%I A005409 M3418
%S A005409 1,1,4,11,28,69,168,407,984,2377,5740,13859,33460,80781,195024,
%T A005409 470831,1136688,2744209,6625108,15994427,38613964,93222357,
%U A005409 225058680,543339719,1311738120,3166815961,7645370044,18457556051
%N A005409 Number of polynomials of height n: a(n)=2a(n-1)+a(n-2)+2.
%C A005409 Starting with n=1, the sum of the antidiagonals of the array in a comment 
               from Cloitre regarding A002002 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Aug 12 2004
%D A005409 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005409 Marjorie Bicknell, A Primer on the Pell Sequence and related sequences, 
               Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
%D A005409 R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, 
               1941, p. 103.
%D A005409 A. F. Horadam, Special Properties of the Sequence W(n){a, b; p, q}, Fibonacci 
               Quarterly, Vol. 5, No. 5, 1967, pp. 424-434.
%D A005409 Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the 
               conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 
               55-64 (see p. 63).
%H A005409 T. D. Noe, <a href="b005409.txt">Table of n, a(n) for n=1..300</a>
%H A005409 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005409 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A005409 {[ (1+sqrt(2))^(n+2) - (1-sqrt(2))^(n+2) ]/2*sqrt(2)}-1.
%F A005409 a(n+3)_1 = A048745(n+1)_0 - A048739(n)_0 Note: FAMP returns A048739 with 
               an initial term of 0. Indeed, if A048739(-1) were 0, then a(n+2)_1 
               = A048745(n)_0 - A048739(n-1)_0. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Feb 22 2005
%F A005409 G.f.: x*(1-2*x+2*x^2+x^3)/(1-3*x+x^2+x^3). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Feb 22 2005
%p A005409 A005409:=(1-2*z+2*z**2+z**3)/(z-1)/(z**2+2*z-1); [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%o A005409 Floretion Algebra Multiplication Program, FAMP Code: 2jesforseq[ - .5'j 
               + 'k - .5j' + .5k' - .5'ii' - .5'ij' - 'ik' - .5'ji' - .5'ki' + .5'kj']; 
               ForType: 1A, LoopType: jes (first iteration)
%o A005409 (PARI) a(n)=polcoeff(1+x*(1+x)/(1-3*x+x^2+x^3)+x*O(x^n),n) (Hanna)
%Y A005409 Equals A000129 - 1.
%Y A005409 Cf. A001333, A000129, A048654, A048655.
%Y A005409 Cf. A048745.
%Y A005409 Sequence in context: A003230 A099326 A127985 this_sequence A020964 A113067 
               A152689
%Y A005409 Adjacent sequences: A005406 A005407 A005408 this_sequence A005410 A005411 
               A005412
%K A005409 nonn,easy,nice
%O A005409 1,3
%A A005409 N. J. A. Sloane (njas(AT)research.att.com), S. M. Diano.
%E A005409 Additional comments from Barry E. Williams