A057353 -id:A057353 - cp's OEIS frontend

A245230 Triangle T(m,n), 1<=n<=m, read by rows: maximum frustration of complete bipartite graph K(m,n).

0, 0, 1, 0, 1, 2, 0, 2, 3, 4, 0, 2, 3, 5, 7, 0, 3, 4, 7, 9, 11, 0, 3, 5, 8, 10, 13, 16, 0, 4, 6, 10, 12, 16, 19, 22

Offset: 1

Robert Israel, Jul 14 2014

The maximum frustration of a graph is the maximum cardinality of a set of edges that contains at most half the edges of any cut-set.  Another term that is used is "line index of imbalance".  It is also equal to the covering radius of the coset code of the graph.
T(m,n) is symmetric in m and n, so only m>=n is listed here.
T(m,1) = 0.
T(m,2) = floor(m/2) = A004526(m).
T(m,3) = floor(3*m/4) = A057353(m).
T(m,4) = A245231(m).
T(m,5) = A245227(m).
T(m,6) = A245239(m).
T(m,7) = A245314(m).

			For m=n=3 a set of edges attaining the maximum cardinality T(3,3)=2 is {(1,4),(2,5)}.
Triangle starts
  0;
  0, 1;
  0, 1, 2;
  0, 2, 3, 4;
  0, 2, 3, 5, 7;
  0, 3, 4, 7, 9,  11;
  0, 3, 5, 8, 10, 13, 16;
  0, 4, 6, 10, 12, 16, 19, 22.

G. S. Bowlin, Maximum Frustration in Bipartite Signed Graphs, Electr. J. Comb. 19(4) (2012) #P10.
R. L. Graham and N. J. A. Sloane, On the Covering Radius of Codes, IEEE Trans. Inform. Theory, IT-31(1985), 263-290.
P. Solé and T. Zaslavsky, A Coding Approach to Signed Graphs, SIAM J. Discr. Math 7 (1994), 544-553.

Cf. A245231 (row/column 4), A245227 (row/column 5), A245239 (row/column 6), A245314 (row/column 7)

A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Apr 23 2019

The old definition was "Lexicographically earliest sequence starting with a(1) = 0 such that a(n) is the number of pairs of contiguous terms whose sum is a(n)".
From Paul Curtz, Apr 27 2019: This can be written as a triangle:
                    0
                  1   1
                1   2   1
              2   2   2   2
            2   3   2   3   2
          3   3   3   3   3   3
        3   4   3   4   3   4   3
...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..11326

Cf. A002024.
Cf. also A007590, A057353, A106466 and A238410.
For other versions see A307720 and A378117.

m = 107; a[1]=0;
a24[n_] := Ceiling[(Sqrt[8n+1]-1)/2];
Array[a, m] /. Solve[Table[a[n] + a[n+1] == a24[n], {n, 1, m-1}]][[1]] (* Jean-François Alcover, Jun 02 2019, after Rémy Sigrist's formula *)

v=0; rem=wanted=1; for (n=1, 107, print1 (v", "); v=wanted-v; if (rem--==0, rem=wanted++)) \\ Rémy Sigrist, Apr 23 2019

a(n) + a(n+1) = A002024(n). - Rémy Sigrist, Apr 24 2019

Let t_m = m*(m+1)/2. Write n = t_m - i with m >= 1 and 0 <= i < m. Then a(n) = m/2 if m is even, or if m is odd, a(n) = (m-1)/2 + (i-1 mod 2). - N. J. A. Sloane, Nov 16 2024

Definition clarified by Rémy Sigrist and N. J. A. Sloane, Nov 17 2024

A047379 Numbers that are congruent to {0, 2, 4, 5} mod 7.

Original entry on oeis.org

0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 25, 26, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 46, 47, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 95, 96

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047215, A057353.

[Floor((7*n-5)/4) : n in [1..100]]; // Wesley Ivan Hurt, Dec 03 2014

A047379:=n->floor((7*n-5)/4): seq(A047379(n), n=1..100); # Wesley Ivan Hurt, Dec 03 2014

Select[Range[0,100],MemberQ[{0,2,4,5},Mod[#,7]]&] (* Harvey P. Dale, Apr 10 2014 *)

a(n) = floor(floor((7n + 2)/2)/2).
a(n) = floor((7n-5)/4). - Gary Detlefs, Mar 07 2010
G.f.: x^2*(2+2*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Dec 03 2014: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5), n>5;
a(n) = (14*n-13-(-1)^n+2*i^n*(-1)^((3+(-1)^n)/4))/8, where i = sqrt(-1);
a(n) = A047215(n-1) - A057353(n-1). (End)

A090221 Array used for numerators of g.f.s for column sequences of array A090214 ((4,4)-Stirling2).

Original entry on oeis.org

1, 96, 72, 14400, 16, 38400, 3456000, 1, 27000, 22104000, 1270080000, 7200, 34905600, 16111872000, 682795008000, 856, 21154176, 48248363520, 15279164006400, 516193026048000, 48, 6064128, 54644474880, 78083415244800

Offset: 4

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is A037915(k-4)= floor(3*(k-4)/4)+1, k>=4: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].

