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

%I A363609 #45 Sep 03 2023 10:23:35
%S A363609 21,30,40,40,51,52,54,48,60,62,65,60,72,74,77,72,78,74,86,84,91,88,92,
%T A363609 88,95,93,90,102,105,102,106,104,107,110,109,106,118,120,121,120,125,
%U A363609 123,126,125,122,128,127,124,136,138,139,140,141,138,145,144,143
%N A363609 Minimum sum of the visible pips on a polycube made from n dice.
%C A363609 This sequence is calculated using standard six-sided dice of the same chirality. Opposite sides sum to seven.
%H A363609 Matt Donahoe, <a href="https://github.com/mdonahoe/oeis/blob/main/A363609/dice.py">Python program</a>
%H A363609 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dice">Dice</a>
%F A363609 Conjecture: a(k^3) = 6*(k+2)*k for k > 1.
%F A363609 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
%F A363609 A193416(n) <= lb(n) <= a(n) <= ub(n) <= 6*A193416(n), where:
%F A363609   lb(n) = Sum_{i=1..A193416(n)} S(i, n),
%F A363609   ub(n) = Sum_{i=1..A193416(n)} S(6*n+1-i, n), and
%F A363609   S(i, j) = 1 + floor((i-1)/j). - _Michael S. Branicky_, Jun 11 2023
%e A363609 For n = 2, two dice are conjoined to hide both their 6-pip faces, so a(2) = 30.
%e A363609 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.
%o A363609 (Python) # see linked program
%Y A363609 Conceptually similar to A193416.
%K A363609 nonn
%O A363609 1,1
%A A363609 _Matt Donahoe_, Jun 11 2023
%E A363609 a(22)-a(24) corrected by _Michael S. Branicky_, Jun 18 2023