A331211 Number of green nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and one green node.

1, 15, 117, 891, 6777, 51543, 392013, 2981475, 22675761, 172461663, 1311666021, 9975943179, 75872547369, 577052549415, 4388802753213, 33379264377459, 253867706760033, 1930803860947887, 14684827767302997, 111686210555580315, 849435201142733529, 6460422977475127287

Offset: 0

Author

George Strand Vajagich, Mar 01 2020

Keywords

• nonn
• easy

Examples

			For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 2 to get 102 add 15 to get 117.
For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 2 to get 774 add 177 to get 891.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Thezebraherd user, Graph W multiplication, Youtube video.
• Index entries for linear recurrences with constant coefficients, signature (8,-3).

Crossrefs

Cf. A332936 (number of blue nodes).

Similar sequences with a cycle size 3..6 are: A007483, A048876, A189274(n+1), A054490.

Programs

PARI

Vec((1 + 7*x) / (1 - 8*x + 3*x^2) + O(x^20)) \\ 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(gl)

Formula

a(n) = a(n-1) + 2*b(n-1), b(n) = 2*a(n-1) + 7*b(n-1) with a(0) = 1 and b(0) = 7 where b(n) = A332936(n).

From Colin Barker, Mar 03 2020: (Start)

G.f.: (1 + 7*x) / (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*(-11 + sqrt(13)) + (4 + sqrt(13))^n*(11 + sqrt(13)))/(2*sqrt(13)).

E.g.f.: exp(4*x)*cosh(sqrt(13)*x) + (11*exp(4*x)*sinh(sqrt(13)*x))/sqrt(13).

(End)

Extensions

a(14)-a(21) from Stefano Spezia, Mar 03 2020

Typo in a(14) fixed by Colin Barker, Apr 26 2020