A056914 a(n) = Lucas(4*n+1).

1, 11, 76, 521, 3571, 24476, 167761, 1149851, 7881196, 54018521, 370248451, 2537720636, 17393796001, 119218851371, 817138163596, 5600748293801, 38388099893011, 263115950957276, 1803423556807921, 12360848946698171

Offset: 0

Author

Barry E. Williams, Jul 11 2000

Keywords

• easy
• nonn

Comments

a(n) = (t(i+4n+1) - t(i))/(t(i+2n+1) - t(i+2n)), where (t) is any sequence of the form t(n+2) = 4t(n+1) - 4t(n) + t(n-1) or of the form t(n+1) = 3t(n) - t(n-1) without regard to initial values as long as t(i+2n+1) - t(i+2n) != 0. - Klaus Purath, Jun 24 2024

References

• V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers, A Publication of the Fibonacci Association, Houghton Mifflin Co., 1969, pp. 27-29.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (7,-1).

Crossrefs

Cf. (A056914) = sqrt{5*(A033889)^2 - 4}.

Cf. quadrisection of A000032: A056854 (first), this sequence (second), A246453 (third, without 11), A288913 (fourth).

Programs

GAP

List([0..30], n-> Lucas(1,-1,4*n+1)[2] ); # G. C. Greubel, Jan 16 2020

Magma

[Lucas(4*n+1): n in [0..30]]; // G. C. Greubel, Dec 24 2017

Maple

with(combinat); seq(fibonacci(4*n+2)+fibonacci(4*n), n = 0..30); # G. C. Greubel, Jan 16 2020

Mathematica

LucasL[4*Range[0,30]+1] (* or *) LinearRecurrence[{7,-1}, {1,11}, 30] (* G. C. Greubel, Dec 24 2017 *)

PARI

my(x='x+O('x^30)); Vec((1+4*x)/(1-7*x+x^2)) \\ G. C. Greubel, Dec 24 2017

Sage

[lucas_number2(4*n+1,1,-1) for n in (0..30)] # G. C. Greubel, Jan 16 2020

Formula

a(n) = 7*a(n-1) - a(n-2), with a(0)=1, a(1)=11.

a(n) = (11*(((7+3*sqrt(5))/2)^n - ((7-3*sqrt(5))/2)^n) - (((7+3*sqrt(5))/2)^(n-1) - ((7-3*sqrt(5))/2)^(n-1)))/3*sqrt(5).

G.f.: (1+4*x)/(1-7*x+x^2). - Philippe Deléham, Nov 02 2008