>1: return 0
s, c, d = bin(g)[2:], 1, 0
w = (bin(k)[2:]).zfill(l:=len(s))
for i in range(0, l-1):
r, t = s[i:i+2], w[i:i+2]
if (x:=int(r, 2)) < (y:=int(t, 2)):
d += (t[0]>r[0])+(t[1]>r[1])
else:
c = c*comb(x, y)&3
d -= sum(1 for i in range(1, l-1) if w[i]>s[i])
return (c<<d)&3 # Chai Wah Wu, Jul 19 2025
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