This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n = 6 the illustration of the shell model with 6 shells shows an imbalance of largest parts (see below): ------------------------------------------------------ Partitions Tree Table 1.0 of 6. A194805 A135010 ------------------------------------------------------ 6 6 6 . . . . . 3+3 3 3 . . 3 . . 4+2 4 4 . . . 2 . 2+2+2 2 2 . 2 . 2 . 5+1 1 5 5 . . . . 1 3+2+1 1 3 3 . . 2 . 1 4+1+1 4 1 4 . . . 1 1 2+2+1+1 2 1 2 . 2 . 1 1 3+1+1+1 1 3 3 . . 1 1 1 2+1+1+1+1 2 1 2 . 1 1 1 1 1+1+1+1+1+1 1 1 1 1 1 1 1 ------------------------------------------------------ The sum of largest parts > 1 on the left hand side is 23 and the sum of largest parts > 1 on the right hand side is 11, so a(6) = -23 + 11 = -12. On the other hand for n = 6 we have that 0 together with the first n-1 terms > 1 of A138137 are 0, 2, 3, 6, 8, 15 so a(6) = 0-2+3-6+8-15 = -12.
Written as a triangle: 1; 1,2; 1,1,3; 1,1,1,2,2,4; 1,1,1,1,1,2,3,5; 1,1,1,1,1,1,1,2,2,2,2,3,3,4,6; 1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,5,7; 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,5,6,8;
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
end:
g:= proc(n) option remember;
add(add(b(k, d)*b(n-d^2-k, d),
k=0..n-d^2)*d, d=1..floor(sqrt(n)))
end:
a:= n-> g(n)-g(n-1):
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; g[n_] := Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Sqrt[n]}]; Table[g[n], {n, 0, 70}] // Differences (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)
For row n = 5 of triangle we have: ------------------------------------- Column Zone The 5th Total sum k shell of parts ------------------------------------- 8 <> 7 (5) 15 7 <> 6 (3... 10 6 = 6 ...2) 7 5 = 5 (1) 5 4 = 4 (1) 4 3 = 3 (1) 3 2 = 2 (1) 2 1 = 1 (1) 1 . Triangle begins: 1; 1,3; 1,2,5; 1,2,3,5,7,11; 1,2,3,4,5,7,10,15; 1,2,3,4,5,6,7,9,11,13,15,19,22,25,31; 1,2,3,4,5,6,7,8,9,10,11,13,15,18,20,25,28,32,39;
---------------------------------------------- . Expanded Geometric Compositions arrangement model ---------------------------------------------- 1; 1; |*| ---------------------------------------------- 2; 2 .; |* *| 1,1; 1,1; |*|o| ---------------------------------------------- 3; . . 3; |* * *| 1,1,1; 1,1,1; |o|o|*| 2,1; 2 .,1; |o o|*| ---------------------------------------------- 4,; 4 . . .; |* * * *| 2,2; 2 .,2 .; |* *|* *| 1,2,1; 1,2 .,1; |*|o o|o| 1,1,1,1,; 1,1,1,1; |*|o|o|o| 1,3; 1,. . 3; |*|o o o| ---------------------------------------------- 5; . . . . 5; |* * * * *| 3,2; . . 3,. 2; |* * *|* *| 1,3,1; 1,. . 3,1; |o|o o o|*| 1,1,1,1,1; 1,1,1,1,1; |o|o|o|o|*| 1,2,1,1; 1,2 .,1,1; |o|o o|o|*| 2,2,1; 2 .,2 .,1; |o o|o o|*| 4,1; 4 . . .,1; |o o o o|*| ---------------------------------------------- 6; 6 . . . . .; |* * * * * *| 3,3; 3 . .,3 . .; |* * *|* * *| 2,4; 2 .,4 . . .; |* *|* * * *| 2,2,2; 2 .,2 .,2 .; |* *|* *|* *| 1,4,1; 1,4 . . .,1; |*|o o o o|o| 1,2,2,1; 1,2 .,2 .,1; |*|o o|o o|o| 1,1,2,1,1; 1,1,2 .,1,1; |*|o|o o|o|o| 1,1,1,1,1,1; 1,1,1,1,1,1; |*|o|o|o|o|o| 1,1,3,1; 1,1,. . 3,1; |*|o|o o o|o| 1,3,2; 1,. . 3,. 2; |*|o o o|o o| 1,5; 1,. . . . 5; |*|o o o o o| ------------------------------------------------ Note that * is a unitary element of every part of the last section of j.
