A332936 Number of blue nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and 1 green node.

7, 51, 387, 2943, 22383, 170235, 1294731, 9847143, 74892951, 569602179, 4332138579, 32948302095, 250590001023, 1905875101899, 14495230812123, 110244221191287, 838468077093927, 6377011953177555, 48500691394138659, 368874495293576607, 2805493888166196879, 21337327619448845211

Offset: 0

Author

George Strand Vajagich, Mar 02 2020

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• George Strand Vajagich, Youtube video explaining graph W multiplication, YouTube video.
• Index entries for linear recurrences with constant coefficients, signature (8,-3).

Crossrefs

Cf. A331211.

Programs

PARI

Vec((1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2) + O(x^40)) \\ Colin Barker, Mar 03 2020

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)

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)

a(n) = 7*A190976(n+1) -5*A190976(n). - R. J. Mathar, Apr 30 2020