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Search: id:A111588
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| A111588 |
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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. |
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2
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| 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, 141, 2, 8, 2, 8, 1, 33, 2, 33, 2, 2, 1, 126, 1, 2, 8, 71, 2, 13, 1, 8, 2, 13, 1, 350, 1, 2, 8, 8, 2, 13, 1, 140, 10, 2, 1, 123, 2, 2, 2, 33, 1, 118, 2
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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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?
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REFERENCES
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M. Gardner, "Penrose Tiles to Trapdoor Ciphers", p. 266.
D. J. Newman, A Problem Seminar, Springer; see Problem #88.
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LINKS
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Matthew M. Conroy, Home page (listed instead of email address)
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EXAMPLE
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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.
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CROSSREFS
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Sequence in context: A077191 A050363 A166974 this_sequence A070972 A075997 A161309
Adjacent sequences: A111585 A111586 A111587 this_sequence A111589 A111590 A111591
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2005
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EXTENSIONS
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Edited and extended by Matthew Conroy (list1(AT)madandmoonly.com), Jan 16 2006
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