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A280054 Index of first occurrence of n in A280053, the nachos numbers based on squares.

%I A280054 #13 Jan 30 2017 12:04:30
%S A280054 1,2,3,4,9,23,53,193,1012,11428,414069,89236803,281079668014,
%T A280054 49673575524946259,3690344289594918623401179,
%U A280054 2363083530686659576336864121757607550,1210869542685904980187672572977511794639836071291151196
%N A280054 Index of first occurrence of n in A280053, the nachos numbers based on squares.
%C A280054 Analysis from _Lars Blomberg_, Jan 08 2017 (Start)
%C A280054 Consider the sequence of sums of squares, q(n), n=1,2,3,... (A000330):
%C A280054 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, ...
%C A280054 which has formula q(n) = n*(n+1)*(2*n+1)/6.
%C A280054 The term A280053(x) can be computed by repeatedly subtracting the largest q(n)<=x from x until 0 is reached. For example, 8 = 5+1+1+1, so A280053(8)=4
%C A280054 Note that A280054 is strictly increasing. Let r be the last term so far in A280054, and s the next term. We must find the smallest term in q such that s-q(n-1) = r, or s=q(n-1)+r.  Therefore s will have one more phase than r, and it will be the smallest possible s.
%C A280054 We also require that s<q(n), otherwise we must pick a larger n.  In other words, r must be less than the interval between q(n-1) and q(n), that is r < q(n-1)-q(n) = n^2
%C A280054 Calculate n=floor(sqrt(r))+1 and from this we get s=q(n-1)+r.
%C A280054 Note that the q sequence need not be explicitly calculated and stored.
%C A280054 Examples:
%C A280054 r.........n....q(n-1).......q(n)........s..phases
%C A280054 4.........3.........5........14.........9.......5
%C A280054 9.........4........14........30........23.......6
%C A280054 23........5........30........55........53.......7
%C A280054 53........8.......140.......204.......193.......8
%C A280054 193......14.......819......1015......1012.......9
%C A280054 1012.....32.....10416.....11440.....11428......10
%C A280054 11428...107....402641....414090....414069......11
%C A280054 414069..644..88822734..89237470..89236803......12
%C A280054 ...
%C A280054 The above values were confirmed by direct calculation.
%C A280054 (End)
%H A280054 Lars Blomberg, <a href="/A280054/b280054.txt">Table of n, a(n) for n = 1..24</a>
%Y A280054 Cf. A280053.
%K A280054 nonn
%O A280054 1,2
%A A280054 _N. J. A. Sloane_, Jan 07 2017
%E A280054 More terms from _Lars Blomberg_, Jan 08 2017