A138137 -id:A138137 - cp's OEIS frontend

A144118 Number of non-Fibonacci parts in the last section of the set of partitions of n.

0, 0, 0, 1, 0, 2, 2, 4, 5, 9, 11, 20, 22, 37, 45, 68, 83, 122, 149, 210, 259, 353, 436, 585, 717, 941, 1161, 1497, 1835, 2344, 2862, 3612, 4403, 5496, 6678, 8279, 10010, 12314, 14857, 18148, 21811, 26503, 31739, 38356, 45803, 55066, 65553, 78488, 93129

Offset: 1

Omar E. Pol, Sep 11 2008

nonn

First differences of A144116.

Cf. A001690, A135010, A138121, A138137, A144115, A144116, A144117, A144121.

b:= proc(n) option remember; true end: l:= [0, 1]: for k to 100 do b(l[1]):= false; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2] -aa(n-1, n-1)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009

Clear[b]; b[] = True; l = {0, 1}; For[k = 1, k <= 100, k++, b[l[[1]]] = False; l = {l[[2]], l[[1]] + l[[2]]}]; a[n, i_] := aa[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i == 0 || n < 0, {0, 0}, g = aa[n, i-1]; h = aa[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]]; a[n_] := aa[n, n][[2]] - aa[n-1, n-1][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 05 2016 after Alois P. Heinz *)

a(n) = A138137(n)-A144117(n) = A144116(n)-A144116(n-1).

More terms from Alois P. Heinz, Jul 28 2009

A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 4, 3, 2, 1, 1, 4, 4, 4, 3, 2, 1, 1, 7, 7, 6, 5, 3, 2, 1, 1, 8, 8, 8, 6, 5, 3, 2, 1, 1, 12, 12, 11, 10, 7, 5, 3, 2, 1, 1, 14, 14, 14, 12, 10, 7, 5, 3, 2, 1, 1, 21, 21, 20, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Mar 10 2012

Note that for n >= 2 the tail of the last section of n starts at the second column and the second column contains only one part of size 1, thus both the first and the second columns contain the same number of parts. For more information see A135010 and A182703.

			Illustration of initial terms. First six rows of triangle as numbers of parts in the columns from the last sections of the first six natural numbers:
.                                       6
.                                       3 3
.                                       4 2
.                                       2 2 2
.                           5             1
.                           3 2             1
.                 4           1             1
.                 2 2           1             1
.         3         1           1             1
.   2       1         1           1             1
1     1       1         1           1             1
---------------------------------------------------
1,  1,1,  1,1,1,  2,2,1,1,  2,2,2,1,1,  4,4,3,2,1,1
...
Triangle begins:
1;
1,   1;
1,   1,  1;
2,   2,  1,  1;
2,   2,  2,  1,  1;
4,   4,  3,  2,  1,  1;
4,   4,  4,  3,  2,  1,  1;
7,   7,  6,  5,  3,  2,  1,  1;
8,   8,  8,  6,  5,  3,  2,  1,  1;
12, 12, 11, 10,  7,  5,  3,  2,  1,  1;
14, 14, 14, 12, 10,  7,  5,  3,  2,  1,  1;
21, 21, 20, 18, 14, 11,  7,  5,  3,  2,  1,  1;

Column 1 is A187219. Row sums give A138137. Reversed rows converge to A000041.

Cf. A002865, A058399, A135010, A138135, A181187, A207031, A207380.

A210945 Triangle read by rows: T(n,k) = number of parts in the k-th column of the mirror of the last shell of the partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 1, 7, 1, 11, 3, 1, 15, 3, 1, 22, 6, 3, 1, 30, 7, 4, 1, 42, 11, 7, 3, 1, 56, 13, 9, 4, 1, 77, 20, 15, 8, 3, 1, 101, 23, 18, 10, 4, 1, 135, 33, 27, 17, 8, 3, 1, 176, 40, 34, 22, 11, 4, 1, 231, 54, 47, 33, 18, 8, 3, 1, 297, 65, 58, 42, 24, 11, 4, 1

Offset: 1

Omar E. Pol, Apr 21 2012

For another version see A207379.

