A166536 A product of consecutive doubled Fibonacci numbers.

1, 3, 6, 16, 40, 105, 273, 715, 1870, 4896, 12816, 33553, 87841, 229971, 602070, 1576240, 4126648, 10803705, 28284465, 74049691, 193864606, 507544128, 1328767776, 3478759201, 9107509825, 23843770275, 62423800998, 163427632720

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Formula

G.f.: (1 - 3*x^2 + x^3)/(1 - 3*x + 3*x^3 - x^4).

a(n) = F(n+1)*F(n+2) + (1 - (-1)^n)/2, where F = A000045.

a(n) = (F(n+2)*(1 + (-1)^n)/2 + F(n)*(1 - (-1)^n)/2)*(F(n+3)*(1 - (-1)^n)/2 + F(n+1)*(1 + (-1)^n)/2).

a(n)*a(n+2) - a(n+1)^2 = (-1)^n*(F(2*n+4) - 1).

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - G. C. Greubel, May 16 2016

a(n) = Sum_{i=0..n+1} F(i+1)*F(i-1), where F(-1) = 1. - Bruno Berselli, Feb 16 2017

Mathematica

LinearRecurrence[{3, 0, -3, 1}, {1, 3, 6, 16}, 30] (* G. C. Greubel, May 16 2016 *)

Prog

(Magma) /* From the sixth formula: */ F:=Fibonacci; [&+[F(i+1)*F(i-1): i in [0..n+1]]: n in [0..30]]; // Bruno Berselli, Feb 15 2017
(PARI) my(x='x+O('x^30)); Vec((1-3*x^2+x^3)/(1-3*x+3*x^3-x^4)) \\ G. C. Greubel, Jan 09 2019
(Sage) ((1-3*x^2+x^3)/(1-3*x+3*x^3-x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 09 2019
(GAP) a:=[1, 3, 6, 16];; for n in [5..30] do a[n]:=3*a[n-1]-3*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 09 2019

Crossrefs

Cf. A000045, A166516.

Sequence in context: A205770 A301959 A018022 * A128720 A096745 A293949

Adjacent sequences: A166533 A166534 A166535 * A166537 A166538 A166539

Keyword

nonn,easy

Author

Paul Barry, Oct 16 2009

Status

approved