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A005314
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For n = 0, 1, 2, a(n) = n; thereafter, a(n) = 2*a(n-1)-a(n-2)+a(n-3).
(Formerly M0709)
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16
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0, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151, 265, 465, 816, 1432, 2513, 4410, 7739, 13581, 23833, 41824, 73396, 128801, 226030, 396655, 696081, 1221537, 2143648, 3761840, 6601569, 11584946, 20330163, 35676949, 62608681, 109870576, 192809420, 338356945, 593775046
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of compositions of n into parts congruent to {1,2} mod 4. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 10 2005
a(n)/a(n-1) tends to 1.75487766625...; an eigenvalue of the matrix M and a root to the characteristic polynomial. - Gary W. Adamson, May 25 2007
Starting with offset 1 = INVERT transform of (1, 1, 0, 0, 1, 1, 0, 0,...). [From Gary W. Adamson, May 04 2009]
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REFERENCES
| R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..400
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 426
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
| G.f.: x/(1-2*x+x^2-x^3). a(n) = Sum_{k=0..[(2n-1)/3]} binomial(n-1-[k/2], k), where [x]=floor(x). - Paul D. Hanna, Oct 22 2004
a(n) = Sum [k=0..n, C(n-k, 2k+1) ].
23*a_n = 3*P_{2n+2} + 7*P_{2n+1} - 2*P_{2n}, where P_n are the Perrin numbers, A001608. - D. E. Knuth, Dec 09 2008
G.f. (z-1)*(1+z**2)/(-1+2*z+z**3-z**2) for the augmented version 1, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151,... was given in S. Plouffe's thesis of 1992.
a(n) = a(n-1)+a(n-2)+a(n-4) = a(n-2)+A049853(n-1) = a(n-1)+A005251(n) = sum_{i <= n} (A005251(i)).
a(n) = Sum(binomial(n-k, 2k+1), {k=0...(n-1)/3}) - Richard Ollerton (r.ollerton(AT)uws.edu.au), May 12 2004
M^n*[1,0,0] = [a(n-2), a(n-1), a]; where M = the 3 X 3 matrix [0,1,0; 0,0,1; 1,-1,2]. Example M^5*[1,0,0] = [3,5,9]. - Gary W. Adamson, May 25 2007
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MATHEMATICA
| LinearRecurrence[{2, -1, 1}, {0, 1, 2}, 100] (* From Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
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PROG
| (PARI) a(n)=sum(k=0, (2*n-1)\3, binomial(n-1-k\2, k))
(Haskell)
a005314 n = a005314_list !! n
a005314_list = 0 : 1 : 2 : zipWith (+) a005314_list
(tail $ zipWith (-) (map (2 *) $ tail a005314_list) a005314_list)
-- Reinhard Zumkeller, Oct 14 2011
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CROSSREFS
| Equals row sums of triangle A099557. - Paul D. Hanna, Oct 22 2004
Cf. A099557, A005251.
Sequence in context: A134009 A018160 A079960 * A099529 A088352 A002572
Adjacent sequences: A005311 A005312 A005313 * A005315 A005316 A005317
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms and additional formulae from Henry Bottomley (se16(AT)btinternet.com), Jul 21 2000
Plouffe's g.f. edited by R. J. Mathar, May 12 2008
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