A275204 -id:A275204 - cp's OEIS frontend

A275599 Triangle read by rows: T(n,k) = number of right-skewed domino towers with n dominoes having a base of k dominoes placed end-to-end.

1, 3, 1, 7, 4, 1, 15, 12, 4, 1, 31, 27, 13, 4, 1, 63, 61, 34, 13, 4, 1, 127, 124, 77, 35, 13, 4, 1, 255, 258, 165, 86, 35, 13, 4, 1, 511, 513, 348, 185, 87, 35, 13, 4, 1, 1023, 1039, 698, 399, 196, 87, 35, 13, 4, 1, 2047, 2062, 1410, 811, 423, 197, 87, 35, 13, 4, 1

Offset: 2

Tricia Muldoon Brown, Aug 03 2016

Domino towers are created by stacking domino blocks horizontally on a convex base of k dominoes. A right-skewed domino tower is a parallelogram domino tower such that at least one column of the polyomino is to the right of the base.

			Triangle begins:
1;
3, 1;
7, 4, 1;
15, 12, 4, 1;
...
For n = 5 and k = 3, each tower has a convex base of three dominoes. The fourth domino may be placed directly above the rightmost domino of the base, in which case the fifth domino must be placed on the fourth domino so its right end is not above the base. Alternately, the fourth domino may be placed so its right end is not above the base, leaving three choices for the fifth domino: directly above, above and to the right, or directly to the left on the same level. Thus T(5,3) = 4.

T. M. Brown Convex domino towers, arXiv:1608.01562 [math.CO], (2016)
Tricia Muldoon Brown, Examples of right- or left-skewed domino towers of 10 dominoes having a base of 4 dominoes

Column 1: A000225, n>=1.

Cf. A275204.

T(n,k) = Sum_{i=1..k} 2*T(n-k,i)+A(n-k,i) where A(n,k) is given by A275204 and with initial conditions T(n+1,n)=1 and T(n,k)=0 if n<2 and k<1, or n

G.f.: x^k/(1-2x^k) Sum_{i=1..k}*A_k(x)*(Sum_{Subsets S of {i,i+1,..,k-1}} (Product_{j in S} 2x^j/(1-2x^k)) where A_k(x) is the generating function in A275204.

A357261 a(n) is the number of blocks in the bottom row after adding n blocks to the preceding structure of rows. See Comments and Example sections for more details.

Original entry on oeis.org

1, 3, 3, 3, 4, 1, 3, 1, 5, 4, 3, 3, 4, 6, 1, 3, 6, 3, 1, 7, 5, 3, 2, 2, 3, 5, 8, 1, 3, 6, 1, 6, 3, 1, 9, 6, 3, 1, 10, 7, 4, 2, 1, 1, 2, 4, 7, 11, 1, 3, 6, 10, 3, 9, 4, 12, 5, 11, 5, 13, 5, 11, 4, 12, 7, 3, 14, 8, 2, 12, 8, 5, 3, 2, 2, 3, 5

Offset: 1

John Tyler Rascoe, Oct 08 2022

A structure of rows is built up successively where each n blocks are added onto the preceding structure. The first row has an initial width of 3. After n = 1, n blocks are first added filling in the last row where n - 1 left off.
Once a row is filled a new row is started below it. After adding n blocks, if the final row reached is filled exactly, then the width of the next row is increased by one. Otherwise the width of the next row is the same as the previous.
Assuming the rows are built according to the given algorithm, a(n) is the number of blocks tagged 'n' in the last row that includes a block tagged 'n'." - Peter Luschny, Dec 20 2022
Will this sequence ever reach a point after which a(n) grows linearly? This is the case using an initial width of 2 instead of 3.

			After blocks 1 and 2, the initial row width 3 is exactly filled and hence the next row (of 3's and 4) is 1 longer.
  |1|2|2|       initial row
  |3|3|3|4|
  |4|4|4|5|
  |5|5|5|5|
  |6|6|6|6|6|
  |6|_|_|_|_|   last row after n=6
For n=6, the structure after blocks 1 through 6 have been added is as shown above and its final row has just one block (one 6) so that a(6) = 1.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A002024, A057176, A064434, A096535, A104647, A275204.

