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A097356 Number of partitions of n into parts not greater than sqrt(n). 15
1, 1, 1, 1, 3, 3, 4, 4, 5, 12, 14, 16, 19, 21, 24, 27, 64, 72, 84, 94, 108, 120, 136, 150, 169, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 16475, 18138, 19928, 21873, 23961 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
a(n^2) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.1582087202672504149766310999238... [see A206226, constant c(1)]. The upper bound of a(n) is c * d^sqrt(n) / n, see graph. For the lower bound, the constant c = 0.088154883798697116... (conjectured). - Vaclav Kotesovec, Jan 08 2024
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> b(n, (r-> `if`(r*r>n, r-1, r))(isqrt(n))):
seq(a(n), n=0..100); # Alois P. Heinz, Aug 02 2018
MATHEMATICA
Table[Length[IntegerPartitions[n, Floor[Sqrt[n]]]], {n, 70}] (* Harvey P. Dale, May 11 2011 *)
f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; Table[ f[n, Floor[Sqrt[n]]], {n, 53}] (* Robert G. Wilson v, Aug 13 2011 *)
PROG
(Haskell)
a097356 n = p [1..a000196 n] n where
p [] _ = 0
p _ 0 = 1
p ks'@(k:ks) m | m < k = 0
| otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011
(PARI) a(n, k=sqrtint(n))=if(min(n, k)<2, 1, sum(i=1, min(k, n), a(n-i, i))) \\ Charles R Greathouse IV, Aug 12 2011
CROSSREFS
Sequence in context: A120188 A369217 A369218 * A083522 A355067 A108942
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Aug 08 2004
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)