id:A078477 - OEIS Search Results

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Number of knots with n crossings and rational unknotting number = 1 (chiral pairs counted only once).

+0
5

1, 1, 1, 3, 3, 6, 7, 15, 15, 30, 31, 63, 63, 126, 127, 255, 255, 510, 511, 1023, 1023, 2046, 2047, 4095, 4095, 8190, 8191, 16383, 16383, 32766, 32767, 65535, 65535, 131070, 131071, 262143, 262143, 524286, 524287, 1048575, 1048575, 2097150, 2097151 (list; graph; listen)

	OFFSET	`2,4`
	COMMENT	`Contribution from Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 16 2009: (Start)` `For n>1 a(2n) = 2^(n-1) - 1 = A000225(n-1).` `For n>1 a(4n-1) = a(4n) - 1 = 2^(2n-1) - 2.` `For n>0 a(4n+1) = a(4n+2) = 2^(2n) - 1. (End)`
	LINKS	`A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots`
	FORMULA	`G.f.: x^3+x^4(x+1)(2/(1-2x^2)+1/(x^2-1))+x^8/(x^4-1) = x^3(2x^7+2x^6-x^5+x^3-x^2+x+1)/((x-1)(x+1)(x^2+1)(2x^2-1)).`
	CROSSREFS	`Sequence in context: A027187 A056508 A050065 this_sequence A098832 A107985 A114999` `Cf. A000225. [From Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 16 2009]` `Adjacent sequences: A078474 A078475 A078476 this_sequence A078478 A078479 A078480`
	KEYWORD	`nonn,new`
	AUTHOR	`Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 03 2003`

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Last modified November 25 20:09 EST 2009. Contains 167514 sequences.

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