A094926 A hexagonal spiral Fibonacci sequence.

0, 1, 1, 2, 3, 5, 8, 14, 23, 38, 63, 102, 168, 272, 445, 720, 1173, 1898, 3084, 5004, 8102, 13143, 21268, 34472, 55841, 90376, 146382, 237028, 383578, 621046, 1005341, 1626832, 2633338, 4262063, 6896574, 11161708, 18063264, 29233060, 47301328, 76547494, 123870067

Offset: 0

Author

Yasutoshi Kohmoto

Keywords

• nonn
• easy

Comments

Consider the following spiral:

..........a(5)..a(6)..a(7)

.......a(4)..a(0)..a(1)..a(8)

....a(13).a(3)..a(2)..a(9)

.......a(12).a(11).a(10)

Then a(0)=0, a(1)=1, a(n)=a(n-1)+Sum{a(i), a(i) is adjacent to a(n-1)}; 6 terms around a(m) touch a(m).

From Manfred Scheucher, Jun 03 2015: (Start)

Since a(n-1)+a(n-2) <= a(n) <= a(n-1)+a(n-2)+a(n-k)+a(n-k-1) holds for some k where k=Theta(sqrt(n)), and also 2^n >= a(n) >= F(n) holds, I believe that a(n) = (a(n-1)+a(n-2))/(1-c*d^(-sqrt(n))) can be proofen properly. This would lead to a similar asymptotic behavior as F(n), i.e., a(n) ~ c*phi^n where phi=1.61803... denotes the golden ratio and c=0.54172... is a constant.

Actually, the terms in the b-file seem to confirm this conjecture because exp(log(a(n))/n) seem to converge to phi. In particular, g(100)=1.60..., g(1000)=1.616..., g(10000)=1.6178..., g(30602)=1.61800..., where g(n):=exp(log(a(n))/n).

(End)

Examples

			Spiral with 2 rings:
... ..5 ... ..8 ... .14 ...
..3 ... ..0 ... ..1 ... .23
... ..2 ... ..1 ... .38 ...
... ... 102 ... .63 ... ...
Spiral with 3 rings:
...... ...... ..1173 ...... ..1898 ...... ..3084 ...... ..5004 ...... ......
...... ...720 ...... .....5 ...... .....8 ...... ....14 ...... ..8102 ......
...445 ...... .....3 ...... .....0 ...... .....1 ...... ....23 ...... .13143
...... ...272 ...... .....2 ...... .....1 ...... ....38 ...... .21268 ......
...... ...... ...168 ...... ...102 ...... ....63 ...... .34472 ...... ......
...... ...... ...... 146382 ...... .90376 ...... .55841 ...... ...... ......
		

Links

• Manfred Scheucher, Table of n, a(n) for n = 0..1322
• N. Fernandez, Spiro-Fibonacci numbers
• Vaclav Kotesovec, Graph - the asymptotic ratio
• Manfred Scheucher, Sage Script

Crossrefs

Cf. A094925, A078510, A079421, A079422.

Formula

a(n) ~ c*phi^n with phi=1.61803... being the golden ratio and c = A258639 = 0.54172002195814443386932... (conjectured). - Manfred Scheucher, Jun 03 2015

Extensions

Offset changed and more terms from Manfred Scheucher, Jun 03 2015