A006048 Number of entries in first n rows of Pascal's triangle not divisible by 3.

1, 3, 6, 8, 12, 18, 21, 27, 36, 38, 42, 48, 52, 60, 72, 78, 90, 108, 111, 117, 126, 132, 144, 162, 171, 189, 216, 218, 222, 228, 232, 240, 252, 258, 270, 288, 292, 300, 312, 320, 336, 360, 372, 396, 432, 438, 450, 468, 480, 504, 540, 558, 594, 648, 651, 657, 666, 672, 684, 702, 711, 729, 756, 762, 774, 792, 804, 828, 864, 882, 918, 972, 981, 999, 1026, 1044, 1080, 1134, 1161, 1215, 1296

Offset: 0

Author

Jeffrey Shallit

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 53.
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024. See p. 1.
• Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & graphics 13.1 (1989): 59-62.
• Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-62. (Annotated scanned copy)
• Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-60. (Annotated scanned copy)
• A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.

Crossrefs

Partial sums of A006047.

Programs

Python

from math import prod
from gmpy2 import digits
def A006048(n): return sum(prod(int(d)+1 for d in digits(m,3)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

Python

from math import prod
from gmpy2 import digits
def A006048(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,3)[::-1]))
    return sum((b-1)*prod(d[a:])*6**a for a, b in enumerate(d))>>1  # Chai Wah Wu, Aug 13 2025

Extensions

More terms from N. J. A. Sloane, Apr 23 2025