|
| |
|
|
A111588
|
|
Crazy Dice: number of ways to design a pair of n-sided dice with positive integers on their faces, so that the sums when they are tossed occur with the same probabilities as if a pair of standard n-sided dice were tossed.
|
|
1
| |
|
|
1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 8, 1, 2, 2, 10, 1, 8, 1, 8, 2, 2, 1, 33, 2, 2, 4, 8, 1, 13, 1, 26, 2, 2, 2, 57, 1, 2, 2, 33, 1, 13, 1, 8, 8, 2, 1, 140, 2, 8, 2, 8, 1, 33, 2, 33, 2, 2, 1, 125, 1, 2, 8, 71, 2, 13, 1, 8, 2, 13, 1, 348, 1, 2, 8, 8, 2, 13, 1, 140, 10, 2, 1, 122, 2, 2, 2, 33, 1, 118, 2
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,4
|
|
|
COMMENTS
| It is not required that the two dice be identical, it is not required that the entries be bounded by n and we do not ask that the entries be distinct from one another on each cube.
We pretend for the purpose of this sequence that regular n-sided dice exist for all n.
In other words, how many (unordered) pairs of polynomials B(x) = x^b_1 + x^b_2 + ... + x^b_n, C(x) = x^c_1 + x^c_2 + ... + x^c_n, are there with all exponents positive integers, such that B(x)*C(x) = (x+x^2+x^3+...+x^n)^2?
|
|
|
REFERENCES
| M. Gardner, "Penrose Tiles to Trapdoor Ciphers", p. 266.
D. J. Newman, A Problem Seminar, Springer; see Problem #88.
|
|
|
LINKS
| Matthew M. Conroy, Home page (listed instead of email address)
|
|
|
EXAMPLE
| The first nontrivial example is for n=4: {1,2,2,3} and {1,3,3,5} together have the same sum probabilities as a pair of {1,2,3,4}. That is, (x+2x^2+x^3)(x+2x^3+x^5)=(x+x^2+x^3 +x^4)^2.
|
|
|
CROSSREFS
| Sequence in context: A077191 A050363 A166974 * A070972 A180229 A075997
Adjacent sequences: A111585 A111586 A111587 * A111589 A111590 A111591
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2005
|
|
|
EXTENSIONS
| Edited and extended by Matthew Conroy (list1(AT)madandmoonly.com), Jan 16 2006
Correction to some terms, thanks to Adam Chalcraft. - Matthew Conroy (list1(AT)madandmoonly.com), Apr 04 2010
|
| |
|
|
