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 A229935 #23 Nov 01 2013 13:24:48 %S A229935 0,0,0,2,8,28,77,202,490,1152,2624,5869,12913,28116,60660,130004, %T A229935 277065,587859,1242540,2617942,5500394,11528284,24109349,50321442, %U A229935 104844426,218086957,452963310,939496802,1946122511,4026488387,8321444573,17179801049,35433395265 %N A229935 Total number of parts in all compositions of n with at least two parts in increasing order. %C A229935 Total number of parts in all compositions of n that are not partitions of n (see example). %F A229935 a(n) = A001792(n-1) - A006128(n), n >= 1. %e A229935 For n = 4 the table shows both the compositions and the partitions of 4. There are three compositions of 4 that are not partitions of 4. %e A229935 ---------------------------------------------------- %e A229935 Compositions Partitions Number of parts %e A229935 ---------------------------------------------------- %e A229935 [1, 1, 1, 1] = [1, 1, 1, 1] %e A229935 [2, 1, 1] = [2, 1, 1] %e A229935 [1, 2, 1] 3 %e A229935 [3, 1] = [3, 1] %e A229935 [1, 1, 2] 3 %e A229935 [2, 2] = [2, 2] %e A229935 [1, 3] 2 %e A229935 [4] = [4] %e A229935 ---------------------------------------------------- %e A229935 Total 8 %e A229935 . %e A229935 A partition of a positive integer n is any nonincreasing sequence of positive integers which sum to n, ence the compositions of 4 that are not partitions of 4 are [1, 2, 1], [1, 1, 2] and [1, 3]. The total number of parts of these compositions is 3 + 3 + 2 = 8. On the other hand the total number of parts in all compositions of 4 is A001792(4-1) = 20, and the total number of parts in all partitions of 4 is A006128(4) = 12, so a(4) = 20 - 12 = 8. %Y A229935 Cf. A000041, A001792, A006128, A001782, A056823, A229936. %K A229935 nonn %O A229935 0,4 %A A229935 _Omar E. Pol_, Oct 14 2013