%I #25 Jun 17 2017 03:04:33
%S 1,1,2,4,5,6,16,17,18,20,21,22,25,26,27,29,30,31,36,37,38,40,41,42,43,
%T 45,46,47,52,53,54,56,57,58,61,62,63,65,66,67,77,78,79,81,82,83,256,
%U 257,258,260,261,262,272,273,274,276,277,278,281,282,283,285,286,287,289,290,291
%N Smallest number that is sum of squares of distinct earlier terms.
%C Up to a(99999680)=10^8, the largest number not in the sequence is 892. I also computed, up to a(99934078)=10^8, the similar sequence which starts with 1,2 instead of 1,1. The largest number not in that sequence seems to be 134179 - _Giovanni Resta_, Oct 06 2011
%C Resta's conjecture is correct. Let x = floor(sqrt(n) - 12). For n > 1935, x^2 > n/2. For n > 1853, n - x^2 > 892. So n > 1935 can be decomposed into x^2 plus a number greater than 892. Since the other number is smaller than x^2, any decomposition into squares will use only numbers smaller than x. By induction, all numbers greater than 1935 (and hence greater than 892) are in this sequence. - _Charles R Greathouse IV_, Oct 06 2011
%D Mihaly Bencze [Beneze], Smarandache Recurrence Type Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 231-236.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Definitions-book.pdf">Definitions, Solved and Unsolved Problems, Conjectures, ... </a>
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheSequences.html">Smarandache Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F For n > 572, a(n) = n + 320. - _Charles R Greathouse IV_, Oct 06 2011
%K nonn,easy,nice
%O 1,3
%A R. Muller
%E More terms from _David W. Wilson_