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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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+0
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: A066389 A077191 A050363 this_sequence A070972 A075997 A029196
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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njas, 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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