A357261_list := proc(maxn) local A, g, c, n, r;
A := []; g := 3; c := 0;
for n from 1 to maxn do
    r := irem(n + c, g);
    c := r;
    if r = 0 then
        r := g; g := g + 1;
    fi;
    A := [op(A), r];
od; return A end:
A357261_list(77); # Peter Luschny, Dec 21 2022

lista(nn) = my(nbc=3, col=0, list=List()); for (n=1, nn, col = lift(Mod(col+n, nbc)); listput(list, if (col, col, nbc)); if (!col, nbc++);); Vec(list); \\ Michel Marcus, Oct 17 2022

def A357261_list(maxn):
    """Returns a list of the first maxn terms"""
    A = []
    g = 3
    c = 0
    for n in range(1,maxn+1):
        if (n + c)%g == 0:
            A.append(g)
            g += 1
            c = 0
        else:
            A.append((n + c)%g)
            c = A[-1]
    return A

A358073 a(n) is the row position of the n-th number n after adding the number n, n times to the preceding triangle. A variant of A357261, see Comments and Examples for more details.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 4, 3, 3, 4, 6, 9, 13, 6, 21, 16, 33, 15, 34, 18, 3, 25, 12, 36, 25, 51, 18, 46, 15, 45, 16, 48, 21, 55, 30, 6, 43, 21, 60, 40, 81, 24, 67, 12, 57, 4, 51, 99, 49, 99, 3, 55, 108, 15, 70, 126, 36, 94, 6, 66, 127, 42, 105, 22, 87, 6, 73, 141, 63

Offset: 1

John Tyler Rascoe, Oct 29 2022

A triangle is built up successively where n appears n times within the triangle. Each row has a set width before n is added, and the first row begins with a width of 1.
Numbers n are added to the first open position within the triangle or where the previous n left off so that no gaps are left in the rows of the triangle. If the row position of the n-th number n placed is the rightmost position within that row, then the width of the next row is increased by n. Otherwise, the width of the next row stays the same as the previous one.
The next row's width can only increase after a given n is added all n times. So when a row is filled after adding fewer than n n's, the next row, by definition, will have the same width.

			After 5 is added 5 times, the fifth 5 falls in the rightmost row position. So the width of the next row is increased by 5.
  |1|       initial row
  |2|2|
  |3|3|3|4|
  |4|4|4|5|
  |5|5|5|5|
  |6|6|6|6|6|6|7|7|7|
  |7|7|7|7|_|_|_|_|_|
a(7) = 4 because the row position of the seventh 7 added is 4.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
John Tyler Rascoe, Scatterplot of a(n) for n = 1...50000

Cf. A002024, A057176, A064434, A096535, A104647, A275204, A357261.

A358073_list := proc(maxn)  local A, g, c, n, r;
A := []; g := 1; c := 0;
for n from 1 to maxn do
    r := irem(n + c, g);
    c := r;
    if r = 0 then
        r := g;
        g := g + n;
    fi;
    A := [op(A), r];
od; return A end:
A358073_list(69); # Peter Luschny, Dec 21 2022

def A358073_list(maxn):
    """Returns a list of the first maxn terms"""
    A = []
    g = 1
    c = 0
    for n in range(1,maxn+1):
        if (n + c)%g ==0:
            A.append(g)
            g += n
            c = 0
        else:
            A.append((n + c)%g)
            c = A[-1]
    return A

A275662 Triangle read by rows: T(n,k) = number of convex domino towers with n dominoes having widest row with k dominoes.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 15, 18, 7, 1, 31, 48, 17, 9, 1, 63, 109, 49, 20, 11, 1, 127, 240, 115, 52, 24, 13, 1, 255, 498, 258, 122, 61, 28, 15, 1, 511, 1026, 551, 261, 136, 71, 32, 17, 1, 1023, 2065, 1163, 531, 298, 157, 81, 36, 19, 1

Offset: 1

Tricia Muldoon Brown, Aug 04 2016

A domino tower is built by placing dominoes horizontally on a convex horizontal base. A domino tower is convex if all its columns and rows are convex.

