A104647 -id:A104647 - cp's OEIS frontend

A105859 Numbers m such that A104647(m) = A104647(m-1) + A104647(m-2).

2, 3, 4, 5, 6, 9, 12, 17, 21, 25, 30, 37, 38, 39, 42, 46, 47, 51, 55, 56, 61, 66, 73, 74, 79, 86, 89, 93, 94, 102, 110, 111, 117, 118, 120, 124, 130, 136, 141, 144, 148, 154, 155, 158, 162, 170, 171, 172, 173, 177, 185, 186, 187, 188, 189, 190, 195, 201, 206, 212

Offset: 1

Reinhard Zumkeller, Apr 23 2005

nonn

			n=12: A104647(12)=17 = A104647(11)+A104647(10)=9+8=7, therefore 12 is a term;
n=42: A104647(42)=60 = A104647(41)+A104647(40)=31+29=60, therefore 42 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A105860.

N:= 10^3: # for all terms <= N
A104647:= Array(0..N):
A104647[1]:= 1:
for n from 2 to N do A104647[n]:= (A104647[n-1] mod n) + (A104647[n-2] mod n) od:
select(t -> (A104647[t] = A104647[t-1]+A104647[t-2]), [$2..N]); # Robert Israel, Aug 14 2018

A105857 Where records occur in A104647.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 9, 12, 17, 21, 30, 42, 47, 51, 61, 74, 79, 102, 124, 136, 148, 195, 206, 218, 252, 279, 300, 314, 323, 346, 416, 475, 503, 532, 574, 589, 596, 711, 731, 744, 778, 785, 818, 850, 862, 873, 929, 950, 1015, 1088, 1117, 1141, 1152, 1230, 1245

Offset: 0

Reinhard Zumkeller, Apr 23 2005

nonn

A105858(n) = A104647(a(n)).

A105858 Records in A104647.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 12, 17, 28, 32, 50, 60, 66, 69, 106, 107, 133, 200, 221, 224, 254, 363, 369, 386, 448, 484, 491, 517, 620, 680, 723, 844, 865, 900, 918, 988, 1030, 1084, 1221, 1255, 1260, 1409, 1446, 1467, 1513, 1638, 1737, 1796, 1840, 1930, 1975, 2012, 2036

Offset: 0

Reinhard Zumkeller, Apr 23 2005

nonn

a(n) = A104647(A105857(n)).

A105860 Numbers m such that A104647(m) < A104647(m-1) + A104647(m-2).

Original entry on oeis.org

7, 8, 10, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 40, 41, 43, 44, 45, 48, 49, 50, 52, 53, 54, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 91, 92, 95, 96, 97, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Apr 23 2005

nonn

			n=11: A104647(11)=9 < A104647(10)+A104647(9)=8+12=20, therefore 11 is a term;
n=58: A104647(58)=60 < A104647(57)+A104647(56)=52+66=118, therefore 58 is a term.

Cf. A105859.

A105855 A104647(n+1) - A104647(n).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, -2, 0, 6, -4, 1, 8, -4, 3, -2, 0, 14, -4, -10, 4, 14, -4, -14, 4, 14, -8, 5, -4, 0, 25, -6, -14, 11, -4, -29, 1, 8, 9, 17, -14, 2, 29, -12, -28, 3, 20, 23, -5, -32, 11, 29, -12, -37, 3, 20, 23, -14, 8, -7, 0, 53, -9, -20, -31, 12, 46, -9, -32, 26, -7, -53, 10, 29, 39, -7

Offset: 0

Reinhard Zumkeller, Apr 23 2005

sign

A105856 a(n) = a(n-1) + A104647(n), a(0) = 0.

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 20, 26, 32, 44, 52, 61, 78, 91, 107, 121, 135, 163, 187, 201, 219, 251, 279, 293, 311, 343, 367, 396, 421, 446, 496, 540, 570, 611, 648, 656, 665, 682, 708, 751, 780, 811, 871, 919, 939, 962, 1005, 1071, 1132, 1161, 1201, 1270, 1327, 1347

Offset: 0

Reinhard Zumkeller, Apr 23 2005

nonn

a(n)/(n^2) ---> 1/2.

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

cp's OEIS Frontend

A105859 Numbers m such that A104647(m) = A104647(m-1) + A104647(m-2).

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A105857 Where records occur in A104647.

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A105858 Records in A104647.

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A105860 Numbers m such that A104647(m) < A104647(m-1) + A104647(m-2).

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A105855 A104647(n+1) - A104647(n).

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A105856 a(n) = a(n-1) + A104647(n), a(0) = 0.

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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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Maple

Python