A385285 - OEIS

A385285

a(n) = Sum_{k=0..n} (binomial(n, k) mod 8).

1, 2, 4, 8, 16, 16, 32, 32, 16, 32, 24, 48, 40, 64, 40, 64, 16, 32, 40, 64, 48, 48, 88, 96, 40, 80, 88, 112, 72, 112, 112, 128, 16, 32, 40, 64, 64, 96, 120, 160, 48, 96, 72, 144, 104, 176, 160, 192, 40, 80, 104, 176, 120, 176, 176, 224, 72, 144, 160, 224, 176

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) is a multiple of A001316(n). - Chai Wah Wu, Jun 26 2025

LINKS

John Tyler Rascoe, Table of n, a(n) for n = 0..8192

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.

Chai Wah Wu, Sum of rows of Pascal's triangle modulo 8, blog post.

MAPLE

A385285 := n -> local k; add(modp(binomial(n, k), 8), k = 0..n): seq(A385285(n), n = 0..60);

MATHEMATICA

A385285[n_] := Sum[Mod[Binomial[n, k], 8], {k, 0, n}];

Array[A385285, 100, 0] (* Paolo Xausa, Jun 26 2025 *)

PROG

(PARI) a(n) = sum(k=0, n, binomial(n, k) % 8); \\ Michel Marcus, Jun 26 2025

(Python)

def A385285(n):

if n==0: return 1

s = '0'+bin(n)[2:]

n1 = n>>1

n2 = n1>>1

n3 = n2>>1

np = ~n

n1111 = (n3&n2&n1&n).bit_count()

n11 = (n1&n).bit_count()

n101 = (n2&(~n1)&n).bit_count()

n100 = (n2&(~n1)&np).bit_count()

n110 = (n2&n1&np).bit_count()

n10 = (n1&np).bit_count()

n1100 = (n3&n2&(~n1)&np).bit_count()

m11 = s.count('0110')

m111 = s.endswith('0111')

c = (n100<<2)+n110+(n10*(n10-1)>>1)

if n11==0:

c += n10+(n100<<1)

elif n11==1:

c += (n10-n1100<<1)+n110

else:

c += n10<<1

if not (n1111 or n11 or n101):

c += 1

elif not (n1111 or n11) and n101:

c += 3

elif m111:

c += 4

elif not (n1111 or n101 or m11) and n11:

c += 2

else:

c += 4

return c<<n.bit_count() # Chai Wah Wu, Jun 26 2025

CROSSREFS

Cf. A034930 (row sums of), A001316 (mod 2), A384715 (mod 4).

Sequence in context: A362896 A316749 A008381 * A083780 A368056 A309529

Adjacent sequences: A385282 A385283 A385284 * A385286 A385287 A385288

KEYWORD

nonn,easy

AUTHOR

Peter Luschny, Jun 26 2025

STATUS

approved