A220945 Number of partitions of n into non-consecutive distinct squares.

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 2, 1, 2, 3, 0, 0, 2, 0, 1, 1, 0, 2, 3, 0, 0, 0, 0, 1, 3, 3, 1, 0, 2, 2, 1, 0, 0, 2, 2, 1, 0, 1

Offset

0,54

Comments

a(n) <= A033461(n).

The golden ratio equals the limit, as n approaches infinity, of the following quotient: (number of partitions of order <= n consisting of distinct squares with no consecutive squares and no 1-part) / Sum_{i=1..n} a(i). - John M. Campbell, Aug 14 2021

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..1000 from Paul Tek)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i^2>n, 0, b(n-i^2, i-2))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..200);  # Alois P. Heinz, Apr 15 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2>n, 0, b[n - i^2, i-2]]]]; a[n_] := b[n, Floor[Sqrt[n]]]; Table[a[n], {n, 0, 200}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Prog

(PARI) a(n) = local(t=0, d=0, nd=0); for(k=1, sqrt(n), nd=(1+t)*x^k^2; t=t+d; d=nd); return(polcoeff(1+t+d, n))

Crossrefs

Cf. A000290, A003714, A033461.

Sequence in context: A327686 A274472 A283669 * A089224 A344987 A359325

Adjacent sequences: A220942 A220943 A220944 * A220946 A220947 A220948

Keyword

nonn

Author

Paul Tek, Apr 14 2013

Status

approved