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%I #141 Jul 22 2020 02:25:23
%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 Wipawee Tangjai, <a href="http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/4478">A Non-standard Ternary Representation of Integers</a>, Thai J. Math (2020) Special Issue: Annual Meeting in Mathematics 2019, 269-283.
%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
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