A334187 -id:A334187 - cp's OEIS frontend

A212964 Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.

0, 0, 0, 2, 6, 14, 26, 44, 68, 100, 140, 190, 250, 322, 406, 504, 616, 744, 888, 1050, 1230, 1430, 1650, 1892, 2156, 2444, 2756, 3094, 3458, 3850, 4270, 4720, 5200, 5712, 6256, 6834, 7446, 8094, 8778, 9500, 10260, 11060, 11900, 12782, 13706

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.
Magic numbers of nucleons in a biaxially deformed nucleus at oscillator ratio 1:2 (oblate ellipsoid) under the simple harmonic oscillator model. - Jess Tauber, May 14 2013
a(n) is the number of Sidon subsets of {1,...,n+1} of size 3. - Carl Najafi, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
Ikuko Hamamoto, One-particle motion in nuclear many-body problem, Division of Mathematical Physics, LTH, University of Lund, Sweden.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002623, A212959, A334187.

First differences: A007590, is first differences of 2*A001752(n-4) for n > 3; partial sums: 2*A001752(n-3) for n > 2, is partial sums of A007590(n-1) for n > 0. - Guenther Schrack, Mar 19 2018

[(2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Jul 25 2014

A212964:=n->add(floor(i^2/2) - 2*floor(i/2), i=1..n): seq(A212964(n), n=0..50); # Wesley Ivan Hurt, Jul 23 2014

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] < Abs[x - y] < Abs[y - w], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A212964 *)
m/2 (* essentially A002623 *)
CoefficientList[Series[2 x^3/((1 + x) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *)

a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8;
vector (100, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: f(x)/g(x), where f(x)=2*x^3 and g(x)=(1+x)(1-x)^4.
a(n+3) = 2*A002623(n).
a(n) = Sum_{k=0..n} floor((k-1)^2/2). - Enrique Pérez Herrero, Dec 28 2013
a(n) = Sum_{i=1..n} floor(i^2/2) - 2*floor(i/2). - Wesley Ivan Hurt, Jul 23 2014
a(n) = (2*n-1)*(2*n^2-2*n-3)/24 - (-1)^n/8. - Robert Israel, Jul 23 2014
E.g.f.: (x*(2*x^2 + 3*x - 3)*cosh(x) + (2*x^3 + 3*x^2 - 3*x + 3)*sinh(x))/12. - Stefano Spezia, Jul 06 2021

A051013 Number of nonaveraging subsets on {1,2,...,n}.

Original entry on oeis.org

1, 2, 4, 7, 13, 23, 40, 65, 106, 169, 278, 443, 705, 1117, 1760, 2692, 4151, 6314, 9526, 14127, 20944, 30848, 45589, 66495, 96847, 140840, 204380, 293822, 425859, 613446, 880288, 1258349, 1794256, 2545965, 3623774, 5123746, 7207773, 10159163, 14273328, 19925242, 27893419

Offset: 0

Eric W. Weisstein

nonn

			The only subset of s = {1,2,3} that contains a 3-term arithmetic progression is s itself, so a(3) = 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..80
Eric Weisstein's World of Mathematics, Nonaveraging Sequence
Wikipedia, Salem-Spencer set
Index entries related to non-averaging sequences

Cf. A018788.
Row sums of A334187.
First differences give A334893.

a[n_] := a[n] = 2^n - Count[Subsets[Range[n], {3, n}], {_, a_, _, b_, _, c_, _} /; b-a == c-b]; Table[Print[n, " ", a[n]]; a[n], {n, 0, 32}] (* Jean-François Alcover, May 30 2019 *)

# Prints out all such sets
def nonaveragingsets(n):
    avoid=list()
    for skip in range(1,(n+1)//2):
        for start in range (1,n+1-2*skip):
            avoid.append(set({start,start+skip,start+2*skip}))
    s=list()
    for i in range(3):
        for smallset in comb(range(1,n+1),i):
            s.append(smallset)
    for i in range(3,n+1):
        for temptuple in comb(range(1,n+1),i):
            tempset=set(temptuple)
            status=True
            for avoidset in avoid:
                if avoidset <= tempset:
                    status=False
                    break
            if status:
                s.append(tempset)
    return s
# Counts all such sets
def a(n):
    return len(nonaveragingsets(n)) # David Nacin, Mar 03 2012

a(n) = 2^n - A018788(n). - David Nacin, Mar 03 2012

More terms from John W. Layman, Nov 27 2001
a(29)-a(37) from Donovan Johnson, Aug 15 2010
a(38)-a(40) from Alois P. Heinz, Oct 27 2011

A334892 Number T(n,k) of k-element subsets of [n] avoiding 3-term arithmetic progressions and containing n if n>0; triangle T(n,k), n>=0, 0<=k<=A003002(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 4, 1, 0, 1, 5, 8, 3, 0, 1, 6, 12, 6, 0, 1, 7, 18, 15, 0, 1, 8, 24, 26, 4, 0, 1, 9, 32, 47, 20, 0, 1, 10, 40, 67, 40, 7, 0, 1, 11, 50, 102, 80, 18, 0, 1, 12, 60, 140, 140, 53, 6, 0, 1, 13, 72, 194, 236, 110, 16, 1

Offset: 0

Alois P. Heinz, May 14 2020

T(n,k) is defined for all n >= 0 and k >= 0. The triangle contains only elements with 0 <= k <= A003002(n). T(n,k) = 0 for k > A003002(n).

