A262123 a(1) + a(2) + ... + a(n) is the representation as a sum of n squares of the smallest integer needing n squares (using the greedy algorithm).

1, 1, 1, 4, 16, 144, 7056, 13046544, 42600214749456, 453694852221644777216198544, 51459754733114686962148583539748993964925660496781456

Offset

1,4

Links

Table of n, a(n) for n=1..11.

E. Lemoine, Décomposition d'un nombre entier N en ses puissances nièmes maxima, C. R. Acad. Sci. Paris, Vol. 95, pp. 719-722, 1882.

Formula

a(1)=1; for n>1, if s = a(1)+a(2)+...+a(n-1) then a(n+1) = floor((s+1)/2)^2.

a(1)+...+a(n) = A006892(n).

a(1)=a(2)=a(3)=1, a(4)=4; for n>=4, a(n+1) = ( a(n)/2+sqrt(a(n)) )^2.

Example

23 =16+4+1+1+1 is the first number to need 5 squares for its greedy decomposition, so a(1)=1,a(2)=1,a(3)=1,a(4)=4,a(5)=16.

Maple

a:=n->if n=1 then 1 else s:=add(a(k), k=1..n-1); floor((s+1)/2)^2 fi;

Mathematica

a[1] = 1; a[n_] := a[n] = Floor[(Total[Array[a, n-1]]+1)/2]^2; Array[a, 11] (* Jean-François Alcover, Oct 05 2015 *)

Prog

(Python)
def list_a(n):
....list=[1, 1, 1, 4]; root=2; length=4
....while length<n:
........root=root**2//2+root
........list.append(root**2)
........length+=1
....return list
list_a(12)
(PARI) a(n) = if(n<4, 1, if(n==4, 4, (a(n-1)/2 + sqrtint(a(n-1)))^2));
vector(12, n, a(n)) \\ Altug Alkan, Oct 04 2015

Crossrefs

Cf. A006892.

Sequence in context: A304193 A208661 A334746 * A005749 A005739 A279887

Adjacent sequences: A262120 A262121 A262122 * A262124 A262125 A262126

Keyword

nonn

Author

Robert FERREOL, Sep 11 2015

Status

approved