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 A378126 #20 Dec 12 2024 23:08:17 %S A378126 0,1,1,1,2,1,2,2,2,2,2,3,3,3,2,2,3,3,3,3,2,3,3,4,4,4,3,3,3,4,4,4,4,4, %T A378126 4,3,3,4,4,4,5,4,4,4,3,3,4,5,5,5,5,5,5,4,3,4,4,5,5,5,5,5,5,5,4,4,4,5, %U A378126 5,5,6,6,6,6,5,5,5,4,4,5,5,6,6,6,6 %N A378126 Array read by antidiagonals: T(n, m) is the maximal size of partitions of (n, m) into sums of distinct pairs of nonnegative integers, excluding (0, 0). %C A378126 A378379(n) is the least number x such that T(x, x) >= n, and as the growth rate of A378379(n) is Theta(n^(3/2)), the growth rate of T(n, m) is O((n+m)^(2/3)). %H A378126 Jimin Park, <a href="/A378126/b378126.txt">Table of n, a(n) for n = 0..1000</a> %F A378126 T(n, m) = T(m, n). %F A378126 T(n, m) >= T(n, 0) + T(0, m). %F A378126 T(n, m) = A086435(p^n * q^m) for any distinct primes p and q. %F A378126 T(n, 0) = A003056(n). %F A378126 T(n, 1) = T(n, 0) + 1. %F A378126 T(n, 2) = T(n-1, 0) + 2, for n >= 1. %F A378126 T(n, m) = O((n+m)^(2/3)). %e A378126 Table begins: %e A378126 0 1 1 2 2 2 3 3 3 3 ... %e A378126 1 2 2 3 3 3 4 4 4 4 ... %e A378126 1 2 3 3 4 4 4 5 5 5 ... %e A378126 2 3 3 4 4 4 5 5 5 6 ... %e A378126 2 3 4 4 5 5 5 6 6 6 ... %e A378126 2 3 4 4 5 5 6 6 6 7 ... %e A378126 3 4 4 5 5 6 6 6 7 7 ... %e A378126 3 4 5 5 6 6 6 7 7 7 ... %e A378126 3 4 5 5 6 6 7 7 7 8 ... %e A378126 3 4 5 6 6 7 7 7 8 8 ... %e A378126 ... %e A378126 T(9, 5) = 7, as (9, 5) can be expressed as the sum (0, 1) + (0, 2) + (1, 0) + (1, 1) + (2, 0) + (2, 1) + (3, 0), which is the longest for (9, 5). %o A378126 (Python) %o A378126 import functools %o A378126 @functools.cache %o A378126 def A378126(n: int, m: int, t: tuple[int, int] = (0, 0)) -> int: %o A378126 if (n, m) <= t: return 0 %o A378126 v = 1 %o A378126 for nn in range(t[0], n//2+1): %o A378126 for nm in range(m+1): %o A378126 if (nn, nm) <= t: continue %o A378126 rn, rm = n-nn, m-nm %o A378126 if (rn, rm) <= (nn, nm): continue %o A378126 nv = 1 + A378126(rn, rm, (nn, nm)) %o A378126 if nv > v: v = nv %o A378126 return v %Y A378126 Maximal size among partitions considered by A054242 and A201377. %Y A378126 The first row is T(n, 0) = A003056(n). %Y A378126 Cf. A086435, A378379. %K A378126 nonn,tabl %O A378126 0,5 %A A378126 _Jimin Park_, Nov 17 2024