A331211 - OEIS

%I #42 Apr 26 2020 15:17:17

%S 1,15,117,891,6777,51543,392013,2981475,22675761,172461663,1311666021,

%T 9975943179,75872547369,577052549415,4388802753213,33379264377459,

%U 253867706760033,1930803860947887,14684827767302997,111686210555580315,849435201142733529,6460422977475127287

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

%H Colin Barker, <a href="/A331211/b331211.txt">Table of n, a(n) for n = 0..1000</a>

%H Thezebraherd user, <a href="https://youtu.be/o7_L_Mo-xpU">Graph W multiplication</a>, Youtube video.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-3).

%F 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).

%F From _Colin Barker_, Mar 03 2020: (Start)

%F G.f.: (1 + 7*x) / (1 - 8*x + 3*x^2).

%F a(n) = 8*a(n-1) - 3*a(n-2) for n>1.

%F (End)

%F From _Stefano Spezia_, Mar 03 2020: (Start)

%F a(n) = ((4 - sqrt(13))^n*(-11 + sqrt(13)) + (4 + sqrt(13))^n*(11 + sqrt(13)))/(2*sqrt(13)).

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

%F (End)

%e For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 2 to get 102 add 15 to get 117.

%e For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 2 to get 774 add 177 to get 891.

%o (Python)

%o g=1

%o b=7

%o sg=0

%o sb=0

%o bl=[]

%o gl=[]

%o for int in range(1,20):

%o   sg=g*1+b*2

%o   sb=b*7+g*2

%o   g=sg

%o   b=sb

%o   gl.append(g)

%o   bl.append(b)

%o print(gl)

%o (PARI) Vec((1 + 7*x) / (1 - 8*x + 3*x^2) + O(x^20)) \\ _Colin Barker_, Mar 03 2020

%Y Cf. A332936 (number of blue nodes).

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

%K nonn,easy

%O 0,2

%A _George Strand Vajagich_, Mar 01 2020

%E a(14)-a(21) from _Stefano Spezia_, Mar 03 2020

%E Typo in a(14) fixed by _Colin Barker_, Apr 26 2020