			Triangle begins:
1;
3, 1;
7, 6, 1;
15, 18, 7, 1;
...
If n = 3 and k = 2, the widest row of the domino tower has two dominoes. Thus the third domino may be found supporting the row of two dominoes in one way or being supported by the row of two dominoes in 5 ways, so T(3,2) = 6.

T. M. Brown, Convex domino towers, arXiv:1608.01562 [math.CO] (2016).

Column 1: A000225, n>=1.

Cf. A117468, A275204, A275599.

G.f.: (2*A_k(x)+B_k(x))*(C_{k-1}(x)+1) where A_k(x) is the generating function on right-skewed domino towers with a base of k dominoes from the sequence A275599, B_k(x) is the generating function on domino stacks with a base of k dominoes associated with the sequence A275204, and C_k(x) is the generating function on flat partitions whose largest part is k-1 given by the sequence A117468.

A338531 a(n) is the number of row-convex domino towers with n bricks (rows need not be offset).

Original entry on oeis.org

1, 4, 16, 61, 225, 815, 2923, 10428, 37097, 131776, 467732, 1659537, 5886945, 20880912, 74060619, 262672473, 931615218, 3304121816, 11718561425, 41561571533

Offset: 1

Alexander M. Haupt, Nov 01 2020

A domino tower is a stack of bricks, where (1) the bottom row is contiguous, and (2) each brick is supported from below by at least half of a brick. Note, that in this definition of domino towers, rows need not be offset by half a brick. The number of domino towers with n bricks is given by 4^(n-1).

In this sequence we want all rows to be convex, rather than just the bottom row.

			For n=2, the a(2) = 4 domino towers are:
+-------+-------+
|       |       |
+-------+-------+
+-------+
|       |
+---+---+---+
    |       |
    +-------+
+-------+
|       |
+-------+
|       |
+-------+
    +-------+
    |       |
+---+---+---+
|       |
+-------+
For n=4, the 4^(n-1)-a(n)=64-61=3 domino towers, which have non-convex rows are:
+-------+   +-------+
|       |   |       |
+-------+---+---+---+
|       |       |
+-------+-------+
+-------+   +-------+
|       |   |       |
+---+---+---+-------+
    |       |       |
    +-------+-------+
+-------+       +-------+
|       |       |       |
+---+---+---+---+---+---+
    |       |       |
    +-------+-------+

T. M. Brown, Convex domino towers, J. of Integer Sequences, 20 (2017), p.17.

Cf. A275204.

f[n_, l_] := (f[n, l] =
Sum[(3 - 2 i + 2 l) f[n - i, i], {i, 1, Min[n, l + 1]}]);
f[0, l_] := 1;
Table[Sum[f[n - l, l], {l, 1, n}], {n, 1, 20}]

G.f.: G(x) := [ Sum_{l>0} z^l (z^3 T(3,l)+(2 z^2-1) T(2,l)+(2 z+1) T(1,l)) ] / (z^5 T(2,3)+(3 z-1) z^3 T(1,3)+(4 z^3-3 (z+1) z+1) T(1,2)) , where
T(i,j) := A(i)B(j)-A(j)B(i),
A(l) := Sum_{n>=0} (z^(l n+n^2+n) (-z;z)_n)/((z;z)_n)^2,
B(l) := Sum_{n>=0} (z^(l n+n^2+n) (-z;z)n)/((z;z)_n)^2 * (l+n+Sum{m=1,...,n} (3 z^m+1)/(1-z^(2 m))), and
(a;q)_n is the q-Pochhammer symbol

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A275599 Triangle read by rows: T(n,k) = number of right-skewed domino towers with n dominoes having a base of k dominoes placed end-to-end.

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A357261 a(n) is the number of blocks in the bottom row after adding n blocks to the preceding structure of rows. See Comments and Example sections for more details.

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A358073 a(n) is the row position of the n-th number n after adding the number n, n times to the preceding triangle. A variant of A357261, see Comments and Examples for more details.

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A275662 Triangle read by rows: T(n,k) = number of convex domino towers with n dominoes having widest row with k dominoes.

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A338531 a(n) is the number of row-convex domino towers with n bricks (rows need not be offset).

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