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A094925 A hexagonal spiral Fibonacci sequence.

%I A094925 #20 May 04 2021 01:06:59
%S A094925 1,1,2,4,7,12,20,34,55,90,148,240,394,638,1043,1688,2750,4450,7232,
%T A094925 11736,19002,30827,49884,80856,130978,211982,343348,555964,899706,
%U A094925 1456702,2358089,3815834,6176654,9996926,16176330,26180456,42368468,68567892
%N A094925 A hexagonal spiral Fibonacci sequence.
%C A094925 Consider the following spiral:
%C A094925 .
%C A094925           a(6)----a(7)----a(8)
%C A094925           /                 \
%C A094925          /                   \
%C A094925         /                     \
%C A094925       a(5)    a(1)----a(2)    a(9)
%C A094925         \             /       /
%C A094925          \           /       /
%C A094925           \         /       /
%C A094925   a(14)   a(4)----a(3)    a(10)
%C A094925      \                    /
%C A094925       \                  /
%C A094925        \                /
%C A094925       a(13)---a(12)---a(11)
%C A094925 .
%C A094925 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).
%H A094925 Manfred Scheucher, <a href="/A094925/b094925.txt">Table of n, a(n) for n = 1..1323</a>
%H A094925 N. Fernandez, <a href="http://www.borve.org/primeness/spirofib.html">Spiro-Fibonacci numbers</a>
%H A094925 Manfred Scheucher, <a href="/A094925/a094925.sage.txt">Sage Script</a>
%F A094925 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
%e A094925 a(2) = a(1) = 1,
%e A094925 a(3) = a(1) + a(2) = 2,
%e A094925 a(4) = a(1) + a(2) + a(3) = 4,
%e A094925 a(5) = a(1) + a(3) + a(4) = 7,
%e A094925 a(6) = a(1) + a(4) + a(5) = 12,
%e A094925 a(7) = a(1) + a(5) + a(6) = 20, etc.
%e A094925 Thus:
%e A094925          12----20----34
%e A094925          /             \
%e A094925         /               \
%e A094925        7     1-----1    55
%e A094925         \         /     /
%e A094925          \       /     /
%e A094925   638     4-----2    90
%e A094925      \               /
%e A094925       \             /
%e A094925      394---240---148
%Y A094925 Cf. A094926, A078510, A079421, A079422.
%K A094925 nonn,easy
%O A094925 1,3
%A A094925 _Yasutoshi Kohmoto_
%E A094925 a(15)-a(38) from _Nathaniel Johnston_, Apr 26 2011