This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094926 #27 Apr 19 2016 01:07:34
%S A094926 0,1,1,2,3,5,8,14,23,38,63,102,168,272,445,720,1173,1898,3084,5004,
%T A094926 8102,13143,21268,34472,55841,90376,146382,237028,383578,621046,
%U A094926 1005341,1626832,2633338,4262063,6896574,11161708,18063264,29233060,47301328,76547494,123870067
%N A094926 A hexagonal spiral Fibonacci sequence.
%C A094926 Consider the following spiral:
%C A094926 ..........a(5)..a(6)..a(7)
%C A094926 .......a(4)..a(0)..a(1)..a(8)
%C A094926 ....a(13).a(3)..a(2)..a(9)
%C A094926 .......a(12).a(11).a(10)
%C A094926 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).
%C A094926 From _Manfred Scheucher_, Jun 03 2015: (Start)
%C A094926 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.
%C A094926 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).
%C A094926 (End)
%H A094926 Manfred Scheucher, <a href="/A094926/b094926.txt">Table of n, a(n) for n = 0..1322</a>
%H A094926 N. Fernandez, <a href="http://www.borve.org/primeness/spirofib.html">Spiro-Fibonacci numbers</a>
%H A094926 Vaclav Kotesovec, <a href="/A094926/a094926.jpg">Graph - the asymptotic ratio</a>
%H A094926 Manfred Scheucher, <a href="/A094926/a094926.sage.txt">Sage Script</a>
%F A094926 a(n) ~ c*phi^n with phi=1.61803... being the golden ratio and c = A258639 = 0.54172002195814443386932... (conjectured). - _Manfred Scheucher_, Jun 03 2015
%e A094926 Spiral with 2 rings:
%e A094926 ... ..5 ... ..8 ... .14 ...
%e A094926 ..3 ... ..0 ... ..1 ... .23
%e A094926 ... ..2 ... ..1 ... .38 ...
%e A094926 ... ... 102 ... .63 ... ...
%e A094926 Spiral with 3 rings:
%e A094926 ...... ...... ..1173 ...... ..1898 ...... ..3084 ...... ..5004 ...... ......
%e A094926 ...... ...720 ...... .....5 ...... .....8 ...... ....14 ...... ..8102 ......
%e A094926 ...445 ...... .....3 ...... .....0 ...... .....1 ...... ....23 ...... .13143
%e A094926 ...... ...272 ...... .....2 ...... .....1 ...... ....38 ...... .21268 ......
%e A094926 ...... ...... ...168 ...... ...102 ...... ....63 ...... .34472 ...... ......
%e A094926 ...... ...... ...... 146382 ...... .90376 ...... .55841 ...... ...... ......
%Y A094926 Cf. A094925, A078510, A079421, A079422.
%K A094926 nonn,easy
%O A094926 0,4
%A A094926 _Yasutoshi Kohmoto_
%E A094926 Offset changed and more terms from _Manfred Scheucher_, Jun 03 2015