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A005993
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G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2).
(Formerly M1576)
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27
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1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146, 182, 231, 280, 344, 408, 489, 570, 670, 770, 891, 1012, 1156, 1300, 1469, 1638, 1834, 2030, 2255, 2480, 2736, 2992, 3281, 3570, 3894, 4218, 4579, 4940, 5340, 5740, 6181, 6622, 7106, 7590, 8119
(list;
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OFFSET
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0,2
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COMMENTS
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Alkane (or paraffin) numbers l(6,n).
Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.
Also multidigraphs with loops on 2 nodes with n arcs (see A136564). - Vladeta Jovovic, Dec 27 1999
Euler transform of finite sequence [2,3,0,-1]. - Michael Somos, Mar 17 2004
a(n-2) is the number of plane partitions with trace 2. - Michael Somos, Mar 17 2004
With offset 4, a(n) is the number of bracelets with n beads, 3 of which are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). - Washington Bomfim, Aug 05 2008
Equals (1,2,3,4...) convolved with (1,0,3,0,5,...). - Gary W. Adamson, Feb 16 2009
Equals row sums of triangle A177878.
Equals (1/2)*((1, 4, 10, 20, 35, 56,...) + (1, 0, 2 0, 3, 0, 4,...)).
a(n) = length of run n+2 of consective 1s in A254338. - Reinhard Zumkeller, Feb 27 2015
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REFERENCES
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S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 96.
L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250. See page 218. MR1433171 (98i:13009)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Washington Bomfim, The 19 bracelets with 8 beads - one blue, three reds and four blacks. [From Washington Bomfim, Aug 05 2008]
Dragomir Z. Djokovic, Poincare series of some pure and mixed trace algebras of two generic matrices. See Table 8.
N. J. A. Sloane, Classic Sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
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FORMULA
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l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1+x^2) / ((1+x)^2*(x-1)^4) = (1/(1-x)^4 +1/(1-x^2)^2)/2.
a(2n)=(n+1)(2n^2+4n+3)/3, a(2n+1)=(n+1)(n+2)(2n+3)/3. a(-4-n)=-a(n).
From Yosu Yurramendi, Sep 12 2008: (Start)
a(n+1)= a(n) + A008794(n+3) with a(1)=1
a(n)= A027656 (n) + 2*A006918 (n)
a(n+2)= a(n) + A000982 (n+2) with a(1)=1, a(2)=2. (End)
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = (n^3 + 6*n^2 + 11*n + 6)/12 + ((n+2)/4)[n even] (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012
a(n) = (1/12)*n*(n+1)*(n+2) + (1/4)*(n+1)*(1/2)*(1-(-1)^n), with offset 1. - Yosu Yurramendi, Jun 20 2013
a(n) = sum( ceiling(i/2)*round(i/2), i=0..n+1 ) = sum( floor(i/2)^2, i=0..n+2 ). - Bruno Berselli, Aug 30 2013
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EXAMPLE
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a(2)=6 since ( x1*y1, x2*y2, x1*x1+y1*y1, x2*x2+y2*y2, x1*x2+y1*y2, x1*y2+x2*y1 ) are a basis for homogeneous quadratic invariant polynomials.
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MAPLE
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g := proc(n) local i; add(floor(i/2)^2, i=1..n+1) end: # Joseph S. Riel (joer(AT)k-online.com), Mar 22 2002
a:= n-> (Matrix([[1, 0$3, -1, -2]]).Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..44); # Alois P. Heinz, Jul 31 2008
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MATHEMATICA
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CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^2)^2), {x, 0, 44}], x] (* Jean-François Alcover, Apr 08 2011 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 2, 6, 10, 19, 28}, 50] (* Harvey P. Dale, Feb 20 2012 *)
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PROG
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(PARI) a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n), n)
(PARI) a(n) = (binomial(n+3, n) + (1-n%2)*binomial((n+2)/2, n>>1))/2. \\ Washington Bomfim, Aug 05 2008]
(Haskell) Following Gary W. Adamson.
import Data.List (inits, intersperse)
a005993 n = a005994_list !! n
a005993_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
tail $ inits [1..]
-- Reinhard Zumkeller, Feb 27 2015
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CROSSREFS
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Cf. A177878.
Partial sums of A008794 (without 0). - Bruno Berselli, Aug 30 2013
Cf. A254338, A002260, A005408.
Sequence in context: A054273 A127567 A169643 * A028247 A209535 A065054
Adjacent sequences: A005990 A005991 A005992 * A005994 A005995 A005996
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)
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STATUS
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approved
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