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 A229541 #12 Sep 30 2013 16:57:18 %S A229541 1,1,0,0,1,1,0,1,0,0,1,0,0,1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0,1,1,1,0,1, %T A229541 1,1,2,1,1,2,1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,1,2,1,4,1,1,4,2,1,4,2, %U A229541 1,2,1,1,2,1,1,2,1,0,1,1,0,1,1,0,1,0,0,1,0,0,1 %N A229541 Number T(n,k) of partitions of n^2 into squares with each number of parts k; irregular triangle T(n,k), 1 <= k <= n^2. %C A229541 Row sums give A037444. %H A229541 Christopher Hunt Gribble, <a href="/A229541/b229541.txt">Rows 1..21 flattened</a> %H A229541 Christopher Hunt Gribble, <a href="/A229541/a229541.cpp.txt">C++ program</a> %F A229541 It appears that T(n+1,g(n+1):(n+1)^2) = T(n,f(n):n^2) where f(1) = 1, f(2) = 1, f(n) = Sum(floor(n/2)), n >= 3, g(2) = 4, g(3) = 6, g(n) = Sum(floor((n+3)/2)) + 5, n >= 4. In addition, g(n+1) - f(n) = 2n + 1 for all n. %e A229541 The irregular triangle begins: %e A229541 \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ... %e A229541 n %e A229541 1 1 %e A229541 2 1 0 0 1 %e A229541 3 1 0 1 0 0 1 0 0 1 %e A229541 4 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 %e A229541 5 1 0 1 1 1 2 1 1 2 1 0 1 1 0 1 1 0 1 ... %e A229541 6 1 0 1 2 1 4 1 1 4 2 1 4 2 1 2 1 1 2 ... %e A229541 7 1 0 1 2 2 3 4 5 3 6 6 2 5 5 2 5 4 2 ... %e A229541 8 1 0 0 1 5 2 7 9 5 11 8 5 12 8 6 12 8 6 ... %e A229541 9 1 0 3 2 2 10 9 9 16 16 14 17 16 14 19 18 13 20 ... %e A229541 Length of row n is n^2. %e A229541 For n = 3, the 4 partitions are: %e A229541 Square side 1 2 3 Number of Parts %e A229541 9 0 0 9 %e A229541 5 1 0 6 %e A229541 1 2 0 3 %e A229541 0 0 1 1 %e A229541 As each partition has a different number of parts, %e A229541 T(3,1) = 1, T(3,3) = 1, T(3,6) = 1, T(3,9) = 1. %Y A229541 Cf. A037444. %K A229541 nonn,tabf %O A229541 1,37 %A A229541 _Christopher Hunt Gribble_, Sep 25 2013