A008642 -id:A008642 - cp's OEIS frontend

A333647 Number of nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

1, 1, 1, 2, 4, 8, 16, 34, 74, 169, 397, 953, 2319, 5732, 14370, 36466, 93468, 241767, 630499, 1656372, 4380128, 11652459, 31168689, 83788315, 226272531, 613632359, 1670604607, 4564607998, 12513715526, 34412992018, 94912212872, 262484672621, 727770127583

Offset: 0

Alois P. Heinz, Mar 31 2020

nonn

The maximal height in all paths of length n is floor(ceil(n/2)^2/4) = A008642(n-3) for n>2.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Alois P. Heinz, Animation of a(9) = 169 paths
Wikipedia, Counting lattice paths

Cf. A008642, A225338, A333678, A333679, A333680, A333682.

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(
      b(x-1, y+j, j), j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);

b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Sum[
     b[x-1, y+j, j], {j, Max[t-1, -y], Min[x(x-1)/2-y, t+1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12

Offset: 0

Wolfdieter Lang, Apr 04 2007

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.

			The triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6   7   8   9  10
   0:  1
   1:  1   1
   2:  0   1   1
   3:  0  -1   1   1
   4:  1  -1  -2   1   1
   5:  1   2  -2  -3   1   1
   6:  0   2   4  -3  -4   1   1
   7:  0  -2   4   7  -4  -5   1   1
   8:  1  -2  -6   7  11  -5  -6   1   1
   9:  1   3  -6 -13  11  16  -6  -7   1   1
  10:  0   3   9 -13 -24  16  22  -7  -8   1   1
... reformatted by _Wolfdieter Lang_, Oct 16 2012
Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
From _Wolfdieter Lang_, Oct 16 2012: (Start)
S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)

Wolfdieter Lang, First 15 rows
Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.
Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1), The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
Cf. A128495 for S(2; n, x) coefficient table.
The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
From Wolfdieter Lang, Oct 16 2012: (Start)
a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 5, 8, 7, 12, 11, 16, 15, 21, 20, 27, 25, 33, 32, 40, 38, 48, 46, 56, 54, 65, 63, 75, 72, 85, 83, 96, 93, 108, 105, 120, 117, 133, 130, 147, 143, 161, 158, 176, 172, 192, 188, 208, 204, 225, 221, 243, 238, 261, 257, 280, 275

Offset: 0

Gus Wiseman, May 15 2019

nonn

			The a(7) = 1 through a(15) = 12 partitions (A = 10, B = 11, C = 12):
  (421)  (521)  (432)  (631)  (542)  (543)  (643)   (653)   (654)
                (531)  (721)  (632)  (732)  (652)   (842)   (753)
                (621)         (641)  (741)  (742)   (851)   (762)
                              (731)  (831)  (751)   (932)   (843)
                              (821)  (921)  (832)   (941)   (852)
                                            (841)   (A31)   (861)
                                            (931)   (B21)   (942)
                                            (A21)           (951)
                                                            (A32)
                                                            (A41)
                                                            (B31)
                                                            (C21)

Cf. A000041, A001399, A005044, A008642, A069905, A124278.

Cf. A325686, A325690, A325691, A325694, A325696.

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^7*(1 + x + 2*x^2) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9.
(End)
a(n) = A325696(n)/6. - Alois P. Heinz, Jun 18 2020

A333679 Sum of the heights of all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 20, 53, 137, 375, 1035, 2878, 7988, 22308, 62642, 176692, 499818, 1418228, 4035568, 11512449, 32916181, 94313011, 270757747, 778694171, 2243200705, 6471953522, 18699169766, 54098598824, 156706773404, 454457344755, 1319382151919, 3834346819731

Offset: 0

Alois P. Heinz, Apr 01 2020

nonn

The maximal height in all paths of length n is floor(ceil(n/2)^2/4) = A008642(n-3) for n>2.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Alois P. Heinz, Animation of A333647(9) = 169 paths with height sum a(9) = 375
Wikipedia, Counting lattice paths

Cf. A008642, A333647, A333678, A333680.

b:= proc(x, y, t, h) option remember;
      `if`(x=0, h, add(b(x-1, y+j, j, max(h, y)),
         j=max(t-1, -y)..min(x*(x-1)/2-y, t+1)))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..32);

b[x_, y_, t_, h_] := b[x, y, t, h] =
     If[x == 0, h, Sum[b[x - 1, y + j, j, Max[h, y]],
     {j, Max[t - 1, -y], Min[x(x - 1)/2 - y, t + 1]}]];
a[n_] := b[n, 0, 0, 0];
a /@ Range[0, 32] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

A128498 Fourth column (m=3) of triangle A128494.

Original entry on oeis.org

1, 1, -3, -3, 7, 7, -13, -13, 22, 22, -34, -34, 50, 50, -70, -70, 95, 95, -125, -125, 161, 161, -203, -203, 252, 252, -308, -308, 372, 372, -444, -444, 525, 525, -615, -615, 715, 715, -825, -825, 946, 946, -1078, -1078, 1222, 1222, -1378, -1378, 1547, 1547, -1729, -1729, 1925, 1925, -2135, -2135

Offset: 0

Wolfdieter Lang, Apr 04 2007

Unsigned, this is the repeated sequence A002623.

