A274626 a(n) = Product_{i=0..2} (2^floor((n+i)/3)-1).

0, 0, 0, 1, 3, 9, 27, 63, 147, 343, 735, 1575, 3375, 6975, 14415, 29791, 60543, 123039, 250047, 504063, 1016127, 2048383, 4112895, 8258175, 16581375, 33227775, 66585855, 133432831, 267126783, 534776319, 1070599167, 2142244863, 4286583807, 8577357823, 17158905855, 34326194175, 68669157375

Offset: 0

Author

N. J. A. Sloane, Jul 05 2016

Keywords

• nonn

Comments

From Tom Karzes, Jul 05 2016: (Start)

This is a three-dimensional analog of the holes-in-sheet-of-paper sequence A274230.

In d dimensions, assuming the axes for folding are selected in a round-robin fashion, the number of times a given dimension is folded is:

floor((n+i)/d)

where i runs from 0 (for the last dimension to be folded) through d-1 (for the first dimension to be folded).

The corresponding number of internal dividing lines/planes/etc. is (2^floor((n+i)/d)-1). The number of internal d-way intersections, which corresponds to the number of holes, is:

Product_{i=0..d-1}(2^floor((n+i)/d)-1)

where d is the number of dimensions and n is the total number of folds.

Note that the first several nonzero entries in these sequences are the powers of 3. Specifically, in d dimensions, the first d entries are 0, followed by the first (d+1) powers of 3.

It's not hard to see why this is so. The first nonzero entry occurs at d folds, and the value is 1. This is when you've folded once along each dimension.

After that, the next d folds each divide 2 old partitions into 4 new ones, i.e., they change the internal folds from 1 to 3. So for the next d entries you just multiply the previous entry by 3 (or more generally, by 3/1).

After that you multiply by 7/3 for the next d entries, then 15/7, then 31/15, etc. Each time you're just replacing one of the old factors with a new one, where each factor is one less than a power of two.

Here's an alternative formulation that avoids the iterated product.

For a given number of folds, there are only two factors, each raised to some exponent (with the sum of the exponents being the dimension d):

v1 = 2^(n/d)-1

v2 = 2^(n/d+1)-1

p1 = d-mod(n,d)

p2 = mod(n,d)

holes = (v1^p1)*(v2^p2)

This flattens to:

((2^(n/d)-1)^(d-mod(n,d))) * ((2^(n/d+1)-1)^(mod(n,d)))

(End)

References

• Tom Karzes, Posting to Math Fun Mailing List, Jul 05 2016.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A274230, A274627.

Programs

Maple

f:=(n,d) ->  mul(2^floor((n+i)/d)-1, i = 0 .. d-1);
[seq(f(n,3),n=0..40)];

Mathematica

Table[Product[(2^Floor[(n+i)/3]-1),{i,0,2}],{n,0,40}] (* Harvey P. Dale, Jun 29 2025 *)

PARI

a(n) = prod(i=0, 2, 2^floor((n+i)/3)-1) \\ Colin Barker, Jul 06 2016

Formula

Empirical g.f.: x^3*(1+2*x^2) / ((1-x)*(1-2*x)*(1-2*x^3)*(1-4*x^3)). - Colin Barker, Jul 06 2016