George Strand Vajagich - cp's OEIS frontend

User: George Strand Vajagich

George Strand Vajagich has authored 2 sequences.

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

George Strand Vajagich, Mar 02 2020

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.

			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.

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

Cf. A331211.

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

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)

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

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

Original entry on oeis.org

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

Offset: 0

George Strand Vajagich, Mar 01 2020

			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.

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).

Cf. A332936 (number of blue nodes).

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

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

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)

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)

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

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

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User: George Strand Vajagich

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

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