The g.f. G(k,x) for the k-th column (with leading zeros) of array A090214 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,4-r)*fallfac(4,4-r)*G(k-r,x),r=1..4))/(1-fallfac(k,4)*x), k>=4, with inputs G(k,x)=0 for k=1,2,3 and G(4,x)=x/(1-4!*x); where fallfac(n,m) := A008279(n,m) (falling factorials with fallfac(n,0) := 1). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=4: recurrence for S_{4,4}(n,k).

			[1]; [96]; [72,14400]; [16,38400,3456000]; [1,27000,22104000,1270080000]; ...
G(5,x)/x^2 = 96/((1-4!*x)*(1-5*4*3*2*x)). kmax(5)=0, hence P(5,x)=a(5,0)=96; x^2 from x^ceiling(5/4).

W. Lang, First 8 rows.

a(k, n) from: sum(a(k, n)*x^n, n=0..kmax(k)) = G(k, x)* product(1-fallfac(p, 4)*x, p=4..k)/x^ceiling(k/4), k>=4, with G(k, x) defined from the recurrence given above and kmax(k) := A057353(k-4)= floor(3*(k-4)/4)= A037915(k-4)-1.

A187228 Rank transform of the sequence floor(3n/4); complement of A187229.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 11, 12, 14, 16, 18, 19, 20, 22, 24, 25, 27, 28, 30, 31, 32, 34, 35, 36, 38, 40, 42, 43, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 72, 73, 75, 76, 78, 79, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 97, 99, 100, 102, 103, 104, 106, 108, 109, 111, 112, 114, 115, 116, 118, 120, 121, 123, 124, 126, 127, 128, 130, 131, 132, 134, 136, 138, 139, 140, 142, 143, 144, 146, 148, 149, 150

Offset: 1

Clark Kimberling, Mar 07 2011

nonn

See A187224.

Cf. A187224, A187229.

seqA=Table[Floor[3n/4],{n,1,220}] (*A057353*)
seqB=Table[n,{n,1,220}];(*A000027*)
jointRank[{seqA_,seqB_}]:={Flatten@Position[#1,{,1}],Flatten@Position[#1,{,2}]}&[Sort@Flatten[{{#1,1}&/@seqA,{#1,2}&/@seqB},1]];
limseqU=FixedPoint[jointRank[{seqA,#1[[1]]}]&,jointRank[{seqA,seqB}]][[1]] (*A187228*)
Complement[Range[Length[seqA]],limseqU] (*A187229*)
(*by Peter J. C. Moses, Mar 07 2011*)

A187322 a(n) = floor(n/2) + floor(3*n/4).

Original entry on oeis.org

0, 0, 2, 3, 5, 5, 7, 8, 10, 10, 12, 13, 15, 15, 17, 18, 20, 20, 22, 23, 25, 25, 27, 28, 30, 30, 32, 33, 35, 35, 37, 38, 40, 40, 42, 43, 45, 45, 47, 48, 50, 50, 52, 53, 55, 55, 57, 58, 60, 60, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 75, 77, 78, 80, 80, 82, 83, 85, 85, 87, 88, 90, 90, 92, 93, 95, 95, 97

Offset: 0

Clark Kimberling, Mar 08 2011

List of quadruples [5*k, 5*k, 5*k+2, 5*k+3]. - Luce ETIENNE, Aug 14 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008587, A016873, A016885, A004526, A057353.

Table[Floor[n/2]+Floor[3n/4], {n,0,120}]
LinearRecurrence[{1,0,0,1,-1},{0,0,2,3,5},80] (* Harvey P. Dale, Dec 05 2018 *)

a(n) = n\2 + 3*n\4; \\ Altug Alkan, Aug 14 2017

concat(vector(2), Vec(x^2*(2 + x + 2*x^2) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^100))) \\ Colin Barker, Aug 14 2017

def A187322(n): return (n>>1)+(3*n>>2) # Chai Wah Wu, Jan 31 2023

a(n) = A004526(n) + A057353(n). - Michel Marcus, Aug 14 2017
a(n) = (10*n-5+3*cos(n*Pi)+2*(cos(n*Pi/2)-sin(n*Pi/2)))/8. - Luce ETIENNE, Aug 14 2017
From Colin Barker, Aug 14 2017: (Start)
G.f.: x^2*(2 + x + 2*x^2) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
(End)

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A245230 Triangle T(m,n), 1<=n<=m, read by rows: maximum frustration of complete bipartite graph K(m,n).

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A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

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A047379 Numbers that are congruent to {0, 2, 4, 5} mod 7.

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A090221 Array used for numerators of g.f.s for column sequences of array A090214 ((4,4)-Stirling2).

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A187228 Rank transform of the sequence floor(3n/4); complement of A187229.

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A187322 a(n) = floor(n/2) + floor(3*n/4).

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