A006048 Number of entries in first n rows of Pascal's triangle not divisible by 3. (Formerly M2519)

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

0,2

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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.

Mathematica

Table[Sum[Times@@(IntegerDigits[m, 3]+1), {m, 0, n}], {n, 0, 80}] (* Vincenzo Librandi, Feb 18 2026 *)

Prog

(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
(Magma) [&+[ &*[ d+1 : d in (m eq 0 select [0] else IntegerToSequence(m, 3))]: m in [0..n]] : n in [0..67]]; // Vincenzo Librandi, Feb 18 2026

Crossrefs

Partial sums of A006047.

Sequence in context: A301283 A288218 A212984 * A245989 A350928 A049827

Adjacent sequences: A006045 A006046 A006047 * A006049 A006050 A006051

Keyword

nonn

Author

Jeffrey Shallit

Extensions

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

Status

approved