A363609 Minimum sum of the visible pips on a polycube made from n dice.

21, 30, 40, 40, 51, 52, 54, 48, 60, 62, 65, 60, 72, 74, 77, 72, 78, 74, 86, 84, 91, 88, 92, 88, 95, 93, 90, 102, 105, 102, 106, 104, 107, 110, 109, 106, 118, 120, 121, 120, 125, 123, 126, 125, 122, 128, 127, 124, 136, 138, 139, 140, 141, 138, 145, 144, 143

Offset: 1

Author

Matt Donahoe, Jun 11 2023

Keywords

• nonn

Comments

This sequence is calculated using standard six-sided dice of the same chirality. Opposite sides sum to seven.

Examples

			For n = 2, two dice are conjoined to hide both their 6-pip faces, so a(2) = 30.
For n = 4, four dice are arranged in a 2 X 2 square such that no 5-pip or 6-pip faces are visible. When the dice can form a cube, such as n = 8, only 1-, 2- and 3-pip faces will be visible.
		

Links

• Matt Donahoe, Python program
• Wikipedia, Dice

Crossrefs

Conceptually similar to A193416.

Programs

Python

# see linked program

Formula

Conjecture: a(k^3) = 6*(k+2)*k for k > 1.

a(i*j*k) <= 48 + 2*((i-2)*(j-2) + (i-2)*(k-2) + (j-2)*(k-2)) + 12*(i+j+k-6), for i, j, k > 1. - Michael S. Branicky, Jun 15 2023

A193416(n) <= lb(n) <= a(n) <= ub(n) <= 6*A193416(n), where:

lb(n) = Sum_{i=1..A193416(n)} S(i, n),

ub(n) = Sum_{i=1..A193416(n)} S(6*n+1-i, n), and

S(i, j) = 1 + floor((i-1)/j). - Michael S. Branicky, Jun 11 2023

Extensions

a(22)-a(24) corrected by Michael S. Branicky, Jun 18 2023