---------------------------------------------- . Expanded Geometric Compositions arrangement model ---------------------------------------------- 1; 1; |*| ---------------------------------------------- 2; . 2; |* *| 1,1; 1,1; |o|*| ---------------------------------------------- 1,2; 1,. 2; |*|o o| 1,1,1; 1,1,1; |*|o|o| 3; 3 . .; |* * *| ---------------------------------------------- 4,; . . . 4; |* * * *| 2,2; . 2,. 2; |* *|* *| 1,2,1; 1,. 2,1; |o|o o|*| 1,1,1,1,; 1,1,1,1; |o|o|o|*| 3,1; 3 . .,1; |o o o|*| ---------------------------------------------- 1,4; 1,. . . 4; |*|o o o o| 1,2,2; 1,. 2,. 2; |*|o o|o o| 1,1,2,1; 1,1,. 2,1; |*|o|o o|o| 1,1,1,1,1; 1,1,1,1,1; |*|o|o|o|o| 1,3,1; 1,3 . .,1; |*|o o o|o| 2,3; 2 .,3 . .; |* *|* * *| 5; 5 . . . .; |* * * * *| ---------------------------------------------- 6; . . . . . 6; |* * * * * *| 3,3; . . 3,. . 3; |* * *|* * *| 4,2; . . . 4,. 2; |* * * *|* *| 2,2,2; . 2,. 2,. 2; |* *|* *|* *| 1,4,1; 1,. . . 4,1; |o|o o o o|*| 1,2,2,1; 1,. 2,. 2,1; |o|o o|o o|*| 1,1,2,1,1; 1,1,. 2,1,1; |o|o|o o|o|*| 1,1,1,1,1,1; 1,1,1,1,1,1; |o|o|o|o|o|*| 1,3,1,1; 1,3 . .,1,1; |o|o o o|o|*| 2,3,1; 2 .,3 . .,1; |o o|o o o|*| 5,1; 5 . . . .,1; |o o o o o|*| ---------------------------------------------- Note that * is a unitary element of every part of the last section of j.
Triangle begins: [1], [1]; [2, 1], [1, 2]; [3, 1, 1], [1, 3, 1]; [4, 2, 2, 1, 1, 1], [1, 2, 4, 1, 2, 1]; [5, 3, 2, 1, 1, 1, 1, 1], [1, 5, 1, 3, 1, 2, 1, 1]; ... Illustration of the first six rows of triangle in an infinite table: |---|---------|-----|-------|---------|-----------|-------------|---------------| | n | | 1 | 2 | 3 | 4 | 5 | 6 | |---|---------|-----|-------|---------|-----------|-------------|---------------| | | | | | | | | 6 | | | | | | | | | 3 3 | | | | | | | | | 4 2 | | P | | | | | | | 2 2 2 | | A | | | | | | 5 | 1 | | R | | | | | | 3 2 | 1 | | T | | | | | 4 | 1 | 1 | | S | | | | | 2 2 | 1 | 1 | | | | | | 3 | 1 | 1 | 1 | | | | | 2 | 1 | 1 | 1 | 1 | | | | 1 | 1 | 1 | 1 | 1 | 1 | |---|---------|-----|-------|---------|-----------|-------------|---------------| | D | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 | 1 2 3 6 | | I | A027750 | | | 1 | 1 2 | 1 3 | 1 2 4 | | V | A027750 | | | | 1 | 1 2 | 1 3 | | I | A027750 | | | | | 1 | 1 2 | | S | A027750 | | | | | 1 | 1 2 | | O | A027750 | | | | | | 1 | | R | A027750 | | | | | | 1 | | S | | | | | | | | |---|---------|-----|-------|---------|-----------|-------------|---------------| . For n = 6 in the upper zone of the above table we can see the parts of the last section of the set of partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A138121. In the lower zone of the table we can see the terms from the 6th row of A336812, these are the divisors of the numbers from the 6th row of A336811. Note that in the lower zone of the table every row gives A027750. The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order. For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A336812 and A338156. The growth of the upper zone of the table is in accordance with the growth of the modular prism described in A221529. The growth of the lower zone of the table is in accordance with the growth of the tower described also in A221529. The number of cubic cells added at n-th stage in each polycube is equal to A138879(10) = 150, hence the total number of cubic cells added at n-th stage is equal to 2*A138879(10) = 300, equaling the sum of the 10th row of this triangle.
Illustration of three arrangements of the last section of the set of partitions of 7 and the zone numbers: -------------------------------------------------------- Zone \ a) b) c) -------------------------------------------------------- 15 (7) (7) (. . . . . . 7) 14 (4+3) (4+3) (. . . 4 . . 3) 13 (5+2) (5+2) (. . . . 5 . 2) 12 (3+2+2) (3+2+2) (. . 3 . 2 . 2) 11 (1) (1) (1) 10 (1) (1) (1) 9 (1) (1) (1) 8 (1) (1) (1) 7 (1) (1) (1) 6 (1) (1) (1) 5 (1) (1) (1) 4 (1) (1) (1) 3 (1) (1) (1) 2 (1) (1) (1) 1 (1) (1) (1) . For n = 7 and k = 12 we can see that in the 12th zone of the last section there are three parts: 3, 2, 2, therefore T(7,12) = 3. Written as a triangle begins: 1; 1,1; 1,1,1; 1,1,1,2,1; 1,1,1,1,1,2,1; 1,1,1,1,1,1,1,3,2,2,1; 1,1,1,1,1,1,1,1,1,1,1,3,2,2,1; 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,3,3,2,2,2,1;
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