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A032766
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Numbers that are congruent to 0 or 1 mod 3.
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40
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0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also (omitting initial 0) number of 1's in n-th row of triangle in A071039. - Hans Havermann (gladhobo(AT)teksavvy.com), May 26 2002
Binomial transform is A053220. - Michael Somos, Jul 10 2003
Smallest number of different people in a set of n-1 photographs which satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
We observe that, without the beginning 0, the sequence 1,3,4,6,7,9,... is the transform of A000034 1,2,1,2,1,2,1,... by the following transform T: T(u_0,u_1,u_2,u_3,...)=(u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In another terms v_p=sum(u_k,k=0..p) and the G.f phi_v of is given by: phi_v=phi_u/(1-z). [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 28 2010]
Starting with 1 = row sums of triangle A171370. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 15 2010]
a(n) is the set of values for m in which 6k +m can be a perfect square(quadratic residues of 6 including trivial case of 0) [From Gary Detlefs (gdetlefs(AT)aol.com), Mar 19 2010]
For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011.
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REFERENCES
| International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest Problem #20.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1,-1)
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FORMULA
| G.f.: x*(1+2*x)/((1-x)*(1-x^2)). a(-n) = -A007494(n).
Formulae from Paul Barry (pbarry(AT)wit.ie), Sep 04 2003: a(n)=(6*n-1+(-1)^n)/4; a(n)=floor((3n+2)/2)-1 = A001651(n)-1; a(n)=sqrt(2)*sqrt((6n-1)(-1)^n+18n^2-6n+1)/4; a(n)=sum{k=0..n, 3/2-2*0^n+(-1)^n/2}.
a(n) = 3*floor(n/2) + n mod 2 = A007494(n)-A000035(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 04 2005
a(n)=2*A004526(n)+A004526(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 07 2006
a(n) = 1 + ceiling(1.5*(n-1)) - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Row sums of triangle A133083. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 08 2007
a(n) = (cos(Pi n) - 1)/4 + 1.5n [From Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008]
A004396(a(n)) = n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 30 2009]
a(n) = floor(n/2)+n [From Gary Detlefs (gdetlefs(AT)aol.com), Mar 19 2010]
a(n)=3*n-a(n-1)-2 (with a(0)=0) [From Vincenzo Librandi, Nov 19 2010]
Contribution from Jaroslav Krizek, Mar 22 2011 (Start):
a(n) = sum of numbers from n to 1 with alternating signs.
a(n) = n + (n-1) - (n-2) + (n-3) - ...
a(n) = A052928(n) + A008619(n-1). (End)
a(n) = a(n-1)+a(n-2)-a(n-3). Robert G. Wilson v, March 28 2011
a(n)=Sum_k>=0 {A030308(n,k)*A003945(k)}. - From DELEHAM Philippe, Oct 17 2011.
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EXAMPLE
| x + 3*x^2 + 4*x^3 + 6*x^4 + 7*x^5 + 9*x^6 + 10*x^7 + 12*x^8 + 13*x^9 + 15*x^10 + ...
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MAPLE
| a[0]:=0:a[1]:=1: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
Digits := 100: t := evalf(1+cos(Pi/3)): A:= n->floor(t*n): seq(floor((t*n)), n=0..69); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 27 2009]
seq(floor(n/2)+n, n=0..69); [From Gary Detlefs (gdetlefs(AT)aol.com), Mar 19 2010]
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MATHEMATICA
| a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, March 28 2011 *)
Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* From Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
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PROG
| (PARI) {a(n) = n + n\2}
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CROSSREFS
| For n>0, a(n)=T(n, 2), array T as in A049615. Column 1 of A026374.
Partial sums are A006578. Partial sums of A000034. Cf. A084056, A047270.
Cf. A001651, A007494, A035360, A132463.
Cf. A133083.
Cf. A032766, A006578, A002717, A070893. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 28 2010]
A171370 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 15 2010]
Cf. A066272 for anti-divisors.
Sequence in context: A026322 A049624 A084056 * A189935 A064717 A109231
Adjacent sequences: A032763 A032764 A032765 * A032767 A032768 A032769
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| Patrick De Geest (pdg(AT)worldofnumbers.com), May 15, 1998.
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EXTENSIONS
| Better description from N. J. A. Sloane (njas(AT)research.att.com) 8/98.
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