Search: id:A001651
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%I A001651 M0957 N0357
%S A001651 1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26,28,29,31,32,34,35,37,
%T A001651 38,40,41,43,44,46,47,49,50,52,53,55,56,58,59,61,62,64,65,67,68,70,71,
%U A001651 73,74,76,77,79,80,82,83,85,86,88,89,91,92,94,95,97,98,100,101,103,104
%N A001651 Not divisible by 3.
%C A001651 Inverse binomial transform of A084858. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 12 2003
%C A001651 Earliest monotonic sequence starting with (1,2) and satisfying the condition 
               : "a(n)+a(n-1) is not in the sequence" - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 25 2004
%C A001651 a(0) = 1; a(n) is least number which is relatively prime to the sum of 
               all the previous terms. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Jun 18 2001
%C A001651 For n>2, numbers having 3 as an anti-divisor. - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), 
               Oct 02 2005
%C A001651 A011655(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 30 2009]
%D A001651 L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related 
               to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
%D A001651 G. Ledin, Jr., Is Eratosthenes out?, Fib. Quart., 6 (No. 4, 1968), 261-265.
%D A001651 M. A. Nyblom, Some curious sequences ..., Am. Math. Monthly 109 (#6, 
               200), 559-564, Ex. 2.2.
%D A001651 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001651 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001651 A. S. Fraenkel, <a href="http://www.integers-ejcnt.org/">New games related 
               to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial 
               Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
%H A001651 G. P. Michon, <a href="http://www.numericana.com/data/polyhedra.htm">
               Counting Polyhedra</a>
%H A001651 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001651 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001651 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RATSSequence.html">RATS Sequence</a>
%H A001651 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A001651 a(n) = 3+a(n-2). a(n) = a(n-1)+a(n-2)-a(n-3). a(2n) = 3n+1, a(2n-1) = 
               3n-1.
%F A001651 G.f.: (1+x+x^2)/((1-x)*(1-x^2)) - Michael Somos, Jun 08, 2000
%F A001651 a(n) = (4-n)*a(n-1)+2*a(n-2)+(n-3)*a(n-3) (from the Carlitz et al. article).
%F A001651 a(n)=Floor[(3n+2)/2]
%F A001651 a(1)=1, a(n) = 2*a(n-1)- 3*floor(a(n-1)/3). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Aug 17 2002
%F A001651 a(n) = 1 + n - n mod 2 + (n + n mod 2)/2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 17 2002
%F A001651 a(0) = 1, a(n+1) = a(n) + a(n) mod 3. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 23 2003
%F A001651 a(0)=1, a(n)=3n-a(n-1). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 12 2003
%F A001651 a(n)=3(2n+1)/4+(-1)^n/4 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 12 2003
%F A001651 Nearest integer to sum(k>n, 1/k^3)/sum(k>n, 1/k^4) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 12 2003
%F A001651 Partial sums of A040001. a(n)=A032766(n)-1. - Paul Barry (pbarry(AT)wit.ie), 
               Sep 02 2003
%F A001651 a(n)=T(n+1, 1)=T(n+1, n), where T is the array in A026386. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Feb 18 2004
%F A001651 a(n)=sqrt(3 A001082(n)+1 ) - Zak Seidov (zakseidov(AT)yahoo.com), Dec 
               12 2007
%F A001651 a(n) = A077043(n+1) - A077043(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 28 2007
%F A001651 a(n) = A001477(n)+ A008619(n) - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), 
               Aug 10 2008
%F A001651 Euler transform of length 3 sequence [ 2, 1, -1]. - Michael Somos Sep 
               06 2008
%F A001651 a(-1 - n) = -a(n).
%e A001651 1 + 2*x + 4*x^2 + 5*x^3 + 7*x^4 + 8*x^5 + 10*x^6 + 11*x^7 + 13*x^8 + 
               ...
%p A001651 A001651 := n -> 3*floor((n+1)/2) + (-1)^n;
%p A001651 A001651:=(1+z+z**2)/(z+1)/(z-1)**2; [S. Plouffe in his 1992 dissertation.]
%p A001651 a[0]:=1:a[1]:=2:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
%o A001651 (PARI) a(n)= 1+n+n\2
%o A001651 (Other) [i for i in range(105) if gcd(3,i) == 1] [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 21 2009]
%Y A001651 Differs from A059564 after 35= a(23)= A059564(24).
%Y A001651 Cf. A026386, A001082.
%Y A001651 Cf. A007494, A032766, A000726, A003105.
%Y A001651 Sequence in context: A054386 A127450 A059564 this_sequence A003253 A119905 
               A161750
%Y A001651 Adjacent sequences: A001648 A001649 A001650 this_sequence A001652 A001653 
               A001654
%K A001651 nonn,easy,new
%O A001651 0,2
%A A001651 N. J. A. Sloane (njas(AT)research.att.com).
%E A001651 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009

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