A097356 Number of partitions of n into parts not greater than sqrt(n).

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

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Vaclav Kotesovec, Graph - the asymptotic ratio

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

Cf. A194020, A000196, A000041, A097355.

Sequence in context: A120188 A369217 A369218 * A083522 A355067 A108942

Adjacent sequences: A097353 A097354 A097355 * A097357 A097358 A097359

Keyword

nonn

Author

Reinhard Zumkeller, Aug 08 2004

Status

approved