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A292290
Number of vertices of type A at level n of the hyperbolic Pascal pyramid.
1
0, 0, 3, 6, 12, 27, 66, 168, 435, 1134, 2964, 7755, 20298, 53136, 139107, 364182, 953436, 2496123, 6534930, 17108664, 44791059, 117264510, 307002468, 803742891, 2104226202, 5508935712, 14422580931, 37758807078, 98853840300, 258802713819, 677554301154
OFFSET
0,3
LINKS
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (1st line of Table 1).
FORMULA
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 4.
From Colin Barker, Sep 17 2017: (Start)
G.f.: 3*x^2*(1 - 2*x) / ((1 - x)*(1 - 3*x + x^2)).
a(n) = 3*(1 + (2^(-1-n)*((7-3*sqrt(5))*(3+sqrt(5))^n - (3-sqrt(5))^n*(7+3*sqrt(5)))) / sqrt(5)) for n>0.
(End)
a(n) = 3*(Fibonacci(2*n - 4) + 1) for n > 0. - Ehren Metcalfe, Apr 18 2019
MATHEMATICA
CoefficientList[Series[3*x^2*(1 - 2*x)/((1 - x)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)
PROG
(PARI) concat(vector(2), Vec(3*x^2*(1 - 2*x) / ((1 - x)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017
CROSSREFS
Cf. A264236.
Sequence in context: A290997 A052103 A072168 * A018011 A025208 A245774
KEYWORD
nonn,easy
AUTHOR
Eric M. Schmidt, Sep 13 2017
STATUS
approved