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A001651 Numbers not divisible by 3.
(Formerly M0957 N0357)
123
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, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Inverse binomial transform of A084858. - Benoit Cloitre, Jun 12 2003

Earliest monotonic sequence starting with (1,2) and satisfying the condition: "a(n)+a(n-1) is not in the sequence." - Benoit Cloitre, Mar 25 2004. [The numbers of the form a(n)+a(n-1) form precisely the complement with respect to the positive integers. - David W. Wilson, Feb 18 2012]

a(1) = 1; a(n) is least number which is relatively prime to the sum of all the previous terms. - Amarnath Murthy, Jun 18 2001

For n>3, numbers having 3 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005

A011655(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009

Also numbers n such that (n+1)*(n+2)/6 = A000292(n)/n is an integer. - Ctibor O. Zizka, Oct 15 2010

Notice the property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 3). - Bruno Berselli, Nov 17 2010

A001651 mod 9 gives A141425. - Paul Curtz, Dec 31 2010. (Correct for the modified offset 1. - M. F. Hasler, Apr 07 2015)

A215879(a(n)) > 0. - Reinhard Zumkeller, Dec 28 2012. (More precisely, A215879 is the characteristic function of A001651. - M. F. Hasler, Apr 07 2015)

The set of natural numbers (1, 2, 3, ...), sequence A000027; represents the numbers of ordered compositions of n using terms in the signed set: (1, 2, -4, -5, 7, 8, -10, -11, 13, 14, ...). This follows from (1, 2, 3, ...) being the INVERT transform of A011655, signed and beginning: (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013

Union of A047239 and A047257. - Wesley Ivan Hurt, Dec 19 2013

Numbers whose sum of digits (and digital root) is != 0 (mod 3). - Joerg Arndt, Aug 29 2014

The number of partitions of 3*(n-1) into at most 2 parts. - Colin Barker, Apr 22 2015

a(n) is the number of partitions of 3*n into two distinct parts. - L. Edson Jeffery, Jan 14 2017

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)

G. Ledin, Jr., Is Eratosthenes out?, Fib. Quart., 6 (No. 4, 1968), 261-265.

G. P. Michon, Counting Polyhedra

M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564, Ex. 2.2.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, RATS Sequence

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

FORMULA

a(n) = 3 + a(n-2) for n>2.

a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.

a(2*n+1) = 3*n+1, a(2*n) = 3*n-1.

G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - Michael Somos, Jun 08 2000

a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article).

a(n) = floor((3*n-1)/2) (corrected by Gary Detlefs).

a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - Benoit Cloitre, Aug 17 2002

a(n+1) = 1 + n - n mod 2 + (n + n mod 2)/2. - Reinhard Zumkeller, Dec 17 2002

a(1) = 1, a(n+1) = a(n) + (a(n) mod 3). - Reinhard Zumkeller, Mar 23 2003

a(1) = 1, a(n) = 3*(n-1) - a(n-1). - Benoit Cloitre, Apr 12 2003

a(n) = 3*(2*n-1)/4 - (-1)^n/4. - Benoit Cloitre, Jun 12 2003

Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003

Partial sums of A040001. a(n) = A032766(n-1)+1. - Paul Barry, Sep 02 2003

a(n) = T(n, 1) = T(n, n-1), where T is the array in A026386. - Emeric Deutsch, Feb 18 2004

a(n) = sqrt(3*A001082(n)+1). - Zak Seidov, Dec 12 2007

a(n) = A077043(n) - A077043(n-1). - Reinhard Zumkeller, Dec 28 2007

a(n) = A001477(n-1) + A008619(n-1). - Yosu Yurramendi, Aug 10 2008

Euler transform of length 3 sequence [2, 1, -1]. - Michael Somos, Sep 06 2008

a(n) = n - 1 + ceiling(n/2). - Michael Somos, Jan 15 2011

a(n) = 3*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i), for n>1. - Bruno Berselli, Nov 17 2010

a(n) = 3*floor(n/2)+(-1)^(n+1). - Gary Detlefs, Dec 29 2011

a(n) = 2n - 1 - floor(n/2). - Wesley Ivan Hurt, Oct 25 2013

a(n) = (3n - 2 + (n mod 2)) / 2. - Wesley Ivan Hurt, Mar 31 2014

a(n) = A000217(n) - A000982(n-1). - Bui Quang Tuan, Mar 28 2015

1/1^3 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + ... = 4 Pi^3/(3 sqrt(3)). - M. F. Hasler, Mar 29 2015

E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - Ilya Gutkovskiy, May 24 2016

a(n) = a(n+k-1) + a(n-k) - a(n-1) for n>k>=0. - Bob Selcoe, Feb 03 2017

EXAMPLE

G.f.: x + 2*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 10*x^7 + 11*x^8 + 13*x^9 + ...

MAPLE

A001651 := n -> 3*floor(n/2) - (-1)^n; # Corrected by M. F. Hasler, Apr 07 2015

A001651:=(1+z+z**2)/(z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation

a[1]:=1:a[2]:=2:for n from 3 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=1..69); # Zerinvary Lajos, Mar 16 2008, offset corrected by M. F. Hasler, Apr 07 2015

MATHEMATICA

Select[Table[n, {n, 200}], Mod[#, 3]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)

Drop[Range[200 + 1], {1, -1, 3}] - 1 (* József Konczer, May 24 2016 *)

PROG

(PARI) {a(n) = n + (n-1)\2}; /* Michael Somos, Jan 15 2011 */

(MAGMA) [3*(2*n-1)/4-(-1)^n/4: n in [1..80]]; // Vincenzo Librandi, Jun 07 2011

(Haskell)

a001651 = (`div` 2) . (subtract 1) . (* 3)

a001651_list = filter ((/= 0) . (`mod` 3)) [1..]

-- Reinhard Zumkeller, Jul 07 2012, Aug 23 2011

(PARI) x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ Altug Alkan, Oct 22 2015

CROSSREFS

Differs from A059564 after 35 = a(24) = A059564(24).

Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A191967.

Sequence in context: A054386 A127450 A059564 * A224999 A274384 A195175

Adjacent sequences:  A001648 A001649 A001650 * A001652 A001653 A001654

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

This is a list, so the offset should be 1. I corrected this and adjusted some of the comments and formulas. Other lines probably also need to be adjusted. - N. J. A. Sloane, Jan 01 2011

Offset of pre-2011 formulas verified or corrected by M. F. Hasler, Apr 07-18 2015 and by Danny Rorabaugh, Oct 23 2015

STATUS

approved

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Last modified March 27 08:41 EDT 2017. Contains 284146 sequences.