			For n = 7 the illustration shows two arrangements of the last shell of the partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 --------
.                  15,3,1
.
We can see that in the right hand picture (the mirror) the number of part for columns 1..3 are 15, 3, 1 therefore row 7 lists 15, 3, 1.
Written as a triangle begins:
1;
2;
3;
5,    1;
7,    1;
11,   3,  1;
15,   3,  1;
22,   6,  3,  1;
30,   7,  4,  1;
42,  11,  7,  3,  1;
56,  13,  9,  4,  1;
77,  20, 15,  8,  3,  1;
101, 23, 18, 10,  4,  1;
135, 33, 27, 17,  8,  3,  1;
176, 40, 34, 22, 11,  4,  1;
231, 54, 47, 33, 18,  8,  3,  1;
297, 65, 58, 42, 24, 11,  4,  1;

Alois P. Heinz, Rows n = 1..70

Column 1 is A000041,n >= 1. Column 2 is A083751. Column 3 is A119907. Row sums give A138137.

Cf. A135010, A138121, A182717, A207379.

More terms from Alois P. Heinz, May 07 2012

A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 6, 9, 11, 12, 8, 14, 17, 19, 20, 15, 23, 29, 32, 34, 35, 19, 34, 42, 48, 51, 53, 54, 32, 51, 66, 74, 80, 83, 85, 86, 42, 74, 93, 108, 116, 122, 125, 127, 128, 64, 106, 138, 157, 172, 180, 186, 189, 191, 192, 83, 147, 189, 221, 240

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.

It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

			For n = 5 the illustration shows five sets containing the last k shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
--------------------------------------------------------
.  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  last       two        three      four       five
.  shell      shells     shells     shells     shells
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.    1        4+1        4+1        4+1        4+1
.      1      2+2+1      2+2+1      2+2+1      2+2+1
.      1        1+1      3+1+1      3+1+1      3+1+1
.        1        1+1      1+1+1    2+1+1+1    2+1+1+1
.          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
. ---------- ---------- ---------- ---------- ----------
.  8         14         17         19         20
.
So row 5 lists 8, 14, 17, 19, 20.
.
Triangle begins:
1;
2,    3;
3,    5,   6;
6,    9,  11,  12;
8,   14,  17,  19,  20;
15,  23,  29,  32,  34,  35;
19,  34,  42,  48,  51,  53,  54;
32,  51,  66,  74,  80,  83,  85,  86;
42,  74,  93, 108, 116, 122, 125, 127, 128;
64, 106, 138, 157, 172, 180, 186, 189, 191, 192;

Mirror of triangle A212000. Column 1 is A138137. Right border is A006128.

Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011

T(n,k) = A006128(n) - A006128(n-k).

T(n,k) = Sum_{j=n-k+1..n} A138137(j).

A206433 Total number of odd parts in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 9, 15, 19, 32, 40, 60, 78, 111, 143, 200, 252, 343, 437, 576, 728, 952, 1190, 1531, 1911, 2426, 3008, 3788, 4664, 5819, 7143, 8830, 10780, 13255, 16095, 19661, 23787, 28881, 34795, 42051, 50445, 60675, 72547, 86859, 103481, 123442, 146548

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A001227.
a(n) is also the total number of odd divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of odd terms in the n-th row of the triangle A207378.
a(n) is also the number of odd terms in the n-th row of the triangle A336812. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums give A066897.

Cf. A001227, A002865, A006128, A135010, A138121, A138137, A182703, A206434, A206435, A206436, A207378, A336811, A336812.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]]+g[[1]], f[[2]]+g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Mar 22 2012

A206434 Total number of even parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 0, 3, 1, 6, 4, 13, 10, 24, 23, 46, 46, 81, 88, 143, 159, 242, 278, 404, 470, 657, 776, 1057, 1251, 1663, 1984, 2587, 3089, 3967, 4742, 6012, 7184, 9001, 10753, 13351, 15917, 19594, 23335, 28514, 33883, 41140, 48787, 58894, 69691, 83680, 98809, 118101

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A183063.
a(n) is also the total number of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of even terms in the n-th row of the triangle A207378.
a(n) is also the number of even terms in the n-th row of the triangle A336812. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums give A066898.

Cf. A002865, A006128, A135010, A138121, A138137, A182703, A183063, A206433, A206435, A206436, A207378, A336811, A336812.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i+1, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[ a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: (Sum_{i>0} (x^(2*i)-x^(2*i+1))/(1-x^(2*i)))/Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 23 2012

More terms from Alois P. Heinz, Mar 22 2012

A211993 A list of ordered partitions of the positive integers.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 3, 2, 5, 6, 3, 3, 4, 2, 2, 2, 2, 5, 1, 3, 2, 1, 4, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 5, 1, 1, 2, 2, 2, 1, 4, 2, 1, 3, 3, 1, 6, 1, 3, 2, 2, 5, 2, 4, 3, 7

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of the odd integers is the same as A211992. The order of the partitions of the even integers is the same as A026792.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
2;             . 2;           |* *|
1,1;           1,1;           |o|*|
--------------------------------------------
1,1,1;         1,1,1;         |o|o|*|
2,1;           . 2,1;         |o o|*|
3;             . . 3;         |* * *|
--------------------------------------------
4,;            . . . 4;       |* * * *|
2,2;           . 2,. 2;       |* *|* *|
3,1;           . . 3,1;       |o o o|*|
2,1,1,;        . 2,1,1;       |o o|o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
--------------------------------------------
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
3,1,1;         . . 3,1,1;     |o o o|o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
4,1;           . . . 4,1;     |o o o o|*|
3,2;           . . 3,. 2;     |* * *|* *|
5;             . . . . 5;     |* * * * *|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
5,1;           . . . . 5,1;   |o o o o o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Rows sums give A036042, n>=1.
Other versions are A026792, A211992, A211994. See also A211983, A211984, A211989, A211999. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 12, 11, 9, 6, 20, 19, 17, 14, 8, 35, 34, 32, 29, 23, 15, 54, 53, 51, 48, 42, 34, 19, 86, 85, 83, 80, 74, 66, 51, 32, 128, 127, 125, 122, 116, 108, 93, 74, 42, 192, 191, 189, 186, 180, 172, 157, 138, 106, 64, 275, 274, 272, 269, 263, 255, 240

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.

It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

			For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
--------------------------------------------------------
.  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  five       four       three      two        last
.  shells     shells     shells     shells     shell
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.  4+1        4+1        4+1        4+1          1
.  2+2+1      2+2+1      2+2+1      2+2+1          1
.  3+1+1      3+1+1      3+1+1        1+1          1
.  2+1+1+1    2+1+1+1      1+1+1        1+1          1
.  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
. ---------- ---------- ---------- ---------- ----------
. 20         19         17         14          8
.
So row 5 lists 20, 19, 17, 14, 8.
.
Triangle begins:
1;
3,     2;
6,     5,   3;
12,   11,   9,   6;
20,   19,  17,  14,  8;
35,   34,  32,  29,  23,  15;
54,   53,  51,  48,  42,  34,  19;
86,   85,  83,  80,  74,  66,  51,  32;
128, 127, 125, 122, 116, 108,  93,  74,  42;
192, 191, 189, 186, 180, 172, 157, 138, 106, 64;

Mirror of triangle A212010. Column 1 is A006128. Right border gives A138137.

Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011

T(n,k) = A006128(n) - A006128(k-1).

T(n,k) = Sum_{j=k..n} A138137(j).

A278602 Sum of the perimeters of all regions of the n-th section of a modular table of partitions.

Original entry on oeis.org

0, 4, 8, 12, 24, 32, 60, 76, 128, 168, 256, 332, 496, 628, 896, 1152, 1580, 2008, 2716, 3416, 4528, 5688, 7388, 9228, 11872, 14708, 18684, 23088, 29004, 35632, 44440, 54288, 67168, 81756, 100384, 121656, 148552, 179192, 217556, 261544, 315836, 378232, 454748, 542584, 649500, 772532, 920912

Offset: 0

Omar E. Pol, Nov 23 2016

nonn

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(k) and the k-th vertical line segment has length A194446(k). Both line segments shares the point (A141285(k),k). For n>=1, the table contains A000041(n) regions which are distributed in n sections. Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first n sections from the table we have a representation of the partitions of n. For an illustration see the example. For the definition of "region" see A206437. For the definition of "section" see A135010. For a visualization of the corner of size n X n of the table see A273140.

a(n) is also the sum of the perimeters of the Ferrers boards of the partitions of n, minus the sum of the perimeters of the Ferrers boards of the partitions of n-1, with n >= 1. For more information see A278355.

			For n = 1..6, consider the modular table of partitions for the first six positive integers as shown below in the fourth quadrant of the square grid (see Figure 1):
|--------------|-----------------------------------------------------|
| Modular table|                      Sections                       |
| of partitions|-----------------------------------------------------|
|  for n=1..6  | 1     2       3         4           5             6 |
1--------------|-----------------------------------------------------|
.  _ _ _ _ _ _   _     _       _         _           _             _
. |_| | | | | | |_|  _| |     | |       | |         | |           | |
. |_ _| | | | |     |_ _|  _ _| |       | |         | |           | |
. |_ _ _| | | |           |_ _ _|  _ _ _| |         | |           | |
. |_ _|   | | |                   |_ _|   |         | |           | |
. |_ _ _ _| | |                   |_ _ _ _|  _ _ _ _| |           | |
. |_ _ _|   | |                             |_ _ _|   |           | |
. |_ _ _ _ _| |                             |_ _ _ _ _|  _ _ _ _ _| |
. |_ _|   |   |                                         |_ _|   |   |
. |_ _ _ _|   |                                         |_ _ _ _|   |
. |_ _ _|     |                                         |_ _ _|     |
. |_ _ _ _ _ _|                                         |_ _ _ _ _ _|
.
.   Figure 1.                         Figure 2.
.
The table contains 11 regions, see Figure 1.
The regions are distributed in 6 sections. The Figure 2 shows the sections separately.
Then consider the following table which contains the diagram of every region separately:
---------------------------------------------------------------------
|         |         |         |                    |       |        |
| Section | Region  |  Parts  |       Region       | Peri- |  a(n)  |
|         |         |(A220482)|       diagram      | meter |        |
---------------------------------------------------------------------
|         |         |         |      _             |       |        |
|    1    |    1    |    1    |     |_|            |   4   |    4   |
---------------------------------------------------------------------
|         |         |         |        _           |       |        |
|         |         |    1    |      _| |          |       |        |
|    2    |    2    |    2    |     |_ _|          |   8   |    8   |
---------------------------------------------------------------------
|         |         |         |          _         |       |        |
|         |         |    1    |         | |        |       |        |
|         |         |    1    |      _ _| |        |       |        |
|    3    |    3    |    3    |     |_ _ _|        |  12   |   12   |
---------------------------------------------------------------------
|         |         |         |      _ _           |       |        |
|         |    4    |    2    |     |_ _|          |   6   |        |
|         |---------|---------|----------------------------|        |
|         |         |         |            _       |       |        |
|         |         |    1    |           | |      |       |        |
|         |         |    1    |           | |      |       |        |
|         |         |    1    |          _| |      |       |        |
|         |         |    2    |      _ _|   |      |       |        |
|    4    |    5    |    4    |     |_ _ _ _|      |  18   |   24   |
---------------------------------------------------------------------
|         |         |         |      _ _ _         |       |        |
|         |    6    |    3    |     |_ _ _|        |   8   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |              _     |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |            _| |    |       |        |
|         |         |    2    |      _ _ _|   |    |       |        |
|    5    |    7    |    5    |     |_ _ _ _ _|    |  24   |   32   |
---------------------------------------------------------------------
|         |         |         |      _ _           |       |        |
|         |    8    |    2    |     |_ _|          |   6   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |          _ _       |       |        |
|         |         |    2    |      _ _|   |      |       |        |
|         |    9    |    4    |     |_ _ _ _|      |  12   |        |
1         |---------|---------|--------------------|-------|        |
|         |         |         |      _ _ _         |       |        |
|         |   10    |    3    |     |_ _ _|        |   8   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |                _   |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |              _| |  |       |        |
|         |         |    2    |             |   |  |       |        |
|         |         |    2    |            _|   |  |       |        |
|         |         |    3    |      _ _ _|     |  |       |        |
|    6    |   11    |    6    |     |_ _ _ _ _ _|  |  34   |   60   |
---------------------------------------------------------------------
.
For n = 1..3, there is only one region in every section. The perimeters of the regions are 4, 8 and 12 respectively, so a(1) = 4, a(2) = 8, and a(3) = 12.
For n = 4, the 4th section contains two regions with perimeters 6 and 18 respectively. The sum of the perimeters is 6 + 18 = 24, so a(4) = 24.
For n = 5, the 5th section contains two regions with perimeters 8 and 24 respectively. The sum of the perimeters is 8 + 24 = 32, so a(5) = 32.
For n = 6, the 6th section contains four regions with perimeters 6, 12, 8 and 34 respectively. The sum of the perimeters is 6 + 12 + 8 + 34 = 60, so a(6) = 60.

