A386953 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 9.

1, 3, 6, 10, 15, 21, 28, 36, 45, 49, 57, 69, 78, 90, 105, 119, 135, 153, 160, 174, 195, 209, 228, 252, 273, 297, 324, 328, 336, 348, 360, 378, 402, 422, 450, 486, 495, 513, 540, 560, 588, 624, 655, 693, 738, 752, 780, 822, 850, 888, 936, 978, 1026, 1080, 1087

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..10000

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 9), Acta Arithmetica, 78 (1997), 331-349.

Maple

P:= [1]; R:= 1:
for i from 1 to 100 do
  P:= [1, op(P[2..-1]+P[1..-2] mod 9), 1];
  v:=i+1 - numboccur(0, P):
  R:= R, v;
od:
ListTools:-PartialSums([R]); # Robert Israel, Sep 10 2025

Prog

(Python)
import re
from gmpy2 import digits
def A386953(n):
    c = 0
    for m in range(n+1):
        s = digits(m, 3)
        n1 = s.count('1')
        n2 = s.count('2')
        n01 = s.count('10')
        n02 = s.count('20')
        n11 = len(re.findall('(?=11)', s))
        n12 = s.count('21')
        c += ((3*((1+n01<<2)+n11)+((n02<<2)+n12<<2))*3**n2<<n1)//3>>2
    return c

Crossrefs

Cf. A001316, A006046-A006048, A194458, A194459, A382720-A382731.

Partial sums of A386952.

Sequence in context: A037123 A062918 A113168 * A341192 A071817 A033442

Adjacent sequences: A386950 A386951 A386952 * A386954 A386955 A386956

Keyword

nonn

Author

Chai Wah Wu, Aug 10 2025

Status

approved