A138121 -id:A138121 - cp's OEIS frontend

A206555 Number of 5's in the last section of the set of partitions of n.

0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 4, 5, 8, 10, 15, 18, 26, 32, 44, 56, 73, 92, 120, 149, 193, 238, 302, 373, 469, 576, 716, 876, 1081, 1316, 1615, 1954, 2383, 2875, 3483, 4188, 5048, 6043, 7253, 8653, 10341, 12293, 14634, 17340, 20567, 24300, 28717, 33830

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024789. Also number of occurrences of 5 in all partitions of n that do not contain 1 as a part. It appears that the sum of five successive terms gives the partition numbers A000041 (see A182703 and A194812).

Column 5 of A182703 and of A194812.

Cf. A135010, A138121, A182712-A182714.

A206555 = lambda n: sum(list(p).count(5) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..5} a(n+j), n >= 0.

More terms from Alois P. Heinz, Feb 20 2012

A206560 Number of 10's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 14, 22, 25, 36, 43, 59, 70, 95, 113, 150, 179, 232, 278, 356, 426, 537, 644, 803, 960, 1189, 1417, 1739, 2072, 2523, 2999, 3631, 4304, 5181, 6130, 7342, 8662, 10330, 12159, 14437, 16958

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024794. Also number of occurrences of 10 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of ten successive terms give the partition numbers A000041.

Column 10 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206555-A206559.

A206560 = lambda n: sum(list(p).count(10) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..10} a(n+j), n >= 0.

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102

Offset: 1

Omar E. Pol, Feb 20 2012

nonn

			For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

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

Row sums of triangle A207034. Partial sums give A196087.

Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {0, 0}, i>n , b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 20 2012

A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043

Offset: 0

Omar E. Pol, Apr 21 2012

nonn

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

			For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
.                                        |  Partitions
.    A194805(7) = 25    A066186(7) = 105 |  of 7
.                                        |
.                   1    * * * * * * 1   |  7
.                 2      * * * 1 * * 2   |  4+3
.               2        * * * * 1 * 2   |  5+2
.             3          * * 1 * 2 * 3   |  3+2+2
.   1       2            * * * * * 1 2   |  6+1
.     2     3            * * 1 * * 2 3   |  3+3+1
.       2   3            * * * 1 * 2 3   |  4+2+1
.         3 4            * 1 * 2 * 3 4   |  2+2+2+1
.           3   1        * * * * 1 2 3   |  5+1+1
.           4 2          * * 1 * 2 3 4   |  3+2+1+1
.       1   4            * * * 1 2 3 4   |  4+1+1+1
.         2 5            * 1 * 2 3 4 5   |  2+2+1+1+1
.           5 1          * * 1 2 3 4 5   |  3+1+1+1+1
.         1 6            * 1 2 3 4 5 6   |  2+1+1+1+1+1
.           7            1 2 3 4 5 6 7   |  1+1+1+1+1+1+1
.   ----------------------------------   |
.                                        |
.   * * * * 1 * * * *                    |
.   * * * 1 2 * * * *                    |
.   * 1 * * 2 1 * * *                    |
.   * * 1 2 2 * * 1 *                    |
.   * * * * 2 2 1 * *                    |
.   1 2 2 3 2 * * * *                    |
.           2 3 2 2 1                    |
.                                        |
.    A194804(7) = 59                     |
.
Note that, as a variant, in this case each part is labeled with its position in the partition.
The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

Other versions: A207380, A210970, A210979, A210990, A210991.

Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606

Offset: 0

Omar E. Pol, Apr 30 2012

nonn

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).

For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.     A182181(11) = 35           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.
.                                A182727(11) = 35
.
The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      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
.

Other versions: A207380, A210970, A210979, A210980, A210990.

Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

a(n) = A182181(n) + A182727(n) + A210692(n).

a(A000041(n)) = 2*A006128(n) + A026905(n).

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

Original entry on oeis.org

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

A182734 Number of parts in all partitions of 2n that do not contain 1 as a part.

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 68, 123, 219, 382, 642, 1055, 1713, 2713, 4241, 6545, 9950, 14953, 22255, 32752, 47774, 69104, 99114, 141094, 199489, 280096, 390836, 542170, 747793, 1025912, 1400425, 1902267, 2572095, 3462556, 4641516, 6196830, 8241460, 10919755, 14416885

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Essentially this is a bisection (even part) of A138135.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A135010, A138121, A138135, A182742.

b:= proc(n,i) option remember; local p,q;
      if n<0 then [0,0]
    elif n=0 then [1,0]
    elif i=1 then [0,0]
    else p, q:= b(n,i-1), b(n-i,i);
         [p[1]+q[1], p[2]+q[2]+q[1]]
      fi
    end:
a:= n-> b(2*n, 2*n)[2]:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 03 2010

Table[Length[Flatten[DeleteCases[IntegerPartitions[2n],?(MemberQ[ #,1]&)]]], {n,0,40}] (* _Harvey P. Dale, Aug 08 2013 *)
b[n_] := DivisorSigma[0, n]-1+Sum[(DivisorSigma[0, k]-1)*(PartitionsP[n-k] - PartitionsP[n-k-1]), {k, 1, n-1}]; a[0] = 0; a[n_] := b[2n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Oct 07 2015 *)

More terms from Alois P. Heinz, Dec 03 2010

A182736 Sum of parts in all partitions of 2n that do not contain 1 as a part.

Original entry on oeis.org

0, 2, 8, 24, 56, 120, 252, 476, 880, 1584, 2740, 4620, 7680, 12428, 19824, 31170, 48224, 73678, 111384, 166364, 246120, 360822, 524216, 755504, 1080912, 1535050, 2165592, 3036096, 4230632, 5861828, 8078820, 11076362, 15112384, 20523492, 27747128

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Essentially this is a bisection (even indices) of A138880.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A135010, A138121, A138880, A182734, A182742.

b:= proc(n,i) option remember; local p,q;
      if n<0 then [0,0]
    elif n=0 then [1,0]
    elif i<2 then [0,0]
    else p, q:= b(n,i-1), b(n-i,i);
        [p[1]+q[1], p[2]+q[2]+q[1]*i]
      fi
    end:
a:= n-> b(2*n,2*n)[2]:
seq(a(n), n=0..34); # Alois P. Heinz, Dec 03 2010

b[n_] := (PartitionsP[n] - PartitionsP[n-1])*n; a[n_] := b[2n]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Oct 07 2015 *)

a(n) = 2*n*A182746(n). - Omar E. Pol, Dec 05 2010

More terms from Alois P. Heinz, Dec 03 2010

A194452 Total number of repeated parts in all partitions of n.

Original entry on oeis.org

0, 0, 2, 3, 8, 12, 24, 35, 60, 87, 136, 192, 287, 396, 567, 773, 1074, 1439, 1958, 2587, 3454, 4514, 5931, 7666, 9951, 12736, 16341, 20743, 26354, 33184, 41807, 52262, 65329, 81144, 100721, 124344, 153390, 188303, 230940, 282063, 344100, 418242, 507762

Offset: 0

Omar E. Pol, Nov 19 2011

nonn

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions             repeated parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 0
2 + 2 + 2 .................. 3
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 24
So a(6) = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006128, A024786, A047967, A135010, A138121, A194544, A264405.

b:= proc(n, i) option remember; local h, j, t;
      if n<0 then [0, 0]
    elif n=0 then [1, 0]
    elif i<1 then [0, 0]
    else h:= [0, 0];
         for j from 0 to iquo(n, i) do
           t:= b(n-i*j, i-1);
           h:= [h[1]+t[1], h[2]+t[2]+`if`(j<2, 0, t[1]*j)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 20 2011
g := add(x^(2*j)*(2-x^j)/(1-x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 02 2016

myCount[p_List] := Module[{t}, If[p == {}, 0, t = Transpose[Tally[p]][[2]]; Sum[If[t[[i]] == 1, 0, t[[i]]], {i, Length[t]}]]]; Table[Total[Table[myCount[p], {p, IntegerPartitions[i]}]], {i, 0, 20}] (* T. D. Noe, Nov 19 2011 *)
b[n_, i_] := b[n, i] = Module[{h, j, t}, Which[n<0, {0, 0}, n==0, {1, 0}, i < 1, {0, 0}, True, h={0, 0}; For[j=0, j <= Quotient[n, i], j++, t = b[n - i*j, i-1]; h = {h[[1]]+t[[1]], h[[2]]+t[[2]] + If[j<2, 0, t[[1]]*j]}]; h] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)
Table[Length[Flatten[Select[Flatten[Split[#]&/@IntegerPartitions[n],1],Length[#]>1&]]],{n,0,60}] (* Harvey P. Dale, Jun 12 2024 *)

a(n) = A006128(n) - A024786(n+1).
a(n) = Sum_{k=2..n} k*A264405(n,k). - Alois P. Heinz, Dec 07 2015
G.f.: g = Sum_{j>0} (x^{2*j}*(2 - x^j)/(1-x^j))/Product_{k>0}(1 - x^k) (obtained by logarithmic differentiation of the bivariate g.f. given in A264405). - Emeric Deutsch, Feb 02 2016

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 17, 23, 33, 46, 64, 86, 121, 161, 217, 291, 388, 507, 671, 870, 1131, 1458, 1872, 2383, 3042, 3840, 4841, 6076, 7605, 9460, 11765, 14544, 17950, 22073, 27077, 33092, 40395, 49113, 59611, 72162, 87185, 105035, 126366

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The physical model shows each part represented by an object, for example using a cube or a cuboid. In this case the small version of the model shows each part as a cube of side 1 which is labeled with the size of the part. On the same way the large version of the model shows each part as a cuboid of sides 1 x 1 x L where L is the size of the part. The cuboid is labeled with the level of the part. For the sum of parts see A194804. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
1) Small version:
.
Level
1        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
2                     3 2 2 1 2 2 3
3                         2 1 2
4                           1
5      Table 2.0            1            Table 2.1
6                           1
7                           1
.
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
2) Large version:
.
.                   . . . . 1 . . . .
.                   . . . 1 2 . . . .
.                   . 1 . . 2 1 . . .
.                   . . 1 2 2 . . 1 .
.                   . . . . 2 2 1 . .
.                   1 2 2 3 2 . . . .
.                           2 3 2 2 1
.
The large version shows the parts labeled with the level of the part where "the level of a part" is its position in the partition. In both versions there are 23 parts that are visible, so a(7) = 23. Also using the formula we have a(7) = 7+8+8 = 23.

Cf. A006128, A096541, A138135, A135010, A138121, A141285, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194804, A194805, A210979.

a(n) = n + A138135(n-1) + A138135(n), if n >= 2.

cp's OEIS Frontend

A206555 Number of 5's in the last section of the set of partitions of n.

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A206560 Number of 10's in the last section of the set of partitions of n.

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A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

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A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

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A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

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

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A182734 Number of parts in all partitions of 2n that do not contain 1 as a part.

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A182736 Sum of parts in all partitions of 2n that do not contain 1 as a part.

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