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 A028310 Expansion of (1 - x + x^2) /(1 - x)^2 in powers of x. 42

%I

%S 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,

%T 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,

%U 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71

%N Expansion of (1 - x + x^2) /(1 - x)^2 in powers of x.

%C 1 followed by the natural numbers

%C Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).

%C Engel expansion of e (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000

%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

%H E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (<a href="http://neilsloane.com/doc/self.txt">Abstract</a>, <a href="http://neilsloane.com/doc/self.pdf">pdf</a>, <a href="http://neilsloane.com/doc/self.ps">ps</a>).

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (2,-1).

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%F Binomial transform is A005183. - Paul Barry, Jul 21 2003

%F 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.

%F Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - Michael Somos Jul, 30 2006

%F G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Apr 05 2012

%F G.f. of A112934(x) = 1 / (1 - a(0)*x / (1 - a(1)*x / ...)). - Michael Somos, Apr 05 2012

%F a(n) = A000027(n) unless n=0.

%F a(n) = Sum_{k, 0<=k<=n} A123110(n,k). [From Philippe DELEHAM, Oct 06 2009]

%F a(n) = a(a(n-1)+a(n-a(n-3)) [From Roger L. Bagula, Feb 13 2010]

%F E.g.f: 1+x*exp(x). [From Wolfdieter Lang, May 03 2010]

%F a(n)=sqrt(floor[A204503(n+3)/9]). - M. F. Hasler, Jan 16 2012

%e 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 + ...

%p a:= n-> `if` (n=0, 1, n):

%p seq (a(n), n=0..60);

%t Contribution from Roger L. Bagula, Feb 13 2010: (Start)

%t f[0] = 0; f[1] = 1; f[2] = 1; f[3] = 1;

%t f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 3]];

%t Table[f[n], {n, 0, 50}] (End)

%o (PARI) {a(n) = (n==0) + max(n, 0)} /* Michael Somos, Feb 02 2004 */

%o (PARI) A028310(n)=n+!n \\ - M. F. Hasler, Jan 16 2012

%o a028310 n = 0 ^ n + n

%o a028310_list = 1 : [1..] -- _Reinhard Zumkeller_, Nov 06 2012

%Y Cf. A000027, A112934.

%Y Cf. A004001, A005229. [From Roger L. Bagula, Feb 13 2010]

%Y Cf. A212393.

%K nonn,easy,mult

%O 0,3

%A _N. J. A. Sloane_.

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