A094925 A hexagonal spiral Fibonacci sequence.

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

1,3

Comments

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).

Links

Manfred Scheucher, Table of n, a(n) for n = 1..1323

N. Fernandez, Spiro-Fibonacci numbers

Manfred Scheucher, Sage Script

Formula

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

Example

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

Crossrefs

Cf. A094926, A078510, A079421, A079422.

Sequence in context: A005182 A329397 A172524 * A289168 A340217 A289000

Adjacent sequences: A094922 A094923 A094924 * A094926 A094927 A094928

Keyword

nonn,easy

Author

Yasutoshi Kohmoto

Extensions

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

Status

approved