Index entries for linear recurrences with constant coefficients, signature (1,-4,4,-6,6,-4,4,-1,1).

Cf. A008642 (unsigned column m=2). A128499 (column m=4).

CoefficientList[Series[1/((1-x)(1+x^2)^4),{x,0,60}],x] (* or *) LinearRecurrence[{1,-4,4,-6,6,-4,4,-1,1},{1,1,-3,-3,7,7,-13,-13,22},60] (* Harvey P. Dale, Jul 04 2021 *)

Vec(1/((1-x)*(1+x^2)^4) + O(x^50)) \\ Michel Marcus, Mar 16 2015

G.f.: 1/((1-x)*(1+x^2)^4).
a(2*k) = a(2*k+1)= ((-1)^k)*A002623(n), k>=0.
a(n) = (-1)^((2*n-1+(-1)^n)/4)*((n+2)*(n+7)*(2*n+9)+3*(n+3)*(n+6)*(-1)^n+12*(-1)^((2*n-1+(-1)^n)/4))/192. - Luce ETIENNE, Mar 13 2015

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++ ]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A355880 a(n) is a maximum of tuv such that 2tu + 2tv + 2uv <= n, where t,u,v are positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 6, 6, 8, 8, 8, 8, 8, 8, 9, 9, 12, 12, 12, 12, 12, 12, 12, 12, 16, 16, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 24, 24, 27, 27, 27, 27, 27, 27, 27, 27, 30, 30, 32, 32, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 40

Offset: 1

Gleb Ivanov, Jul 20 2022

nonn

The largest volume of a rectangular cuboid with a surface area less than or equal to n.

Cf. A008642.

for k in range(1, 200):
    max_v = 0
    for a in range(1, k):
        for b in range(1, a+1):
            for c in range(1, b+1):
                if 2*a*b + 2*a*c + 2*b*c <= k and a*b*c > max_v:
                    max_v = a*b*c
    print(max_v, end = ', ')

A345206 Maximum number of unit cubes that can be fully enclosed in n unit cubes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 22, 22, 24, 24, 27

Offset: 8

Abraham Maxfield, Jun 11 2021

nonn

Cubes are assumed to be aligned in a 3D grid. Cubes with an exposed edge or corner are not considered enclosed.

The Moore neighborhood of a cube in a 3-D grid consists of the 26 that share a face, an edge, or a vertex with it. - N. J. A. Sloane, Jul 12 2021

			a(26) = 1 as the number of neighbors in Moore's neighborhood is 26 in 3D.

Cf. A345205. 3D equivalent to A008642.

A359979 Irregular table T(n,k), n >= 0 and k >= 0, read by rows with T(n + 3*k,k) = A008619(n).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 1, 3, 2, 4, 2, 1, 4, 3, 1, 5, 3, 2, 5, 4, 2, 1, 6, 4, 3, 1, 6, 4, 3, 2, 7, 5, 4, 2, 1, 7, 6, 4, 3, 1, 8, 6, 5, 3, 2, 8, 7, 5, 4, 2, 1, 9, 7, 6, 4, 3, 1, 9, 8, 6, 5, 3, 2, 10, 8, 7, 5, 4, 2, 1, 10, 9, 7, 6, 4, 3, 1, 11, 9, 8, 6, 5, 3, 2

Offset: 0

Philippe Deléham, Jan 20 2023

A008620(n) is the length of the n-th row.

			Table: n >= 0, k >= 0.
  1;
  1;
  2;
  2, 1;
  3, 1;
  3, 2;
  4, 2, 1;
  4, 3, 1;
  5, 3, 2;
  5, 4, 2, 1;
  6, 4, 3, 1;
  6, 5, 3, 2;
  7, 5, 4, 2, 1;
  7, 6, 4, 3, 1;
  8, 6, 5, 3, 2;
  8, 7, 5, 4, 2, 1;
  9, 7, 6, 4, 3, 1;
  9, 8, 6, 5, 3, 2;
  ....

Cf. A001399, A008619, A008620, A008642, A008805.

Sum_{k >= 0} T(n,k) = A001399(n).

A373612 Size of the largest polyiamond that can be enclosed in n cells on a triangular lattice.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 6, 6, 6, 7, 7, 10

Offset: 6

Abraham Maxfield, Jun 10 2024

			A single triangular face takes 12 triangles to completely enclose so a(12) = 1.

Cf. A290648. A257594 is the hexagonal tiling equivalent. A008642 is the square tiling equivalent (if prepended with 7 zeros).

cp's OEIS Frontend

A333647 Number of nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

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A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

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A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

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Formula

A333679 Sum of the heights of all nonnegative lattice paths from (0,0) to (n,0) such that slopes of adjacent steps differ by at most one, assuming zero slope before and after the paths.

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A128498 Fourth column (m=3) of triangle A128494.

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A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A355880 a(n) is a maximum of t*u*v such that 2*t*u + 2*t*v + 2*u*v <= n, where t,u,v are positive integers.

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A345206 Maximum number of unit cubes that can be fully enclosed in n unit cubes.

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A359979 Irregular table T(n,k), n >= 0 and k >= 0, read by rows with T(n + 3*k,k) = A008619(n).

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A355880 a(n) is a maximum of tuv such that 2tu + 2tv + 2uv <= n, where t,u,v are positive integers.