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A028242
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Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8.
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31
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1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 12, 11, 13, 12, 14, 13, 15, 14, 16, 15, 17, 16, 18, 17, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22, 24, 23, 25, 24, 26, 25, 27, 26, 28, 27, 29, 28, 30, 29, 31, 30, 32, 31, 33, 32, 34, 33, 35, 34, 36, 35, 37, 36, 38
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| A two-way infinite sequences which is palindromic (up to sign). - Michael Somos, Mar 21, 2003
Number of permutations of [n+1] avoiding the patterns 123, 132 and 231 and having exactly one fixed point. Example: a(0) because we have 1; a(2)=2 because we have 213 and 321; a(3)=1 because we have 3214. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 17 2005
The ring of invariants for the standard action of Quaternions on C^2 is generated by x^4 + y^4, x^2 * y^2, and x * y * (x^4 - y^4). - Michael Somos Mar 14 2011
A000027 and A001477 interleaved. - Omar E. Pol, Feb 06 2012
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REFERENCES
| D. Benson, Polynomial Invariants of Finite Groups, Cambridge, p. 23.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 15.
M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 97.
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 90.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418; Theorem 3.3.
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LINKS
| Index entries for two-way infinite sequences
Index entries for Molien series
aglearner, A question about an application of Molien's formula to find the generators and relations of an invariant ring
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FORMULA
| Expansion of the Molien series for standard action of Quaternions on C^2: (1 + t^6) / (1 - t^4)^2 = (1 - t^12) / ((1 - t^4)^2 * (1 - t^6)) in powers of t^2.
Euler transform of length 6 sequence [ 0, 2, 1, 0, 0, -1]. - Michael Somos Mar 14 2011
a(n) = n-a(n-1) [with a(0) = 1] = A000035(n-1)+A004526(n) [noting that A004526 offset by 1 also satisfies a(n) = n-a(n-1) but with a(0) = 0]. - Henry Bottomley (se16(AT)btinternet.com), Jul 25 2001
G.f.: (1 - x + x^2) / ((1 - x) * (1 - x^2)). a(2*n) = n + 1, a(2*n + 1) = n, a(-1 - n) = -a(n). a(n) = a(n - 1) + a(n - 2) - a(n - 3).
a(n) = floor(n/2) + 1 - n mod 2. a(2*k) = k+1, a(2*k+1) = k; A110657(n) = a(a(n)), A110658(n) = a(a(a(n))); a(n) = A109613(n)-A110654(n) = A110660(n)/A110654(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 05 2005
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EXAMPLE
| 1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + 4*x^9 + 6*x^10 + 5*x^11 + ...
1 + 2*t^4 + t^6 + 3*t^8 + 2*t^10 + 4*t^12 + 3*t^14 + 5*t^16 + 4*t^18 + 6*t^20 + ...
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MAPLE
| series((1+x^3)/(1-x^2)^2, x, 80);
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MATHEMATICA
| lst={}; a=1; Do[a=n-a; AppendTo[lst, a], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 01 2008]
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PROG
| (PARI) {a(n) = (n\2) - (n%2) + 1} /* Michael Somos Oct 02 1999 */
(MAGMA) &cat[ [n+1, n]: n in [0..37] ]; [From Klaus Brockhaus, Nov 23 2009]
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CROSSREFS
| Cf. A000124 "a=1, a=n+a", A028242 "a=1, a=n-a" [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 01 2008]
Partial sums give A004652. A030451(n)=A028242(n+1), n>0.
Cf. A052938 (same sequence except no leading 1,0,2). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 16 2009]
Sequence in context: A026238 A066136 A097140 * A030451 A029162 A005044
Adjacent sequences: A028239 A028240 A028241 * A028243 A028244 A028245
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com)
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EXTENSIONS
| First part of definition adjusted to match offset by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 23 2009
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