A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 48, 50, 53, 56, 58, 61, 64, 66, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 141, 144, 146, 149, 152, 154, 157, 160, 162, 165, 168, 170, 173, 176, 178, 181, 184, 186, 189

Offset: 1

Author

Kival Ngaokrajang, Jan 28 2014

Keywords

• nonn

Comments

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

Links

• Kival Ngaokrajang, Illustration of irregular spirals (Center points 3..9)

Crossrefs

Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points); A236326, A236327 (5 center points).

Formula

Conjecture from Colin Barker, Jul 12 2014: (Start)

a(n) = a(n-1)+a(n-3)-a(n-4).

G.f.: x*(3*x^2+3*x+2) / ((x-1)^2*(x^2+x+1)). (End)