A027973 a(n) = T(n,n) + T(n,n+1) + ... + T(n,2n), T given by A027960.

1, 4, 9, 21, 46, 99, 209, 436, 901, 1849, 3774, 7671, 15541, 31404, 63329, 127501, 256366, 514939, 1033449, 2072676, 4154701, 8324529, 16673534, 33386671, 66837421, 133778524, 267724809, 535721061, 1071881326, 2144473299, 4290096449, 8582053396, 17167117141

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Crossrefs

Cf. A000032, A000045, A027960.

Programs

GAP

List([0..40], n-> 2^(n+2) - Lucas(1,-1,n+2)[2]); # G. C. Greubel, Sep 26 2019

Magma

[2^n-Lucas(n): n in [2..40]]; // Vincenzo Librandi, May 05 2017

Maple

with(combinat): a[0]:=1: for n from 1 to 30 do a[n]:=2*a[n-1]+fibonacci(n+1)-fibonacci(n-3) od: seq(a[n],n=0..30); # Emeric Deutsch, Nov 29 2006

Mathematica

Table[2^n - LucasL[n], {n, 2, 50}] (* Vincenzo Librandi, May 05 2017 *)

PARI

vector(40, n, f=fibonacci; 2^(n+1) - f(n+2) - f(n) ) \\ G. C. Greubel, Sep 26 2019

Sage

[2^(n+2) - lucas_number2(n+2,1,-1) for n in (0..40)] # G. C. Greubel, Sep 26 2019

Formula

With a different offset: recurrence: a(-1)=a(0)=1 a(n+2) = a(n+1) + a(n) + 2^n; formula: a(n-2) = floor(2^n - phi^n) - (1-(-1)^n)/2. - Benoit Cloitre, Sep 02 2002

a(n) = A101220(4, 2, n+1) - A101220(4, 2, n). - Ross La Haye, Aug 05 2005

a(n) = 2*a(n-1) + Fibonacci(n+1) - Fibonacci(n-3) for n>=1; a(0)=1. - Emeric Deutsch, Nov 29 2006

O.g.f.: 4/(1-2*x) - (x+3)/(1-x-x^2). - R. J. Mathar, Nov 23 2007

a(n) = 2^(n+2) + F(n) - F(n+4) with F(n)=A000045(n). - Johannes W. Meijer, Aug 15 2010

Eigensequence of an infinite lower triangular matrix with the Lucas series (1, 3, 4, 7, ...) as the left border and the rest ones. - Gary W. Adamson, Jan 30 2012

a(n) = 2^(n+2) - Lucas(n+2). - Vincenzo Librandi, May 05 2017, corrected by Greg Dresden, Sep 13 2021