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 ...... ...... ......
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Offset changed and more terms from Manfred Scheucher, Jun 03 2015
STATUS
approved