A079421 -id:A079421 - cp's OEIS frontend

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

Yasutoshi Kohmoto

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)

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

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

Cf. A094925, A078510, A079421, A079422.

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

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

A079422 a(n) = number of 1's in the first n^2 Spiro-Fibonacci differences.

Original entry on oeis.org

0, 3, 7, 11, 15, 19, 24, 26, 32, 41, 49, 55, 61, 67, 75, 78, 83, 111, 119, 135, 139, 155, 160, 168, 182, 217, 229, 249, 259, 279, 293, 303, 312, 342, 399, 454, 493, 530, 566, 603, 642, 681, 722, 765, 817, 875, 909, 958, 992, 1044, 1107, 1160, 1215, 1267, 1315

Offset: 1

Neil Fernandez, Jan 07 2003

nonn

			The Spiro-Fibonacci differences are 0,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,...(A079421). Terms are written in square boxes radiating spirally. a(n) = the sum of the first n^2 terms in A079421, i.e. the number of 1's in a spiral of height n and width n.

N. Fernandez, Graphical representations of some Spiro-Fibonacci sequences

Cf. A063826, A078510, A079421.

More terms from Sean A. Irvine, Aug 13 2025

A094925 A hexagonal spiral Fibonacci sequence.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 20, 34, 55, 90, 148, 240, 394, 638, 1043, 1688, 2750, 4450, 7232, 11736, 19002, 30827, 49884, 80856, 130978, 211982, 343348, 555964, 899706, 1456702, 2358089, 3815834, 6176654, 9996926, 16176330, 26180456, 42368468, 68567892

Offset: 1

Yasutoshi Kohmoto

Consider the following spiral:
.
          a(6)----a(7)----a(8)
          /                 \
         /                   \
        /                     \
      a(5)    a(1)----a(2)    a(9)
        \             /       /
         \           /       /
          \         /       /
  a(14)   a(4)----a(3)    a(10)
     \                    /
      \                  /
       \                /
      a(13)---a(12)---a(11)
.
Then a(1)=1, a(n) = a(n-1) + Sum_{a(i) adjacent to a(n-1)} a(i). Here 6 terms around a(m) touch a(m).

			a(2) = a(1) = 1,
a(3) = a(1) + a(2) = 2,
a(4) = a(1) + a(2) + a(3) = 4,
a(5) = a(1) + a(3) + a(4) = 7,
a(6) = a(1) + a(4) + a(5) = 12,
a(7) = a(1) + a(5) + a(6) = 20, etc.
Thus:
         12----20----34
         /             \
        /               \
       7     1-----1    55
        \         /     /
         \       /     /
  638     4-----2    90
     \               /
      \             /
     394---240---148

Manfred Scheucher, Table of n, a(n) for n = 1..1323
N. Fernandez, Spiro-Fibonacci numbers
Manfred Scheucher, Sage Script

Cf. A094926, A078510, A079421, A079422.

a(n) ~ c*phi^n with phi=1.61803... being the golden ratio and c = 0.78529667298898361017570049509486675274402985275383398273772345738007479334754... (conjectured). Cf. A094926. - Manfred Scheucher, Jun 03 2015

a(15)-a(38) from Nathaniel Johnston, Apr 26 2011

cp's OEIS Frontend

A094926 A hexagonal spiral Fibonacci sequence.

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A079422 a(n) = number of 1's in the first n^2 Spiro-Fibonacci differences.

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A094925 A hexagonal spiral Fibonacci sequence.

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