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Search: id:A005409
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| A005409 |
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Number of polynomials of height n: a(n)=2a(n-1)+a(n-2)+2.
(Formerly M3418)
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+0
8
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| 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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Marjorie Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, 1941, p. 103.
A. F. Horadam, Special Properties of the Sequence W(n){a, b; p, q}, Fibonacci Quarterly, Vol. 5, No. 5, 1967, pp. 424-434.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..300
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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{[ (1+sqrt(2))^(n+2) - (1-sqrt(2))^(n+2) ]/2*sqrt(2)}-1.
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
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
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MAPLE
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A005409:=(1-2*z+2*z**2+z**3)/(z-1)/(z**2+2*z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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PROGRAM
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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) (Hanna)
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CROSSREFS
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Equals A000129 - 1.
Cf. A001333, A000129, A048654, A048655.
Cf. A048745.
Sequence in context: A003230 A099326 A127985 this_sequence A020964 A113067 A152689
Adjacent sequences: A005406 A005407 A005408 this_sequence A005410 A005411 A005412
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), S. M. Diano.
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EXTENSIONS
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Additional comments from Barry E. Williams
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