Search: id:A016777
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%I A016777
%S A016777 1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,
               73,
%T A016777 76,79,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,133,
%U A016777 136,139,142,145,148,151,154,157,160,163,166,169,172,175,178,181,184,187
%N A016777 3n+1.
%C A016777 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).
%C A016777 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
%C A016777 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
%C A016777 Number of vertices of squares sharing a common vertex. - Parthasarathy 
               Nambi (PachaNambi(AT)yahoo.com), Nov 11 2005
%C A016777 Hilbert series for twisted cubic curve. - Paul Barry (pbarry(AT)wit.ie), 
               Aug 11 2006
%C A016777 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
%C A016777 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]
%C A016777 A145389(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 10 2008]
%C A016777 Union of A035504, A165333, and A165336. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 17 2009]
%C A016777 Hankel transform of A076025. [From Paul Barry (pbarry(AT)wit.ie), Sep 
               23 2009]
%D A016777 Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.
%D A016777 W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, 
               p. 22
%D A016777 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 
               16.
%H A016777 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A016777 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A016777 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">
               Observatio de summis divisorum</a> p. 9.
%H A016777 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation 
               on the sums of divisors</a> p. 9.
%H A016777 Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
               c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen 
               Reihen</a>, Berlin, J. Springer, 1922. (Original German edition of 
               "Theory and Application of Infinite Series")
%H A016777 T. Mansour, <a href="http://arXiv.org/abs/math.CO/9909019">Permutations 
               avoiding a set of patterns T \subseteq S_3 and a pattern \tau \in 
               S_4</a>
%F A016777 G.f.: (1+2*x)/(1-x)^2. a(n)=3+a(n-1).
%F A016777 sum(n=1, inf, (-1)^n/a(n))=1/3(Pi/sqrt(3)+ln(2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002
%F A016777 (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
%F A016777 E.g.f. : exp(x)(1+3x) - Paul Barry (pbarry(AT)wit.ie), Jul 23 2003
%F A016777 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
%F A016777 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
%F A016777 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]
%F A016777 a(n)=6*n-a(n-1)-7 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 22 2009]
%e A016777 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]
%p A016777 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
%t A016777 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]
%o A016777 (MAGMA) [ 3*n+1 : n in [1..10] ]; - from Sergei Haller (sergei(AT)sergei-haller.de), 
               Dec 21 2006
%o A016777 (Other) sage: [i+1 for i in range(189) if gcd(i,3) == 3] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]
%o A016777 (Other) sage: [crt(1, n, 3, 2) for n in xrange(1, 64)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2009]
%Y A016777 A016789(n)=1+A016777(n).
%Y A016777 Cf. A058183.
%Y A016777 First differences of A000326.
%Y A016777 Cf. A131033.
%Y A016777 Complement of A007494. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 10 2008]
%Y A016777 Sequence in context: A070300 A112335 A145289 this_sequence A143460 A143459 
               A143458
%Y A016777 Adjacent sequences: A016774 A016775 A016776 this_sequence A016778 A016779 
               A016780
%K A016777 nonn,easy,new
%O A016777 0,2
%A A016777 N. J. A. Sloane (njas(AT)research.att.com).
%E A016777 Better description from T. D. Noe (noe(AT)sspectra.com), Aug 15 2002

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