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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

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

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 *)

Prog

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

Crossrefs

Cf. A000032, A000045, A027960.

Sequence in context: A048638 A144527 A117880 * A348407 A103040 A084861

Adjacent sequences: A027970 A027971 A027972 * A027974 A027975 A027976

Keyword

nonn

Author

Clark Kimberling

Status

approved