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A189334
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Expansion of g.f. (1-6*x+x^2)/(1-10*x+5*x^2).
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1
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1, 4, 36, 340, 3220, 30500, 288900, 2736500, 25920500, 245522500, 2325622500, 22028612500, 208658012500, 1976437062500, 18721080562500, 177328620312500, 1679680800312500, 15910164901562500, 150703245014062500, 1427481625632812500, 13521300031257812500, 128075592184414062500
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OFFSET
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0,2
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COMMENTS
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(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 2 0)
(1 0 2 0 1)
(0 2 0 2 0).
Then a(n)=(1/5)*Trace(A^(2*n)). (See also A189317.) (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of A_(N,r).
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LINKS
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FORMULA
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a(n) = 10*a(n-1) - 5*a(n-2), n>2, a(0)=1, a(1)=4, a(2)=36.
a(n) = (1/5)*Sum_{k=1..5) ((w_k)^3-2*w_k)^(2*n), w_k = 2*cos((2*k-1)*Pi/10).
a(n) = 2*((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/5 for n > 0.
E.g.f.: (1 + 4*exp(5*x)*cosh(2*sqrt(5)*x))/5. (End)
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MATHEMATICA
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LinearRecurrence[{10, -5}, {1, 4, 36}, 22] (* Stefano Spezia, Jul 09 2024 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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