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A028310 Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x. 66
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

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

Also the denominators of the series expansion of log(1+x). Numerators are A062157. - Robert G. Wilson v, Aug 14 2015

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

LINKS

Table of n, a(n) for n=0..71.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

OeisWiki, Autosequence

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

M. Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Index entries for Molien series

Index entries for sequences related to Engel expansions

FORMULA

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

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

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

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

a(n) = Sum_{k, 0<=k<=n} A123110(n,k). - Philippe Deléham, Oct 06 2009

E.g.f: 1+x*exp(x). - Wolfdieter Lang, May 03 2010

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

a(n) = A001477(n) + A000007(n). - Miko Labalan, Dec 12 2015 (See the first comment.)

EXAMPLE

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  + ...

MAPLE

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

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

MATHEMATICA

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

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

Table[f[n], {n, 0, 50}] (* Roger L. Bagula, Feb 13 2010 *)

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 *)

PROG

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

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

(Haskell)

a028310 n = 0 ^ n + n

a028310_list = 1 : [1..]  -- Reinhard Zumkeller, Nov 06 2012

CROSSREFS

Cf. A000027, A112934, A004001, A005229, A212393, A000660 (boustrophedon transform).

Sequence in context: A069782 A088480 A061019 * A097045 A118760 A130446

Adjacent sequences:  A028307 A028308 A028309 * A028311 A028312 A028313

KEYWORD

nonn,easy,mult,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 29 15:31 EDT 2017. Contains 284273 sequences.