A173302 -id:A173302 - cp's OEIS frontend

A173303 Row sums of triangle A173302.

1, 2, 3, 6, 9, 14, 21, 32, 45, 65, 90, 125, 170, 231, 307, 411, 539, 707, 917, 1188, 1522, 1950, 2475, 3137, 3949, 4962, 6195, 7725, 9579, 11856, 14610, 17971, 22012, 26919, 32798, 39892, 48367, 58541, 70647, 85123, 102291, 122724, 146891, 175545, 209322

Offset: 0

Gary W. Adamson, Feb 15 2010

nonn

Each term is a sum of partition numbers.

			a(7) = 32 = (1 + 5 + 11 + 15), = p(0) + p(4) + p(6) + p(7).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row sums of A173302.

Cf. A000041, A173304.

a(n) = sum(k=0, logint(n+1, 2), numbpart(n + 1 - 2^k)) \\ Andrew Howroyd, Aug 10 2018

a(n) = Sum_{k >=0, 2^k <= n + 1} A000041(n + 1 - 2^k). - Andrew Howroyd, Aug 10 2018

Terms a(21) and beyond from Andrew Howroyd, Aug 10 2018

A173304 Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 3, 4, 6, 4, 7, 8, 6, 1, 8, 11, 9, 2, 12, 15, 12, 3, 14, 20, 17, 5, 21, 26, 23, 7, 24, 35, 31, 11, 34, 45, 41, 15, 41, 58, 55, 21, 1, 55, 75, 71, 29, 1, 66, 96, 93, 40, 2, 88, 121, 120, 53, 3, 105, 154, 154, 72, 5, 137, 193, 196, 94, 7

Offset: 0

Gary W. Adamson, Feb 15 2010

Row sums = A000041, the partition numbers.

			The finite difference array starts:
  1, 1, 1, 1, 2, 2, 4, 4, 7,  8, 12, 14, 21, 24, ...; = A002865 (a variant)
        1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, ...; = A027336
           1, 1, 2, 3, 4, 6,  9, 12, 17, 23, 31, ...; = A017338
                       1, 1,  2,  3,  5,  7, 11, ...; = A027342
  ...
Last, columns of the array become rows of triangle A173304:
    1;
    1;
    1,   1;
    2,   2,   1;
    2,   3,   2;
    4,   4,   3;
    4,   6,   4,  1;
    7,   8,   6,  1;
    8,  11,   9,  2;
   12,  15,  12,  3;
   14,  20,  17,  5;
   21,  26,  23,  7;
   24,  35,  31, 11;
   34,  45,  41, 15;
   41,  58,  55, 21, 1;
   55,  75,  71, 29, 1;
   66,  96,  93, 40, 2;
   88, 121, 120, 53, 3;
  105, 154, 154, 72, 5;
  137, 193, 196, 94, 7;
  ...

Cf. A000041, A173301, A173302, A173303.

The generating array is in A173302.
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...
     1, 1, 2, 3, 5,  7, 11, 15, 22, 30, 42, 56,  77, 101, 135, ...
           1, 1, 2,  3,  5,  7, 11, 15, 22, 30,  42,  56,  77, ...
                         1,  1,  2,  3,  5,  7,  11,  15,  22, ...
                                                            1, ...
  ...
Take finite differences from the bottom, creating a new array in which rows are A002865 (a slight variant), A027336, A027338, A027342, ...; i.e., the numbers of partitions of n that do not contain (1, 2, 4, 8, ...) as a part.

A173301 a(n) = A000041(2^n - 1).

Original entry on oeis.org

1, 1, 3, 15, 176, 6842, 1505499, 3913864295, 338854264248680, 4216199393504640098482, 59475094770587936660132803278445, 17618334934720173062514849536736413843694654543

Offset: 0

Gary W. Adamson, Feb 15 2010

nonn

The partition numbers have an apparent fractal-like structure starting with every term in A173301.
Let A000041 = row 0, then under every (2^n - 1)-th term, begin a new row with the partition numbers; then take finite differences of each column from below.
The sum of finite difference terms will reproduce the partition numbers, with finite difference rows (starting from the top going down) = number of partitions of n that do not contain (1, 2, 3,...). (Cf. the array shown in A173302).

Refer to tables of the partition numbers.

Amiram Eldar, Table of n, a(n) for n = 0..19

Cf. A000041, A000225, A002865, A027336, A173302.

Table[PartitionsP[2^n - 1], {n, 0 ,10}] (* Amiram Eldar, Feb 26 2020 *)

a(n) = A000041(2^n - 1), n = (0, 1, 2,...).

a(n) = A000041(A000225(n)). - Omar E. Pol, Oct 29 2013

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A173303 Row sums of triangle A173302.

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A173304 Triangle generated from the array in A173302 (partition numbers starting new rows at n = 1, 3, 7, 15, ...).

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A173301 a(n) = A000041(2^n - 1).

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