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A032766 Numbers that are congruent to 0 or 1 mod 3. 62
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; text; internal format)
OFFSET

0,3

COMMENTS

Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A071039. - Hans Havermann, 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). - Richard Choulet, Jan 28 2010

Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, 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). - Gary Detlefs, 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

Sequence is also, the maximum number of floors with 3 elevators and n stops in a "Convenient Building".  See A196592 and Erich Friedman link below. - Robert Price, May 30 2013

a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013

LINKS

Table of n, a(n) for n=0..69.

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. Book's website

Erich Friedman, Problem of the month November 2009

International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20.

Kival Ngaokrajang, Illustration of initial terms (U)

Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1,-1)

FORMULA

G.f.: x*(1+2*x)/((1-x)*(1-x^2)).

a(-n) = -A007494(n).

From Paul Barry, Sep 04 2003: (Start)

a(n) = (6n - 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. (End)

a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005

a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, 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, Sep 08 2007

a(n) = (cos(Pi*n) - 1)/4 + 1.5*n. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008

A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009

a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010

a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010

a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011

a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011

a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - Philippe Deléham, Oct 17 2011

a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013

a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013

a(n) = sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014

a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014

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, Mar 16 2008

seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010

select(n->member(n mod 3, {0, 1}), [$0..103]); # Peter Luschny, Apr 06 2014

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, Mar 28 2011 *)

Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

Flatten[{#, #+1}&/@(3Range[0, 40])] (* or *) LinearRecurrence[{1, 1, -1}, {0, 1, 3}, 100] (* or *) With[{nn=110}, Complement[Range[0, nn], Range[2, nn, 3]]] (* Harvey P. Dale, Mar 10 2013 *)

PROG

(PARI) {a(n) = n + n\2}

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, A001651, A007494, A035360, A132463, A133083, A032766, A006578, A002717, A070893, A171370.

Cf. A066272 for anti-divisors.

Sequence in context: A026322 A049624 A084056 * A189935 A064717 A109231

Adjacent sequences:  A032763 A032764 A032765 * A032767 A032768 A032769

KEYWORD

nonn,easy,nice

AUTHOR

Patrick De Geest, May 15 1998

EXTENSIONS

Better description from N. J. A. Sloane, Aug 01 1998

STATUS

approved

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Last modified October 25 15:47 EDT 2014. Contains 248556 sequences.