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

0,4

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)

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

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

Example

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 ...... ...... ......

Crossrefs

Cf. A094925, A078510, A079421, A079422.

Sequence in context: A306274 A282240 A004692 * A225391 A078843 A018068

Adjacent sequences: A094923 A094924 A094925 * A094927 A094928 A094929

Keyword

nonn,easy

Author

Yasutoshi Kohmoto

Extensions

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

Status

approved