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A274309
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Related to integral generalized binomial coefficients of multiplicative functions (see Comments for precise definition).
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1
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2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1
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OFFSET
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1,1
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COMMENTS
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Let C(n,m)_f = ( Product_{i=1..n} f(i) ) / (( Product_{i=1..n-m} f(i) ) * ( Product_{i=1..m} f(i) )). [I would have expected the lower limits in these products to be 0 rather than 1. - N. J. A. Sloane, Jun 21 2016]
Set m=3, and for any multiplicative function f, construct a binary sequence S by writing 1 if C(n,3)_f is an integer, 0 if not.
The present sequence gives the lengths of the successive runs in B.
For example the initial terms 2,1,3,3,... mean that B begins 1,1,0,1,1,1,0,0,0,...
Periodic with period length 16.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).
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FORMULA
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G.f.: x*(2 - x + 4*x^2 - x^3 + 2*x^4 - x^5 + 4*x^6 + x^7 + 2*x^8 - x^9 + 2*x^10 + x^11 + 2*x^12 - x^13 + 3*x^14) / ((1 - x)*(1 + x^2)*(1 + x^4)*(1 + x^8)). - Colin Barker, Jun 22 2016
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PROG
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(PARI) Vec(x*(2 -x +4*x^2 -x^3 +2*x^4 -x^5 +4*x^6 +x^7 +2*x^8 -x^9 +2*x^10 +x^11 +2*x^12 -x^13 +3*x^14) / ((1 -x)*(1 +x^2)*(1 +x^4)*(1 +x^8)) + O(x^100)) \\ Colin Barker, Jun 22 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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