0,2

Numbers n such that concatenation of first n natural numbers is not divisible by 3. E.g. 16 is in the sequence because we have 123456789101111213141516 = 1 (mod 3).

Ignoring the first term, this sequence represents the number of bonds in a hydrocarbon: a(#of carbon atoms)=number of bonds. - Nathan Savir (thoobik(AT)yahoo.com), Jul 03 2003

n such that sum(k=0,n,binomial(n+k,n-k) mod 2) is even (cf. A007306) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2004

Number of vertices of squares sharing a common vertex. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Nov 11 2005

Hilbert series for twisted cubic curve. - Paul Barry (pbarry(AT)wit.ie), Aug 11 2006

If Y is a 3-subset of an n-set X then, for n>=3, a(n-3) is the number of 3-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 23 2007

a(n) = A144390 (1,9,23,43,69, ...) - A045944 (0,5,16,33,56, ...). From successive spectra of hydrogen atom. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 05 2008]

A145389(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 10 2008]

Union of A035504, A165333, and A165336. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 17 2009]

Hankel transform of A076025. [From Paul Barry (pbarry(AT)wit.ie), Sep 23 2009]

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

W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, p. 22

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

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

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

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

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

T. Mansour, Permutations avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in S_4

G.f.: (1+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 + 4x + 7x^2 + 10x^3...) = (1 + 2x + 3x^2...) / (1 - 2x + 4x^2 - 8x^3...) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2003

E.g.f. : exp(x)(1+3x) - Paul Barry (pbarry(AT)wit.ie), Jul 23 2003

1 - 1/4 + 1/7 - 1/10... = (1/3)*(Pi/(sqrt(3) + ln 2). [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2006

Row sums of triangle A131033: (1; 3,1; 4,2,1; 5,2,2,1;...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 10 2007

a(n)=2*a(n-1)-a(n-2); a(0)=1, a(1)=4. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]

a(n)=6*n-a(n-1)-7 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009]

For n=2, a(2)=6*2-1-7=4; n=3, a(3)=6*3-4-7=7; n=4, a(4)=6*4-7-7=10 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 22 2009]

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+3 od: seq(a[n], n=1..63); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008

f[n_]:=3*n+1; lst={}; Do[a=f[n]; AppendTo[lst, a], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]

(MAGMA) [ 3*n+1 : n in [1..10] ]; - from Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(Other) sage: [i+1 for i in range(189) if gcd(i, 3) == 3] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]

(Other) sage: [crt(1, n, 3, 2) for n in xrange(1, 64)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2009]

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

Cf. A058183.

First differences of A000326.

Cf. A131033.

Complement of A007494. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 10 2008]

Sequence in context: A070300 A112335 A145289 this_sequence A143460 A143459 A143458

Adjacent sequences: A016774 A016775 A016776 this_sequence A016778 A016779 A016780

nonn,easy

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

Better description from T. D. Noe (noe(AT)sspectra.com), Aug 15 2002