			  1;
  0, 1;
  0, 1,  1;
  0, 1,  2;
  0, 1,  3,  2;
  0, 1,  4,  4,   1;
  0, 1,  5,  8,   3;
  0, 1,  6, 12,   6;
  0, 1,  7, 18,  15;
  0, 1,  8, 24,  26,   4;
  0, 1,  9, 32,  47,  20;
  0, 1, 10, 40,  67,  40,   7;
  0, 1, 11, 50, 102,  80,  18;
  0, 1, 12, 60, 140, 140,  53,   6;
  0, 1, 13, 72, 194, 236, 110,  16,  1;
  0, 1, 14, 84, 248, 342, 198,  42,  3;
  0, 1, 15, 98, 326, 532, 377, 100, 10;
  ...

Fausto A. C. Cariboni, Rows n = 0..80, flattened (rows n = 0..40 from Alois P. Heinz)
Eric Weisstein's World of Mathematics, Nonaveraging Sequence
Wikipedia, Arithmetic progression
Wikipedia, Salem-Spencer set
Index entries related to non-averaging sequences

Columns k=0-3 give: A000007, A057427, A000027(n-1), A007590(n-2).
Row sums give A334893.
Last elements of rows give A334894.
Cf. A003002, A334187.

b:= proc(n, s) option remember; `if`(n=0, x, b(n-1, s)+ `if`(
      ormap(j-> 2*j-n in s, s), 0, expand(x*b(n-1, s union {n}))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
            `if`(n=0, 1, b(n-1, {n}))):
seq(T(n), n=0..16);

b[n_, s_] := b[n, s] = If[n == 0, x, b[n-1, s] + If[
     AnyTrue[s, MemberQ[s, 2#-n]&], 0, Expand[x*b[n-1, s ~Union~ {n}]]]];
T[n_] := If[n == 0, {1}, CoefficientList[b[n-1, {n}], x]];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)

T(0,k) = A334187(0,k), T(n,k) = A334187(n,k) - A334187(n-1,k) for n > 0.

A262347 Number of subsets of [1..n] of maximal size that are free of 3-term arithmetic progressions.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 4, 10, 25, 4, 24, 7, 25, 6, 1, 4, 14, 43, 97, 220, 2, 18, 62, 232, 2, 33, 2, 12, 36, 106, 1, 11, 2, 4, 14, 40, 2, 4, 86, 307, 20, 1, 4, 14, 41, 99, 266, 674, 1505, 3510, 7726, 14, 50, 156, 2, 8, 26, 56, 2, 4, 6, 14, 48, 2, 4, 8, 16, 28, 108, 319, 1046, 4, 26, 82, 1, 2

Offset: 0

Nathan McNew, Sep 18 2015

nonn

The sequence A003002 gives the size of the largest subset of the integers up to n that avoids three-term arithmetic progressions. This sequence gives the number of distinct subsets of [1..n] that have that size and are free of three-term arithmetic progressions.

			The largest subset of [1,6] that doesn't have any 3 terms in arithmetic progression has size 4. There are 4 such subsets with this property: {1,2,4,5}, {1,2,5,6}, {1,3,4,6} and {2,3,5,6}, so a(6)=4.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..140
Fausto A. C. Cariboni, All sets that yield a(n) for n = 4..130., Feb 19 2018.
Janusz Dybizbanski, Sequences containing no 3-term arithmetic progressions, The Electronic Journal of Combinatorics, 19, no. 2 (2012).
Index entries related to non-averaging sequences

Cf. A003002, A065825.

Last elements of rows of A334187.

G:= proc(n, cons, t)
option remember;
local consn, consr;
   if n < t or member({},cons) then return {} fi;
   if t = 0 then return {{}} fi;
   consn, consr:= selectremove(has,cons,n);
   consn:= subs(n=NULL,consn);
   procname(n-1,consr,t) union
      map(`union`,procname(n-1,consr union   consn,t-1),{n});
end proc:
F:= proc(n)
local m,cons,R;
   m:= A003002(n-1);
   cons:= {seq(seq({i,i+j,i+2*j},i=1..n-2*j),j=1..(n-1)/2)};
   R:= G(n,cons,m+1);
   if R = {} then
      A003002(n):= m;
      G(n,cons,m);
   else
      A003002(n):= m+1;
      R
   fi
end proc:
A003002(1):= 1:
a[1]:= 1:
for n from 2 to 40 do
  a[n]:= nops(F(n))
od:
seq(a[i],i=1..40); # Robert Israel, Sep 20 2015

A003002 = Cases[Import["https://oeis.org/A003002/b003002.txt", "Table"], {, }][[All, 2]];
a[n_] := a[n] = Count[Subsets[Range[n], {A003002[[n+1]]}], s_ /; !MatchQ[s, {_, n1_, _, n2_, _, n3_, _} /; n2 - n1 == n3 - n2]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, May 30 2020 *)

a(25) to a(44) from Robert Israel, Sep 20 2015
a(45) to a(75) from Fausto A. C. Cariboni, Jan 15 2018
a(0)=1 prepended by Alois P. Heinz, May 16 2020

A300760 Number of ways to select 4 numbers from the set of the first n natural numbers avoiding 3-term arithmetic progressions.

