login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048739 Expansion of 1/((1 - x)*(1 - 2*x - x^2)). 63

%I

%S 1,3,8,20,49,119,288,696,1681,4059,9800,23660,57121,137903,332928,

%T 803760,1940449,4684659,11309768,27304196,65918161,159140519,

%U 384199200,927538920,2239277041,5406093003,13051463048,31509019100,76069501249

%N Expansion of 1/((1 - x)*(1 - 2*x - x^2)).

%C Partial sums of Pell numbers A000129.

%C W(n){1,3;2,-1,1} = Sum_{i=1..n} W(i){1,2;2,-1,0}, where W(n){a,b; p,q,r} implies x(n) = p*x(n-1) - q*x(n-2) + r; x(0)=a, x(1)=b.

%C Number of 2 X (n+1) binary arrays with path of adjacent 1's from upper left to lower right corner. - _R. H. Hardin_, Mar 16 2002

%C Binomial transform of A029744. - _Paul Barry_, Apr 23 2004

%C Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+2, s(0) = 1, s(n+2) = 3. - _Herbert Kociemba_, Jun 16 2004

%C Equals row sums of triangle A153346. - _Gary W. Adamson_, Dec 24 2008

%C Equals the sum of the terms of the antidiagonals of A142978. - _J. M. Bergot_, Nov 13 2012

%C a(p-2) == 0 mod p where p is an odd prime, see A270342. - _Altug Alkan_, Mar 15 2016

%C Also, the lexicographically earliest sequence of positive integers such that for n > 3, {sqrt(2)*a(n)} is located strictly between {sqrt(2)*a(n-1)} and {sqrt(2)*a(n-2)} where {} denotes the fractional part. - _Ivan Neretin_, May 02 2017

%C a(n+1) is the number of weak orderings on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n. - _J. Devillet_, Oct 06 2017

%D Allombert, Bill, Nicolas Brisebarre, and Alain Lasjaunias. "On a two-valued sequence and related continued fractions in power series fields." The Ramanujan Journal 45.3 (2018): 859-871. See Theorem 3, d_{4n+3}.

%H T. D. Noe, <a href="/A048739/b048739.txt">Table of n, a(n) for n = 0..200</a>

%H M. Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/bicknell.pdf">A Primer on the Pell Sequence and related sequences</a>, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.

%H M. Bicknell-Johnson and G. E. Bergum, <a href="http://dx.doi.org/10.1007/978-94-015-7801-1_18">The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K</a>, Applications of Fibonacci Numbers, 1986, pp. 193-205.

%H B. Bradie, <a href="https://projecteuclid.org/euclid.mjms/1312232719">Extensions and Refinements of some properties of sums involving Pell Numbers</a>, Miss. J. Math. Sci 22 (1) (2010) 37-43

%H M. Couceiro, J. Devillet, and J.-L. Marichal, <a href="http://arxiv.org/abs/1709.09162">Quasitrivial semigroups: characterizations and enumerations</a>, arXiv:1709.09162 [math.RA], 2017.

%H Jimmy Devillet, <a href="http://hdl.handle.net/10993/39776">On the single-peakedness property</a>, International summer school "Preferences, decisions and games" (Sorbonne Université, Paris, 2019).

%H I. M. Gessel, Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1065">Encyclopedia of Combinatorial Structures 1065</a>

%H Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee, Jung Hoon Han, <a href="https://arxiv.org/abs/1709.01344">Proposal of a spin-one chain model with competing dimer and trimer interactions</a>, arXiv:1709.01344 [cond-mat.str-el], 2017.

%H Ahmet Öteleş, <a href="https://dx.doi.org/10.1063/1.4992479">On the sum of Pell and Jacobsthal numbers by the determinants of Hessenberg matrices</a>, AIP Conference Proceedings 1863, 310003 (2017).

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1)

%F a(n) = 2*a(n-1) + a(n-2) + 1 with n > 1, a(0)=1, a(1)=3.

%F a(n) = ((2 + (3*sqrt(2))/2)*(1 + sqrt(2))^n - (2 - (3*sqrt(2))/2)*(1 - sqrt(2))^n )/(2*sqrt(2)) - 1/2.

%F a(0)=1, a(n+1) = ceiling(x*a(n)) for n > 0, where x = 1+sqrt(2). - _Paul D. Hanna_, Apr 22 2003

%F a(n) = 3*a(n-1) - a(n-2) - a(n-3). With two leading zeros, e.g.f. is exp(x)(cosh(sqrt(2)x)-1)/2. a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n+2, 2k+2)2^k. - _Paul Barry_, Aug 16 2003

%F -a(-3-n) = A077921(n). - _N. J. A. Sloane_, Sep 13 2003

%F E.g.f.: exp(x)(cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2. - _N. J. A. Sloane_, Sep 13 2003

%F a(n) = floor((1+sqrt(2))^(n+2)/4). - _Bruno Berselli_, Feb 06 2013

%F a(n) = (((1-sqrt(2))^(n+2) + (1+sqrt(2))^(n+2) - 2) / 4). - _Altug Alkan_, Mar 16 2016

%F 2*a(n) = A001333(n+2)-1. - _R. J. Mathar_, Oct 11 2017

%F a(n) = Sum_{k=0..n} binomial(n+1,k+1)*2^floor(k/2). - _Tony Foster III_, Oct 12 2017

%p a:=n->sum(fibonacci(i,2), i=0..n): seq(a(n), n=1..29); # _Zerinvary Lajos_, Mar 20 2008

%t Join[{a=1,b=3},Table[c=2*b+a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)

%t CoefficientList[Series[1/(1-3x+x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,3,8},30] (* _Harvey P. Dale_, Jun 13 2011 *)

%o (PARI) a(n)=local(w=quadgen(8));-1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n

%o (PARI) vector(100, n, n--; floor((1+sqrt(2))^(n+2)/4)) \\ _Altug Alkan_, Oct 07 2015

%o (PARI) Vec(1/((1-x)*(1-2*x-x^2)) + O(x^40)) \\ _Michel Marcus_, May 06 2017

%Y First row of table A083087.

%Y With a different offset, a(4n)=A008843(n), a(4n-2)=8*A001110(n), a(2n-1)=A001652(n).

%Y Cf. A001333, A048654, A048655, A083044, A083047, A083050, A153346.

%K easy,nice,nonn

%O 0,2

%A _Barry E. Williams_

%E Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 11 2002

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 15:09 EST 2019. Contains 329896 sequences. (Running on oeis4.)