OFFSET
0,4
COMMENTS
It can be noticed that A014335/A011782 is an "autosequence", that is a sequence which is identical to its inverse binomial transform, except for alternating signs. - Jean-François Alcover, Jun 15 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).
FORMULA
a(n) = A014334(n)/2.
G.f.: x^2/((1-x)*(1-2*x-4*x^2)). - Vladeta Jovovic, Mar 05 2003
E.g.f.: exp(x)*(cosh(sqrt(5)*x)-1)/5. - Vladeta Jovovic, Sep 01 2004
From Benoit Cloitre, Sep 25 2004: (Start)
a(n+1) = Sum_{i=0..n} A000045(i)*2^(i-1).
a(n) = (1/5)*(2^(n-1)*A000032(n) - 1). (End)
a(n) = 2*a(n-1) + 4*a(n-2) + 1, a(0)=0; a(1)=0. - Zerinvary Lajos, Dec 14 2008
G.f.: G(0)*x^2/(2*(1-x)^2), where G(k)= 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = (A203579(n) - 2)/5. - Vladimir Reshetnikov, Oct 06 2016
MAPLE
a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]+1 od: seq(a[n], n=0..29); # Zerinvary Lajos, Dec 14 2008
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <-4|2|3>>^n)[1, 3]:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 04 2016
MATHEMATICA
LinearRecurrence[{3, 2, -4}, {0, 0, 1}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)
PROG
(Magma) [(2^n*Lucas(n)-2)/10: n in [0..40]]; // G. C. Greubel, Jan 06 2023
(SageMath) [(2^n*lucas_number2(n, 1, -1) -2)/10 for n in range(41)] # G. C. Greubel, Jan 06 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved