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A220945
Number of partitions of n into non-consecutive distinct squares.
2
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
KEYWORD
nonn
AUTHOR
Paul Tek, Apr 14 2013
STATUS
approved