0,1

Except for 1, n such that sum(k=1,n,(k mod 3)*C(n,k)) is a power of 2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 17 2002

The sequence 0,0,2,0,0,5,0,0,8,.. has a(n)=n(1+cos(2pi*n/3+pi/3)-sqrt(3)sin(2pi*n+pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry (pbarry(AT)wit.ie), Jan 28 2004. Artur Jasinski (grafix(AT)csl.pl), Dec 11 2007, remarks that this should read Table[(3n + 2)(1 + Cos[2Pi*(3n + 2)/3 + Pi/3] - Sqrt[3] Sin[2Pi*(3n + 2)/3 + Pi/3])/3, {n, 0,20}] .

Except for 2, exponents e such that x^e+x+1 is reducible.

a(n) = A125199(n+1,1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 24 2006

Primitive roots of 3. - Paolo P. Lava (ppl(AT)spl.at), Feb 29 2008

The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the 1st of the second kind) [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009]

If X=(28+n^3)/9, Y=(28+n^3)/3, Z=(n^4+28n)/9, then X^3+Y^3+Z^3=(X*Y)^2 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jun 30 2009]

Union of A165334 and A165335. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 17 2009]

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 16.

Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

Index entries for sequences related to linear recurrences with constant coefficients

L. Euler, Observatio de summis divisorum p. 9.

L. Euler, An observation on the sums of divisors p. 9.

Tanya Khovanova, Recursive Sequences

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")

G.f.: (2+x)/(1-x)^2. a(n)=3+a(n-1).

sum(n=1, inf, (-1)^n/a(n))=1/3(Pi/sqrt(3)-ln(2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002

1/2 - 1/5 + 1/8 - 1/11...= (1/3)*(Pi/sqrt(3) - ln 2). [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2006

a(n)=2*a(n-1)-a(n-2); a(0)=2, a(1)=5. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]

a(n)=6*n-a(n-1)-5 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009]

For n=2, X=4, Y=12, Z=8, 4^3+12^3+8^3=(4*12)^2: n=5, X=17, Y=51, Z=85, 17^3+51^3+85^3=(17*51)^2: n=8, X=60, Y=180, Z=480, 60^3+180^3+480^3=(60*180)^2 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jun 30 2009]

For n=2, a(2)=6*2-2-5=5; n=3, a(3)=6*3-5-5=8; n=4, a(4)=6*4-8-5=11 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009]

[seq(4*binomial(3*n, 2)/binomial(2*n, 1)/3, n=1..60)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 16 2007

a[1]:=-1:for n from 2 to 100 do a[n]:=a[n-1]+3 od: seq(a[n], n=2..61); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008

f[n_]:=3*n+2; lst={}; Do[a=f[n]; AppendTo[lst, a], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]

(Other) sage: [i+2 for i in range(180) if gcd(i, 3) == 3] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]

(Other) sage: [crt(2, n, 3, 2) for n in xrange(2, 62)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2009]

A016789(n)=1+A016777(n).

First differences of A005449.

a(n)=A124388(n)/9.

Cf. A002939, A017041, A017485, A125202.

Cf. A017233.

Sequence in context: A109232 A064718 A078608 this_sequence A165334 A135677 A163516

Adjacent sequences: A016786 A016787 A016788 this_sequence A016790 A016791 A016792

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).