|
| |
|
|
A014335
|
|
Exponential convolution of Fibonacci numbers with themselves (divided by 2).
|
|
12
|
|
|
|
0, 0, 1, 3, 11, 35, 115, 371, 1203, 3891, 12595, 40755, 131891, 426803, 1381171, 4469555, 14463795, 46805811, 151466803, 490156851, 1586180915, 5132989235, 16610702131, 53753361203, 173949530931, 562912506675, 1821623137075, 5894896300851, 19076285150003
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
Cf. A000032, A000045, A014334, A081057, A203579.
Cf. (partial sums of) A063727.
Column k=2 of A346415.
Sequence in context: A107683 A259400 A320087 * A147474 A247417 A222286
Adjacent sequences: A014332 A014333 A014334 * A014336 A014337 A014338
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
STATUS
|
approved
|
| |
|
|
