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.

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

1,2

Comments

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.

Links

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

Example

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.

Maple

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

Prog

(Python)
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
(PARI) 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

Crossrefs

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

Sequence in context: A098535 A069239 A321080 * A010265 A239963 A084501

Adjacent sequences: A357258 A357259 A357260 * A357262 A357263 A357264

Keyword

nonn,look,easy

Author

John Tyler Rascoe, Oct 08 2022

Status

approved