A096535 -id:A096535 - cp's OEIS frontend

A099536 Sum of the first n digits of Zeta(3) (Apery's constant), including the initial 1.

1, 3, 3, 5, 5, 10, 16, 25, 25, 28, 29, 34, 43, 48, 57, 61, 63, 71, 76, 79, 88, 97, 104, 107, 115, 116, 122, 123, 128, 129, 130, 134, 138, 147, 156, 165, 165, 172, 178, 182, 191, 199, 205, 207, 216, 218, 221, 225, 225, 229, 238, 246, 254, 262, 263, 270, 279, 281

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 22 2004

			Zeta(3)=1.20205690... so sequence begins 1, 1+2, 1+2+0, 1+2+0+2, 1+2+0+2+0,
1+2+0+2+0+5,... which gives 1, 3, 3, 5, 5, 10, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Analogous sequences for other constants: A096535 (log 2), A099534 and A046975 (e), A039918 and A046974 (Pi).

Apéry's number or Apéry's constant zeta(3) is A002117. - N. J. A. Sloane, Jul 11 2023

Accumulate[RealDigits[Zeta[3],10,120][[1]]] (* Harvey P. Dale, Jan 18 2012 *)

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

A096275 a(0) = 0, a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n^2.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 0, 89, 89, 178, 11, 189, 200, 28, 228, 256, 0, 256, 256, 512, 92, 604, 696, 459, 255, 714, 969, 594, 407, 1001, 112, 1113, 1225, 817, 442, 1259, 1701, 1111, 876, 1987, 747, 525, 1272, 1797, 569, 2366, 231, 2597, 2828, 2400, 2092

Offset: 0

Neil Fernandez, Jun 23 2004

Cf. A096535.

a = 0; b = 1; i = 2; Do[c = Mod[a + b, i^2]; If[c == 0, Print[i]]; a = b; b = c; i++, {1000000}]

A125677 a(0) = a(1) = 1; for n>1, a(n) = a(n-2) + a(n-1) (mod n).

Original entry on oeis.org

1, 1, 2, 3, 5, 3, 8, 4, 12, 7, 19, 15, 22, 24, 32, 26, 42, 34, 58, 35, 73, 45, 74, 50, 76, 51, 101, 71, 116, 71, 127, 74, 137, 79, 148, 87, 163, 102, 189, 135, 204, 175, 211, 214, 249, 238, 257, 260, 277, 292, 319, 305, 364, 351, 391, 357, 412, 370, 434, 391, 465, 429

Offset: 0

Leroy Quet, Jan 30 2007

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A096535.

f[l_List] := Append[l, l[[ -2]] + Mod[l[[ -1]], Length[l]]];Nest[f, {1, 1}, 63] (* Ray Chandler, Feb 08 2007 *)

Extended by Ray Chandler, Feb 08 2007

A131971 a(0) = a(1) = a(2) = 1; a(n) = (a(n-1) + a(n-2) + a(n-3)) mod n.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 5, 3, 3, 2, 8, 2, 0, 10, 12, 7, 13, 15, 17, 7, 19, 1, 5, 2, 8, 15, 25, 21, 5, 22, 18, 14, 22, 21, 23, 31, 3, 20, 16, 0, 36, 11, 5, 9, 25, 39, 27, 44, 14, 36, 44, 43, 19, 0, 8, 27, 35, 13, 17, 6, 36, 59, 39, 8, 42, 24, 8, 7, 39, 54, 30

Offset: 0

Jonathan Vos Post, Oct 05 2007

Tribonacci (A000213) analog of A096535. The analogous 3 Klaus Brockhaus conjectures are applicable: (1) All numbers appear infinitely often, i.e., for every number k >= 0 and every frequency f > 0 there is an index i such that a(i) = k is the f-th occurrence of k in the sequence. (2) a(j) = a(j-1) + a(j-2) + a(j-3) and a(j) = a(j-1) + a(j-2) + a(j-3) - j occur approximately equally often, i.e., lim_{n -> infinity} x_n / y_n = 1, where x_n is the number of j <= n such that a(j) = a(j-1) + a(j-2) + a(j-3) and y_n is the number of j <= n such that a(j) = a(j-1) + a(j-2) + a(j-3) - j (cf. A122276). (3) There are sections a(g+1), ..., a(g+k) of arbitrary length k such that a(g+h) = a(g+h-1) + a(g+h-2) + a(g+h-3) for h = 1,...,k, i.e., the sequence is nondecreasing in these sections (cf. A122277, A122278, A122279).

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A000213, A096535.

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==Mod[a[n-1]+a[n-2]+a[n-3],n]},a[n],{n,80}] (* Harvey P. Dale, May 14 2011 *)
Fold[Append[#1, Mod[#1[[-1]] + #1[[-2]] + #1[[-3]], #2]] &, {1, 1, 1}, Range[68] + 2] (* Ivan Neretin, Jun 28 2017 *)

lista(nn) = {va = vector(nn, k, k<=3); for (n=4, nn, va[n] = (va[n-1] + va[n-2] + va[n-3]) % (n-1);); va;} \\ Michel Marcus, Jul 02 2017

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

A099536 Sum of the first n digits of Zeta(3) (Apery's constant), including the initial 1.

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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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A096275 a(0) = 0, a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n^2.

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A125677 a(0) = a(1) = 1; for n>1, a(n) = a(n-2) + a(n-1) (mod n).

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A131971 a(0) = a(1) = a(2) = 1; a(n) = (a(n-1) + a(n-2) + a(n-3)) mod n.

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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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