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

Author

Yasutoshi Kohmoto

Keywords

• nonn
• easy

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).

Examples

			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
		

Links

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

Crossrefs

Cf. A094926, A078510, A079421, A079422.

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

Extensions

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