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 A094926 A hexagonal spiral Fibonacci sequence. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 EXTENSIONS Offset changed and more terms from Manfred Scheucher, Jun 03 2015 STATUS approved

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Last modified January 27 15:47 EST 2021. Contains 340467 sequences. (Running on oeis4.)