OFFSET
0,1
COMMENTS
The series of green nodes in n-th power W exponentiation for all n<6 n blue 1 green, 2 edge per node graphs already corresponds with an existing OEIS sequence (empirical). For example the number of blue nodes in n-th power W exponentiation of a square containing 3 blue nodes and 1 green node corresponds to A163063.
LINKS
George Strand Vajagich, Youtube video explaining graph W multiplication, YouTube video.
Index entries for linear recurrences with constant coefficients, signature (8,-3).
FORMULA
g(n) = g(n-1) + 2*a(n-1), a(n) = 2*g(n-1) + 7*a(n-1) with g(0) = 1 and b(0) = 7, where g(n) = A332211(n).
From Colin Barker, Mar 03 2020: (Start)
G.f.: (1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2).
a(n) = 8*a(n-1) - 3*a(n-2) for n > 1.
(End)
From Stefano Spezia, Mar 03 2020: (Start)
a(n) = ((4 - sqrt(13))^n*(-23 + 7*sqrt(13)) + (4 + sqrt(13))^n*(23 + 7*sqrt(13)))/(2*sqrt(13)).
E.g.f.: exp(4*x)*(91*cosh(sqrt(13)*x) + 23*sqrt(13)*sinh(sqrt(13)*x))/13.
(End)
EXAMPLE
For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 7 to get 357 add 30 to get 387.
For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 7 to get 774 add 234 to get 2943.
PROG
(Python)
g=1
b=7
sg=0
sb=0
bl=[]
gl=[]
for int in range(1, 20):
sg=g*1+b*2
sb=b*7+g*2
g=sg
b=sb
gl.append(g)
bl.append(b)
print(bl)
(PARI) Vec((1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2) + O(x^40)) \\ Colin Barker, Mar 03 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
George Strand Vajagich, Mar 02 2020
STATUS
approved