OFFSET
0,2
COMMENTS
Apart from the initial 1, this sequence is simply twice the Pell numbers, A000129. - Antonio Alberto Olivares, Dec 31 2003
Image of 1/(1-2x) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005
The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators = A052542 and denominators = A001333. - Clark Kimberling, Aug 26 2008
a(n) is the number of generalized compositions of n+1 when there are 2*i-2 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010
Apart from the initial 1, this is the p-INVERT transform of (1,0,1,0,1,0,...) for p(S) = 1 - 2 S. See A291219. - Clark Kimberling, Sep 02 2017
Conjecture: Apart from the initial 1, a(n) is the number of compositions of two types of n having no even parts. - Gregory L. Simay, Feb 17 2018
For n>0, a(n+1) is the length of tau^n(10) where tau is the morphism: 1 -> 101, 0 -> 1. See Song and Wu. - Michel Marcus, Jul 21 2020
The above conjecture is true, as the g.f. can be written as 1/(1 - (2*x)/(1 - x^2)). - John Tyler Rascoe, Jun 01 2024
LINKS
Iain Fox, Table of n, a(n) for n = 0..2500 (first 1001 terms from Vincenzo Librandi)
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
C. P. de Andrade, J. P. de Oliveira Santos, E. V. P. da Silva and K. C. P. Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From N. J. A. Sloane, Feb 20 2013
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 477
S. Kitaev and J. Remmel, The 1-box pattern on pattern avoiding permutations, arXiv:1305.6970 [math.CO], 2013.
Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See p.2 and 4.
Index entries for linear recurrences with constant coefficients, signature (2,1).
FORMULA
G.f.: (1 - x^2)/(1 - 2*x - x^2).
Recurrence: a(0)=1, a(2)=4, a(1)=2, a(n) + 2*a(n+1) - a(n+2) = 0;
a(n) = Sum_{alpha = RootOf(-1+2*x+x^2)} (1/2)*(1-alpha)*alpha^(-n-1).
a(n) = 2*A001333(n-1) + a(n-1), n > 1. A001333(n)/a(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
Binomial transform of A094024. a(n) = 0^n + ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/sqrt(2). - Paul Barry, Apr 22 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, k)2^(n-2k). - Paul Barry, Jan 16 2005
If p[i] = 2*(i mod 2) and if A is Hessenberg matrix of order n defined by A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i=j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det A. - Milan Janjic, May 02 2010
a(n) = round(sqrt(Pell(2n) + Pell(2n-1))). - Richard R. Forberg, Jun 22 2014
MAPLE
spec := [S, {S=Sequence(Prod(Union(Z, Z), Sequence(Prod(Z, Z))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
A052542 := proc(n)
option remember;
if n <=2 then
2^n;
else
2*procname(n-1)+procname(n-2) ;
end if;
end proc: # R. J. Mathar, Sep 23 2016
A052542List := proc(m) local A, P, n; A := [1, 2]; P := [1, 1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-2]]);
A := [op(A), P[-1]] od; A end: A052542List(31); # Peter Luschny, Mar 26 2022
MATHEMATICA
Join[{1}, LinearRecurrence[{2, 1}, {2, 4}, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2012 *)
PROG
(PARI) Vec((1-x^2)/(1-2*x-x^2) +O(x^40)) \\ Charles R Greathouse IV, Nov 20 2011
(Haskell)
a052542 n = a052542_list !! n
a052542_list = 1 : 2 : 4 : tail (zipWith (+)
(map (* 2) $ tail a052542_list) a052542_list)
-- Reinhard Zumkeller, Feb 24 2015
(Magma) I:=[2, 4]; [n le 2 select I[n] else 2*Self(n-1) +Self(n-2): n in [1..40]]; // G. C. Greubel, May 09 2019
(Sage) ((1-x^2)/(1-2*x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019
(GAP) a:=[2, 4];; for n in [3..40] do a[n]:=2*a[n-1]+a[n-2]; od; a; # G. C. Greubel, May 09 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved