cp's OEIS Frontend

Let points 1, 2, 3 & 4 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 3, 4, 1, ... and so on. The form is expanded spiral. See illustration in links.

Let points 2, 1, 3 & 4 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 3, 4, 1, ... and so on. The form is non-expanded loop.

The alternative point order [2, 3, 1, 4] gives the same pattern with reflection, but the sequence will be 2*A047215(n). See illustration in links.

Conjecture: Numbers equivalent 0 or 3 modulo 10. - Ralf Stephan, Jan 13 2014

Let points 1, 2, 3, 4, and 5 be placed on a horizontal straight line at intervals of 1 unit. (See illustration under Links.) Construct a continuous curve from half circles, alternately lying below and above the line, centered at points 1, 2, 3, 4, 5, 1, ... and so on, with the first half circle (centered at point 1) beginning at point 2.

Also floor(a(n)/3) = A047607(n+1).

Conjecture: All forms of 120 permutations of 5 center points are non-expanded loops.

Let points 1, 2, 3, 4 & 5 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle; by selecting radius point on the right hand side of point 1 (pattern 2); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 4, 5, 1, ... and so on.

Conjecture: All forms of 120 permutations 5 center points are non-expanded loops.

Let points 2, 3, & 1 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle; by selecting radius point on the left hand side of point 1 (pattern 1); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 1, 2, 3, ... and so on.

Conjecture: All forms of 3 center points are non-expanded loops.

There are other sets of center points that give the same sequence, e.g.: [2,3,1,4]; [3,2,4,1]; [3,2,4,1,5]; [2,3,1,4,5,7,6]; [2,3,1,7,4,6,5]; [3,4,2,5,1,6,7]; [4,3,5,6,2,7,1]; [4,5,3,2,1,6,7]; [5,4,6,3,2,7,1].

Also, there are some similar patterns that give difference sequences, e.g.:

A047622: [1,2,7,3,4,6,5]; [1,2,7,6,3,5,4]...

A047399: [1,2,7,3,6,4,5]; [1,2,7,6,5,3,4]...

A047395: [2,3,1,4 7,5,6]; [2,3,1,7,6,4,5]...

A047464: [4,5,3,6,2,7,1]; [1,8,2,7,3,6,4,5];

[9,1,8,2,7,3,6,4,5].

See illustration in links.

Appears to be basically a duplicate of A047618. - R. J. Mathar, Feb 03 2014

See A236535 for construction rule.

Conjecture: (1) For odd n all forms are non-expanded loops and equal to n!. (2) For n mod 4 = 0, there are both non-expanded loops and expanded spirals.

See A236535 for construction rule.

Conjecture: (1) For even n; except n mod 4 = 0; all forms are expanded spirals and equal to n!. (2) For n mod 4 = 0, there are both non-expanded loops and expanded spirals.