Original entry on oeis.org

0, 1, 4, 10, 25, 51, 98, 165, 267, 407, 601, 849, 1175, 1580, 2089, 2703, 3452, 4338, 5395, 6622, 8058, 9706, 11606, 13758, 16210, 18963, 22066, 25520, 29379, 33645, 38376, 43571, 49293, 55545, 62391, 69831, 77937, 86710, 96223, 106477, 117550, 129444, 142241

Offset: 4

Heinrich Ludwig, Mar 12 2018

			There are 4 selections of 4 natural numbers from the set {1,2,3,4,5,6} free of 3-term arithmetic progressions: {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {2,3,5,6}.

Heinrich Ludwig, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).

Cf. A212964, A300761.

Column k=4 of A334187.

Array[(#^4 - 12 #^3 + 51 #^2 - 78 # + 32)/24 + Boole[OddQ@ #] (-# + 2)/4 - Boole[Mod[#, 3] == 0]/3 - Boole[Mod[#, 4] == 0] &, 43, 4] (* Michael De Vlieger, Mar 14 2018 *)
LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{0,1,4,10,25,51,98,165,267,407,601},50] (* Harvey P. Dale, Feb 18 2024 *)

concat(0, Vec(x^5*(1 + 2*x + 2*x^2 + 6*x^3 + 5*x^4 + 8*x^5) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Aug 06 2018

a(n) = (n^4 - 12*n^3 + 51*n^2 - 78*n + 32)/24 + b(n) + c(n), where
  b(n) = 0          for n even
  b(n) = (-n + 2)/4 for n odd
  c(n) = 0          for n == 1,2,5,7,10,11 (mod 12)
  c(n) = -1/3       for n == 3,6,9         (mod 12)
  c(n) = -4/3       for n == 0             (mod 12)
  c(n) = -1         for n == 4,8           (mod 12).
a(n) = (n^4 - 12*n^3 + 51*n^2 - 78*n + 32)/24 + (n == 1 (mod 2))*(-n + 2)/4 - (n == 0 (mod 3))/3 - (n == 0 (mod 4)).
From Colin Barker, Mar 12 2018: (Start)
G.f.: x^5*(1 + 2*x + 2*x^2 + 6*x^3 + 5*x^4 + 8*x^5) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-3) - 2*a(n-5) + 2*a(n-6) + a(n-8) - 2*a(n-10) + a(n-11) for n>14.
(End)

A334894 Number of maximal subsets of [n] avoiding 3-term arithmetic progressions and containing n if n>0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 15, 4, 20, 7, 18, 6, 1, 3, 10, 29, 54, 123, 2, 16, 44, 170, 2, 31, 2, 10, 24, 70, 1, 10, 2, 2, 10, 26, 2, 2, 82, 221, 20, 1, 3, 10, 27, 58, 167, 408, 831, 2005, 4216, 14, 36, 106, 2, 6, 18, 30, 2, 2, 2, 8, 34, 2, 2, 4, 8, 12, 80, 211

Offset: 0

Alois P. Heinz, May 14 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..136
Eric Weisstein's World of Mathematics, Nonaveraging Sequence
Wikipedia, Arithmetic progression
Wikipedia, Iverson bracket
Wikipedia, Salem-Spencer set
Index entries related to non-averaging sequences

Last elements of rows of A334892.

Cf. A003002, A262347, A334187.

a(n) = A262347(n) - [n > 0 and A003002(n) = A003002(n-1)] * A262347(n-1).
a(n) = A334892(n,A003002(n)).
a(n) = A334187(n,A003002(n)) - [n > 0] * A334187(n-1,A003002(n)).

cp's OEIS Frontend

A212964 Number of (w,x,y) with all terms in {0,...,n} and |w-x| < |x-y| < |y-w|.

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A051013 Number of nonaveraging subsets on {1,2,...,n}.

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A334892 Number T(n,k) of k-element subsets of [n] avoiding 3-term arithmetic progressions and containing n if n>0; triangle T(n,k), n>=0, 0<=k<=A003002(n), read by rows.

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A262347 Number of subsets of [1..n] of maximal size that are free of 3-term arithmetic progressions.

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Maple

Mathematica

Extensions

A300760 Number of ways to select 4 numbers from the set of the first n natural numbers avoiding 3-term arithmetic progressions.

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Examples

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Programs

Mathematica

PARI

Formula

A334894 Number of maximal subsets of [n] avoiding 3-term arithmetic progressions and containing n if n>0.

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