A132085
Number of partitions of n into distinct parts such that (u+1)^2 <= v for all pairs (u,v) of parts with u
1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
Offset: 1
Keywords
Examples
a(10)=#{10,9+1}=2;
a(20)=#{20,19+1,18+2,17+3}=4;
a(30)=#{30,29+1,28+2,27+3,26+4,25+4+1}=6;
a(40)=#{40,39+1,38+2,37+3,36+4,35+4+1}=6;
a(50)=#{50,49+1,48+2,47+3,46+4,45+5,45+4+1,44+5+1}=8;
a(60)=#{60,59+1,58+2,57+3,56+4,55+5,55+4+1,54+6,54+5+1,53+6+1}=10.
Links
- R. Zumkeller, Table of n, a(n) for n = 1..10000
Formula
a(n) = f(n,1) with f(m,p) = if p=m then 1 else (if p
A132086 Record values in A132085.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 1
Keywords
Comments
Crossrefs
Cf. A132016.
A132042 Number of times record values occur in A132015.
2, 4, 1, 5, 1, 7, 1, 9, 1, 1, 1, 9, 1, 1, 1, 11, 1, 1, 1, 13, 1, 1, 1, 15, 1, 1, 1, 17, 1, 1, 1, 1, 18, 1, 1, 1, 1, 20, 1, 1, 1, 1, 22, 1, 1, 1, 1, 24, 1, 1, 1, 1, 26, 1, 1, 1, 1, 28, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 33, 1, 1, 1, 1, 1, 35, 1, 1, 1, 1, 1, 37, 1, 1, 1, 1, 1, 39, 1, 1, 1, 1, 1
Offset: 1
Keywords
Comments
Crossrefs
Cf. A132088.
A132087 Where record values occur in A132085.
1, 5, 11, 19, 29, 30, 41, 42, 55, 56, 71, 72, 89, 90, 109, 110, 111, 131, 132, 133, 155, 156, 157, 181, 182, 183, 209, 210, 211, 239, 240, 241, 271, 272, 273, 305, 306, 307, 308, 341, 342, 343, 344, 379, 380, 381, 382, 419, 420, 421, 422, 461, 462, 463, 464
Offset: 1
Comments