A049440 - OEIS

A049440

a(n) = Fib(3*n)^2 - 2*Fib(3*n) + 4*Fib(3*n+1) + 5, where Fib = A000045.

9, 17, 105, 1313, 21385, 374833, 6688617, 119863873, 2150198793, 38580909137, 692294282601, 12422665840481, 222915477614473, 4000055027956849, 71778071199324777, 1288005210351490177

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OFFSET

0,1

COMMENTS

Continued fraction for sqrt(a(n)) has period n.

LINKS

Table of n, a(n) for n=0..15.

Index entries for linear recurrences with constant coefficients, signature (22, -71, -36, 73, 14, -1).

FORMULA

G.f.: (-x^5 - 91*x^4 + 534*x^3 + 370*x^2 - 181*x + 9)/((x^2-18*x+1)*(x^2-1)*(x^2+4*x-1)). - Michael Somos

a(n) = 22*a(n-1) - 71*a(n-2) - 36*a(n-3) + 73*a(n-4) + 14*a(n-5) - a(n-6).

MATHEMATICA

#[[1]]^2-2#[[1]]+4#[[2]]+5&/@Table[{Fibonacci[3n], Fibonacci[3n+1]}, {n, 0, 20}] (* or *) LinearRecurrence[{22, -71, -36, 73, 14, -1}, {9, 17, 105, 1313, 21385, 374833}, 20] (* Harvey P. Dale, Jul 07 2017 *)

PROG

(PARI) a(n) = fibonacci(3*n)^2 - 2*fibonacci(3*n) + 4*fibonacci(3*n+1) + 5 \\ Andrew Howroyd, Aug 10 2024

CROSSREFS

Cf. A000045 (Fibonacci numbers), A003285.

Sequence in context: A217965 A121442 A357567 * A177200 A177166 A073221

Adjacent sequences: A049437 A049438 A049439 * A049441 A049442 A049443

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved