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 (seeA006784 for definition) [when offset by 1]. -Henry Bottomley, Dec 18 2000
Also the denominators of the series expansion of log(1+x). Numerators areA062157. -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
G. C. Greubel,Table of n, a(n) for n = 0..10000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner,Generating functions for lattice paths with several forbidden patterns, (2019).
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner,Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Olivia Nabawanda and Fanja Rakotondrajao,The sets of flattened partitions with forbidden patterns, arXiv:2011.07304 [math.CO], 2020.
G. Nebe, E. M. Rains and N. J. A. Sloane,Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Oeis Wiki,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).
Michael Somos,Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
Binomial transform isA005183. -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. ofA112934(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..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
EXAMPLE
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 + ...
MAPLE
a:= n-> `if`(n=0, 1, n):
seq(a(n), n=0..60);
MATHEMATICA
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 *)
Join[{1}, Range[75]] (*G. C. Greubel, Jan 05 2024 *)
LinearRecurrence[{2, -1}, {1, 1, 2}, 80] (*Harvey P. Dale, Jan 29 2025 *)
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
(Python)
defA028310(n): return n|bool(n)^1 #Chai Wah Wu, Jul 13 2023
(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
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
STATUS
approved