cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A385553 Period of {binomial(N,n) mod 6: N in Z}.

Original entry on oeis.org

1, 6, 12, 36, 72, 72, 72, 72, 144, 432, 432, 432, 432, 432, 432, 432, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 2592, 2592, 2592, 2592, 2592, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184
Offset: 0

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 35 (mod 36), binomial(N,3) == {0, 0, 0, 1, 4, 4, 2, 5, 2, 0, 0, 3, 4, 4, 4, 5, 2, 2, 0, 3, 0, 4, 4, 1, 2, 2, 2, 3, 0, 0, 4, 1, 4, 2, 2, 5} (mod 6).
For N == 0, 1, ..., 71 (mod 72), binomial(N,4) == {0, 0, 0, 0, 1, 5, 3, 5, 4, 0, 0, 0, 3, 1, 5, 3, 2, 4, 0, 0, 3, 3, 1, 5, 0, 2, 4, 0, 3, 3, 3, 1, 2, 0, 2, 4, 3, 3, 3, 3, 4, 2, 0, 2, 1, 3, 3, 3, 0, 4, 2, 0, 5, 1, 3, 3, 0, 0, 4, 2, 3, 5, 1, 3, 0, 0, 0, 4, 5, 3, 5, 1} (mod 6).

Links

Crossrefs

Column 6 of A349593. A062383, A064235 (if offset 0), A385552, and A385554 are respectively columns 2, 3, 5, and 10.

Programs

  • PARI
    a(n) = if(n, (2^(logint(n,2)+1)) * (3^(logint(n,3)+1)), 1)

Formula

a(n) = (the smallest power of 2 > n) * (the smallest power of 3 > n) = A062383(n) * A064235(n+1). For the general result, see A349593.

A385554 Period of {binomial(N,n) mod 10: N in Z}.

Original entry on oeis.org

1, 10, 20, 20, 40, 200, 200, 200, 400, 400, 400, 400, 400, 400, 400, 400, 800, 800, 800, 800, 800, 800, 800, 800, 800, 4000, 4000, 4000, 4000, 4000, 4000, 4000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000, 8000
Offset: 0

Views

Author

Jianing Song, Jul 03 2025

Keywords

Examples

			For N == 0, 1, ..., 19 (mod 20), binomial(N,3) == {0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9} (mod 10).
For N == 0, 1, ..., 39 (mod 40), binomial(N,4) == {0, 0, 0, 0, 1, 5, 5, 5, 0, 6, 0, 0, 5, 5, 1, 5, 0, 0, 0, 6, 5, 5, 5, 5, 6, 0, 0, 0, 5, 1, 5, 5, 0, 0, 6, 0, 5, 5, 5, 1} (mod 10).

Links

Crossrefs

Column 10 of A349593. A062383, A064235 (if offset 0), A385552, and A385553 are respectively columns 2, 3, 5, and 6.

Programs

  • PARI
    a(n) = if(n, (2^(logint(n,2)+1)) * (5^(logint(n,5)+1)), 1)

Formula

a(n) = (the smallest power of 2 > n) * (the smallest power of 5 > n) = A062383(n) * A385552(n). For the general result, see A349593.

A349221 Triangle read by rows: T(n, k) = 1 if k divides binomial(n-1, k-1), T(n, k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0
Offset: 1

Views

Author

Richard L. Ollerton, Nov 11 2021

Keywords

Comments

Similar to A054521 as gcd(n, k) = 1 => k divides binomial(n-1, k-1) but not equivalent as the converse is not true, the earliest example being T(10,4) where 4 divides binomial(9,3) = 84 but gcd(10,4) is not 1. Question: What characterizes the cases where this triangle differs from A054521?
The period of the k-th column is given by A349593(k-1, k) = k * Product_{prime p|k} p^(floor(log(k-1)/log(p))). - Jianing Song, Nov 29 2021
{T(n, k)} is the sum of triangles [k|binomial(n-1, k-1) AND gcd(n, k) = j], n >= 1, 1 <= k <= n, j >= 1, where [] is the Iverson bracket. For j > 1, bitmaps of these triangles suggest simpler fractal gaskets that combine to produce the "shadowing" effect observed in the bitmap of {T(n, k)} provided in the LINKS section. For prime j, the bitmaps suggest a fractal (Hausdorff) dimension of log(A000217(j))/log(j) = log(j(j + 1)/2)/log(j), which is the same as that of the gasket formed by taking the Pascal triangle (A007318) mod j (see Bondarenko reference). - Richard L. Ollerton, Dec 10 2021

Examples

			The triangle T(n, k) begins:
n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
1:   1
2:   1  0
3:   1  1  0
4:   1  0  1  0
5:   1  1  1  1  0
6:   1  0  0  0  1  0
7:   1  1  1  1  1  1  0
8:   1  0  1  0  1  0  1  0
9:   1  1  0  1  1  0  1  1  0
10:  1  0  1  1  0  1  1  0  1  0
11:  1  1  1  1  1  1  1  1  1  1  0
12:  1  0  0  0  1  1  1  0  0  0  1  0
13:  1  1  1  1  1  1  1  1  1  1  1  1  0
14:  1  0  1  0  1  0  0  0  1  0  1  0  1  0
15:  1  1  0  1  0  0  1  1  0  0  1  0  1  1  0
...
Differences between this example and that for A054521 occur at (n,k) = (10,4), (10,6), and (12,6).

References

  • Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

Crossrefs

Cf. A007318, A051731, A054521.

Programs

  • Mathematica
    Table[Boole[Mod[Binomial[n - 1, k - 1], k] == 0], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    row(n) = vector(n, k, !(binomial(n-1,k-1) % k)); \\ Michel Marcus, Nov 11 2021

Formula

T(n, k) = [k|binomial(n-1, k-1)] = Sum_{j>=1} [k|binomial(n-1, k-1) AND gcd(n, k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. (The j = 1 case is A054521.)
T(n, k) = T(n, n-k), n > 1, 1 <= k < n.
Previous Showing 11-13 of 13 results.

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