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

0,1

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

Table of n, a(n) for n=0..21.

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)

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

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

Cf. A331211.

Sequence in context: A285880 A147958 A104454 * A222849 A273055 A019472

Adjacent sequences: A332933 A332934 A332935 * A332937 A332938 A332939

Keyword

nonn,easy

Author

George Strand Vajagich, Mar 02 2020

Status

approved