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

0,2

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).

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)

Example

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.

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(gl)
(PARI) Vec((1 + 7*x) / (1 - 8*x + 3*x^2) + O(x^20)) \\ Colin Barker, Mar 03 2020

Crossrefs

Cf. A332936 (number of blue nodes).

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

Sequence in context: A125352 A126510 A328725 * A183475 A253804 A161476

Adjacent sequences: A331208 A331209 A331210 * A331212 A331213 A331214

Keyword

nonn,easy

Author

George Strand Vajagich, Mar 01 2020

Extensions

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

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

Status

approved