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 A005409 Number of polynomials of height n: a(n) = 2a(n-1) + a(n-2) + 2. (Formerly M3418) 20
 1, 1, 4, 11, 28, 69, 168, 407, 984, 2377, 5740, 13859, 33460, 80781, 195024, 470831, 1136688, 2744209, 6625108, 15994427, 38613964, 93222357, 225058680, 543339719, 1311738120, 3166815961, 7645370044, 18457556051, 44560482148, 107578520349, 259717522848 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Starting with n=1, the sum of the antidiagonals of the array in a comment from Cloitre regarding A002002. - Gerald McGarvey, Aug 12 2004 Cumulative sum of A001333. - Sture Sjöstedt, Nov 15 2011 a(n) = number of self-avoiding walks on a 3 rows x n columns grid of squares, starting top-left, ending bottom-left, using moves of R(ight), L(eft), U(p), D(own). E.g., for 3x1 there is just the path (D,D), and a(1) = 1. For 3x2, there are 4 paths (D,D) (D,R,D,L) (R,D,D,L) and (R,D,L,D) and a(2) = 4. - Toby Gottfried, Mar 04 2013 Define a triangle to have T(n,1) = n*(n-1)+1 and T(n,n) = n; the other terms T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1).  The sum of the terms in row(n+1) minus those in row(n) = a(n+2). - J. M. Bergot, Apr 30 2013 REFERENCES R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, 1941, p. 103. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 1..300 M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349. S. M. Diano, Letter to N. J. A. Sloane A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. 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). FORMULA a(n) = ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2))-1 for n>1, a(1)=1. G.f.: x*(1-2*x+2*x^2+x^3)/(1-3*x+x^2+x^3). - Paul D. Hanna, Feb 22 2005 a(n) = 3*a(n-1)-a(n-2)-a(n-3). - Toby Gottfried, Mar 08 2013 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, Feb 22 2005 (1, 4, 11, 28,...) = (1, 2, 2, 2,...) * the Pell sequence starting (1, 2, 5, 12, 29,...); such that, for example: a(5) = (2, 2, 2, 1) dot (1, 2, 5, 12) = (2 + 4 + 10 + 12) = 48. - Gary W. Adamson May 21 2013 MAPLE A005409:=(1-2*z+2*z**2+z**3)/(z-1)/(z**2+2*z-1); # Conjectured by Simon Plouffe in his 1992 dissertation. MATHEMATICA Join[{1}, RecurrenceTable[{a==1, a==4, a[n]==2a[n-1]+a[n-2]+2}, a[n], {n, 30}]] (* Harvey P. Dale, Jul 27 2011 *) Join[{1}, CoefficientList[Series[(x+1)/((x-1)*(x^2+2*x-1)), {x, 0, 40}], x]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *) PROG 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) (PARI) a(n)=polcoeff(1+x*(1+x)/(1-3*x+x^2+x^3)+x*O(x^n), n) \\ Paul D. Hanna (Haskell) a005409 n = a005409_list !! (n-1) a005409_list = 1 : scanl1 (+) (tail a001333_list) -- Reinhard Zumkeller, Jul 08 2012 CROSSREFS Equals A000129 - 1. Cf. A001333, A048654, A048655, A048745. Cf. A214931 (walks on grids with 4 rows), A006189 (grids with 3 columns). Cf. A216211 (grids with 4 columns). Sequence in context: A003230 A099326 A127985 * A245124 A020964 A113067 Adjacent sequences:  A005406 A005407 A005408 * A005410 A005411 A005412 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane, S. M. Diano EXTENSIONS Additional comments from Barry E. Williams STATUS approved

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Last modified June 25 17:47 EDT 2019. Contains 324353 sequences. (Running on oeis4.)