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

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: A120196 A196179 A120188 * A083522 A355067 A108942

Adjacent sequences: A097353 A097354 A097355 * A097357 A097358 A097359

Keyword

nonn

Author

Reinhard Zumkeller, Aug 08 2004

Status

approved