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A273013 Number of different arrangements of nonnegative integers on a pair of n-sided dice such that the dice can add to any integer from 0 to n^2-1. 2
1, 1, 1, 3, 1, 7, 1, 10, 3, 7, 1, 42, 1, 7, 7, 35, 1, 42, 1, 42, 7, 7, 1, 230, 3, 7, 10, 42, 1, 115, 1, 126, 7, 7, 7, 393, 1, 7, 7, 230, 1, 115, 1, 42, 42, 7, 1, 1190, 3, 42, 7, 42, 1, 230, 7, 230, 7, 7, 1, 1158, 1, 7, 42, 462, 7, 115, 1, 42, 7, 115, 1, 3030 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The set of b values (see formula), and therefore also a(n), depends only on the prime signature of n. So, for example, a(24) will be identical to a(n) of any other n which is also of the form p_1^3*p_2, (e.g., 40, 54, 56).

The value of b_1 will always be 1. When n is prime, the only nonzero b will be b_1, so therefore a(n) will be 1.

In any arrangement, both dice will have a 0, and one will have a 1 (here called the lead die). To determine any one of the actual arrangements to numbers on the dice, select one of the permutations of divisors (for the lead die), then select another permutation that is either the same length as that of the lead die, or one less. For example, if n = 12, we might select 2*3*2 for the lead die, and 3*4 for the second die. These numbers effectively tell you when to "switch track" when numbering the dice, and will uniquely result in the numbering: (0,1,6,7,12,13,72,73,78,79,84,85; 0,2,4,18,20,22,36,38,40,54,56,58).

a(n) is the number of (unordered) pairs of polynomials c(x) = x^c_1 + x^c_2 + ... + x^c_n, d(x) = x^d_1 + x^d_2 + ... + x^d_n with nonnegative integer exponents such that c(x)*d(x) = (x^(n^2)-1)/(x-1). - Alois P. Heinz, May 13 2016

a(n) is also the number of principal reversible squares of order n. - S. Harry White, May 19 2018

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

S. Harry White, Reversible Squares

FORMULA

a(n) = b_1^2 + b_2^2 + b_3^2 + ... + b_1*b_2 + b_2*b_3 + b_3*b_4 + ..., where b_k is the number of different permutations of k divisors of n to achieve a product of n.

For example for n=24, b_3 = 9 (6 permutations of 2*3*4 and 3 permutations of 2*2*6).

EXAMPLE

When n = 4, a(n) = 3; the three arrangements are (0,1,2,3; 0,4,8,12), (0,1,4,5; 0,2,8,10), (0,1,8,9; 0,2,4,6).

When n = 5, a(n) = 1; the sole arrangement is (0,1,2,3,4; 0,5,10,15,20).

CROSSREFS

Cf. A111588.

Sequence in context: A210442 A077202 A086665 * A050521 A266724 A247146

Adjacent sequences:  A273010 A273011 A273012 * A273014 A273015 A273016

KEYWORD

nonn,easy

AUTHOR

Elliott Line, May 13 2016

STATUS

approved

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Last modified August 19 13:43 EDT 2018. Contains 313863 sequences. (Running on oeis4.)