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A028310
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Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.
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91
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1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
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OFFSET
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0,3
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COMMENTS
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1 followed by the natural numbers.
Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).
The right-shifted sequence (with a(0)=0) is an autosequence (of the first kind - see definition in links). - Jean-François Alcover, Mar 14 2017
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LINKS
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E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
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FORMULA
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G.f.: (1 - x + x^2) / (1 - x)^2 = (1 - x^6) /((1 - x) * (1 - x^2) * (1 - x^3)) = (1 + x^3) / ((1 - x) * (1 - x^2)). a(0) = 1, a(n) = n if n>0.
Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - Michael Somos Jul 30 2006
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Apr 05 2012
E.g.f.: 1-x + x*E(0), where E(k) = 2 + x/(2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9 + ...
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MAPLE
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a:= n-> `if`(n=0, 1, n):
seq(a(n), n=0..60);
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MATHEMATICA
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Denominator@ CoefficientList[Series[Log[1+x], {x, 0, 75}], x] (* or *)
CoefficientList[ Series[(1 -x +x^2)/(1-x)^2, {x, 0, 75}], x] (* Robert G. Wilson v, Aug 14 2015 *)
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PROG
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(PARI) {a(n) = (n==0) + max(n, 0)} /* Michael Somos, Feb 02 2004 */
(Haskell)
a028310 n = 0 ^ n + n
(Python)
(Magma) [n eq 0 select 1 else n: n in [0..75]]; // G. C. Greubel, Jan 05 2024
(SageMath) [n + int(n==0) for n in range(76)] # G. C. Greubel, Jan 05 2024
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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