Partial sums give A278355.

Cf. A000041, A135010, A138137, A141285, A194446, A206437, A211992, A220482, A233968, A273140.

a(n) = 4 * A138137(n) = 2 * A233968(n), n >= 1 in both cases.

A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 3, 0, 2, 1, 2, 0, 2, 2, 2, 4, 0, 3, 2, 4, 2, 2, 0, 2, 3, 4, 4, 4, 4, 0, 4, 2, 6, 4, 8, 4, 3, 0, 2, 4, 4, 6, 8, 8, 7, 4, 0, 4, 2, 8, 4, 12, 8, 14, 8, 2, 0, 3, 4, 4, 8, 8, 12, 14, 16, 12, 6, 0, 4, 3, 8, 4, 16, 8, 21, 16, 24, 14, 2, 0, 2, 4, 6, 8, 8, 16, 14, 24, 24, 28, 21

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

			Triangle begins:
1;
2, 0;
2, 0, 1;
3, 0, 2, 1;
2, 0, 2, 2, 2;
4, 0, 3, 2, 4, 2;
2, 0, 2, 3, 4, 4,  4;
4, 0, 4, 2, 6, 4,  8,  4;
3, 0, 2, 4, 4, 6,  8,  8,  7;
4, 0, 4, 2, 8, 4, 12,  8, 14,  8;
2, 0, 3, 4, 4, 8,  8, 12, 14, 16, 12;
6, 0, 4, 3, 8, 4, 16,  8, 21, 16, 24, 14;
2, 0, 2, 4, 6, 8,  8, 16, 14, 24, 24, 28, 21;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   4   =   4
2      0   *   2   =   0
3      1   *   3   =   3
4      1   *   2   =   2
5      2   *   2   =   4
6      2   *   1   =   2
.           A000005
--------------------------
The sum of row 6 is 4 + 0 + 3 + 2 + 4 + 2 = 15, equaling A138137(6) = 15.

Row sums give A138137 (conjectured).
Columns 1, 3 and 4 are A000005.
Column 2 gives A000004.
Columns 5 and 6 give A062011.
Columns 7 and 8 give A145154, n >= 1.
Leading diagonal gives A002865.
Cf. A339304 (irregular or expanded version).
Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340426.

f(n) = if (n==0, 1, numbpart(n) - numbpart(n-1)); \\ A002865
T(n, k) = numdiv(n-k+1) * f(k-1); \\ Michel Marcus, Jan 13 2021

cp's OEIS Frontend

A144118 Number of non-Fibonacci parts in the last section of the set of partitions of n.

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A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

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A210945 Triangle read by rows: T(n,k) = number of parts in the k-th column of the mirror of the last shell of the partitions of n.

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A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

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A206433 Total number of odd parts in the last section of the set of partitions of n.

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A206434 Total number of even parts in the last section of the set of partitions of n.

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A211993 A list of ordered partitions of the positive integers.

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A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

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A278602 Sum of the perimeters of all regions of the n-th section of a modular table of partitions.

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