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.
%I A207034 #94 Feb 04 2014 08:57:02 %S A207034 0,1,2,2,3,3,4,3,4,4,5,4,5,5,6,4,5,5,6,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7, %T A207034 7,7,8,7,8,8,8,9,6,7,7,8,7,8,8,9,8,8,9,9,9,10,6,7,7,8,8,8,9,8,9,9,9, %U A207034 10,8,9,9,10,9,10,10,10,11,7,8,8,9,8,9 %N A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j). %C A207034 a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010. %H A207034 Alois P. Heinz, <a href="/A207034/b207034.txt">Table of n, a(n) for n = 1..10143</a> %F A207034 a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n). %F A207034 a(n) = A000120(A194602(n-1)) = A000120(A228354(n)-1). %F A207034 a(n) = i - A193173(i,n), i >= 1, 1<=n<=A000041(i). %e A207034 Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram. %e A207034 ---------------------------------- %e A207034 n Tail a(n) %e A207034 ---------------------------------- %e A207034 15 1 . . . . . . 6 %e A207034 14 1 . . . . . 5 %e A207034 13 1 . . . . . 5 %e A207034 12 1 . . . . 4 %e A207034 11 1 . . . . . 5 %e A207034 10 1 . . . . 4 %e A207034 9 1 . . . . 4 %e A207034 8 1 . . . 3 %e A207034 7 1 . . . . 4 %e A207034 6 1 . . . 3 %e A207034 5 1 . . . 3 %e A207034 4 1 . . 2 %e A207034 3 1 . . 2 %e A207034 2 1 . 1 %e A207034 1 1 0 %e A207034 ---------------------------------- %e A207034 Written as a triangle: %e A207034 0; %e A207034 1; %e A207034 2; %e A207034 2,3; %e A207034 3,4; %e A207034 3,4,4,5; %e A207034 4,5,5,6; %e A207034 4,5,5,6,6,6,7; %e A207034 5,6,6,7,6,7,7,8; %e A207034 5,6,6,7,7,7,8,7,8,8,8,9; %e A207034 6,7,7,8,7,8,8,9,8,8,9,9,9,10; %e A207034 6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11; %e A207034 ... %e A207034 Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j. %e A207034 Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix. %e A207034 --------------------------------------------------------- %e A207034 . j: 1 2 3 4 5 6 %e A207034 n a(n) %e A207034 --------------------------------------------------------- %e A207034 1 0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %e A207034 2 1 | . 2 . 2 1 . 2 1 1 . 2 1 1 1 . 2 1 1 1 1 %e A207034 3 2 | . . 3 . . 3 1 . . 3 1 1 . . 3 1 1 1 %e A207034 4 2 | . . 2 2 . . 2 2 1 . . 2 2 1 1 %e A207034 5 3 | . . . 4 . . . 4 1 . . . 4 1 1 %e A207034 6 3 | . . . 3 2 . . . 3 2 1 %e A207034 7 4 | . . . . 5 . . . . 5 1 %e A207034 8 3 | . . . 2 2 2 %e A207034 9 4 | . . . . 4 2 %e A207034 10 4 | . . . . 3 3 %e A207034 11 5 | . . . . . 6 %e A207034 ... %e A207034 Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix. %e A207034 --------------------------------------------------------- %e A207034 . j: 1 2 3 4 5 6 %e A207034 n a(n) %e A207034 --------------------------------------------------------- %e A207034 1 0 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 %e A207034 2 1 | 2 . 2 1 . 2 1 1 . 2 1 1 1 . 2 1 1 1 1 . %e A207034 3 2 | 3 . . 3 1 . . 3 1 1 . . 3 1 1 1 . . %e A207034 4 2 | 2 2 . . 2 2 1 . . 2 2 1 1 . . %e A207034 5 3 | 4 . . . 4 1 . . . 4 1 1 . . . %e A207034 6 3 | 3 2 . . . 3 2 1 . . . %e A207034 7 4 | 5 . . . . 5 1 . . . . %e A207034 8 3 | 2 2 2 . . . %e A207034 9 4 | 4 2 . . . . %e A207034 10 4 | 3 3 . . . . %e A207034 11 5 | 6 . . . . . %e A207034 ... %Y A207034 Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions. %Y A207034 Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440. %K A207034 nonn,tabf %O A207034 1,3 %A A207034 _Omar E. Pol_, Feb 20 2012