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
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
Keywords
Comments
Crossrefs
Cf. A132017.
A132088 Number of times record values occur in A132085.
4, 6, 8, 10, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 1, 20, 1, 1, 22, 1, 1, 24, 1, 1, 26, 1, 1, 28, 1, 1, 30, 1, 1, 32, 1, 1, 1, 33, 1, 1, 1, 35, 1, 1, 1, 37, 1, 1, 1, 39, 1, 1, 1, 41, 1, 1, 1, 43, 1, 1, 1, 45, 1, 1, 1, 47, 1, 1, 1, 49, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 51, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1
Offset: 1
Keywords
Comments
A132016 Record values in A132015.
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, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81
Offset: 1
Keywords
Comments
Crossrefs
Cf. A132086.
A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.
1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425
Offset: 1
Keywords
Comments
The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
Examples
Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
1 | 1
5 | 2 3
15 | 4 5 6
45 | 7 8 9 10 11
105 | 12 13 14 15 16 17 18
264 | 19 20 21 22 23 24 25 26 27 28 29
555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.
Crossrefs
Programs
-
Mathematica
Module[{s = 0}, Table[s += PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 - s*(s - 1)/2, {n, 1, 30}]]
Comments