A078477 - OEIS

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A078477

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

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; refs; listen; history; internal format)

	OFFSET	`2,4`
	COMMENTS	`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	`Cf. A000225. [From Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 16 2009]` `Sequence in context: A027187 A056508 A050065 * A098832 A107985 A114999` `Adjacent sequences: A078474 A078475 A078476 * A078478 A078479 A078480`
	KEYWORD	`nonn`
	AUTHOR	`Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 03 2003`

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Last modified February 17 14:50 EST 2012. Contains 206050 sequences.