A167430 -id:A167430 - cp's OEIS frontend

A163255 An interspersion: the order array of A163254.

1, 3, 2, 7, 5, 4, 13, 10, 8, 6, 21, 17, 14, 11, 9, 31, 26, 22, 18, 15, 12, 43, 37, 32, 27, 23, 19, 16, 57, 50, 44, 38, 33, 28, 24, 20, 73, 65, 58, 51, 45, 39, 34, 29, 25, 91, 82, 74, 66, 59, 52, 46, 40, 35, 30, 111, 101, 92, 83, 75, 67, 60, 53, 47, 41, 36

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Except for initial terms, rows 1 to 4 are A002061, A002522, A014206, A059100 and columns 1 to 4 are A002620, A024206, A014616, A004116.
This is the interspersion of the fractal sequence A167430; i.e., row n of this array consists of the numbers k such that n=A167430(k). - Clark Kimberling, Nov 03 2009

			Corner:
1....3....7...13
2....5...10...17
4....8...14...22
To obtain A163255 from A163254, replace each term of A163254 by its rank when all the terms of A163254 are arranged in increasing order.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A002061, A002522, A014206, A059100, A002620, A024206, A014616, A004116, A163253, A163254.

Cf. A167430. [From Clark Kimberling, Nov 03 2009]

A269837 Irregular triangle read by rows: even terms of A094728(n+1) divided by 4.

Original entry on oeis.org

1, 2, 4, 3, 6, 4, 9, 8, 5, 12, 10, 6, 16, 15, 12, 7, 20, 18, 14, 8, 25, 24, 21, 16, 9, 30, 28, 24, 18, 10, 36, 35, 32, 27, 20, 11, 42, 40, 36, 30, 22, 12, 49, 48, 45, 40, 33, 24, 13, 56, 54, 50, 44, 36, 26, 14, 64, 63, 60, 55, 48, 39, 28, 15

Offset: 0

Paul Curtz, Mar 06 2016

See A264798 and A261046 for the Hydrogen atom and the Janet periodic table.
a(n) odd terms are again A264798.
Decomposition by multiplication i.e. a(n) = b(n)*c(n) by irregular triangle:
1,                   1                 1,
2,                   1                 2,
4,   3,              2, 1,             2, 3,
6,   4,         =    2, 1,        *    3, 4,
9,   8, 5,           3, 2, 1,          3, 4, 5,
12, 10, 6,           3, 2, 1,          4, 5, 6,
16, 15, 12, 7,       4, 3, 2, 1,       4, 5, 6, 7,
etc.                 etc.              etc.
b(n) is duplicated A004736(n) or mirror of A122197(n+1). c(n) = A138099(n+1).
Decomposition by subtraction, a(n) = d(n) - e(n):
1,                    1                    0,
2,                    2,                   0,
4,   3,               4,  3,               0, 0,
6,   4,         =     6,  5,          -    0, 1,
9,   8, 5,            9,  8,  7,           0, 0, 2,
12, 10, 6,           12, 11, 10,           0, 1, 4,
16, 15, 12, 7,       16, 15, 14, 13,       0, 0, 2, 6,
20, 18, 14, 8,       20, 19, 18, 17,       0, 1, 4, 9,
etc.                 etc.                  etc.
d(n) is the natural numbers A000027 inverted by lines.  e(n) will be studied (see A239873).
Sum of a(n) by diagonals: 1, 5, 13, 27, 48, ... . The third differences have the period 2: repeat 2, 1. See A002717.

Cf. A000027, A000034, A002717, A004736, A094728, A122197, A138099, A167430, A239873, A261046, A264798.

Table[(n + 1)^2 - k^2, {n, 15}, {k, 0, n - 1}]/4 /. Rational -> Nothing // Flatten (* _Michael De Vlieger, Mar 07 2016 *)

A374921 Irregular triangle read by rows: T(n,k), n >= 0, k >= 1, in which if n is even then row n lists the first A008619(n) even indexed terms of A027336 otherwise if n is odd then row n lists the first A008619(n) odd indexed terms of A027336.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 8, 1, 1, 3, 6, 11, 1, 2, 4, 8, 15, 1, 1, 3, 6, 11, 20, 1, 2, 4, 8, 15, 26, 1, 1, 3, 6, 11, 20, 35, 1, 2, 4, 8, 15, 26, 45, 1, 1, 3, 6, 11, 20, 35, 58, 1, 2, 4, 8, 15, 26, 45, 75, 1, 1, 3, 6, 11, 20, 35, 58, 96, 1, 2, 4, 8, 15, 26, 45, 75, 121

Offset: 0

Omar E. Pol, Aug 01 2024

The sum of row n equals the number of partitions of n.

			Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 1, 3;
  1, 2, 4;
  1, 1, 3, 6;
  1, 2, 4, 8;
  1, 1, 3, 6, 11;
  1, 2, 4, 8, 15;
  1, 1, 3, 6, 11, 20;
  1, 2, 4, 8, 15, 26;
  1, 1, 3, 6, 11, 20, 35;
  1, 2, 4, 8, 15, 26, 45;
  1, 1, 3, 6, 11, 20, 35, 58;
  1, 2, 4, 8, 15, 26, 45, 75;
  1, 1, 3, 6, 11, 20, 35, 58, 96;
  1, 2, 4, 8, 15, 26, 45, 75, 121;
  ...
For n = 10 the sum of the 10th row is 1 + 1 + 3 + 6 + 11 + 20 = 42, the same as the number of partitions of 10.

Row sums give A000041.
Row lengths give A008619.
Right border gives A027336.
Columns 1..4: A000012, A000034, A010702, A010724.
Cf. A058695, A058696, A002865, A167430, A176206, A182844, A182845, A336811, A338156.

cp's OEIS Frontend

A163255 An interspersion: the order array of A163254.

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A269837 Irregular triangle read by rows: even terms of A094728(n+1) divided by 4.

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A374921 Irregular triangle read by rows: T(n,k), n >= 0, k >= 1, in which if n is even then row n lists the first A008619(n) even indexed terms of A027336 otherwise if n is odd then row n lists the first A008619(n) odd indexed terms of A027336.

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