A304868 - OEIS

%I #15 May 20 2018 13:48:48

%S 1,5,9,13,14,17,21,25,26,29,30,33,37,41,45,46,49,53,57,58,61,62,65,69,

%T 73,77,78,81,85,89,90,93,94,97,101,105,109,110,113,117,121,122,125,

%U 126,129,133,137,141,142,145,149,153,154,157,158,161,165,169,173,174,177

%N Numbers x satisfying x == 1 (mod 4) or x == 14, 26, 30 (mod 32).

%C The sum of two distinct terms of this sequence is never a square.

%C Sequence has density 11/32, the maximal density that can be attained with such a sequence.

%D J. P. Massias, Sur les suites dont les sommes des termes 2 à 2 ne sont pas des carrés, Publications du département de mathématiques de Limoges, 1982.

%H Colin Barker, <a href="/A304868/b304868.txt">Table of n, a(n) for n = 1..1000</a>

%H J. C. Lagarias, A. M. Odlyzko, J. B. Shearer, <a href="http://dx.doi.org/10.1016/0097-3165(82)90005-X">On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions</a>, Journal of Combinatorial Theory. Series A, 33 (1982), pp. 167-185.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

%F From _Colin Barker_, May 20 2018: (Start)

%F G.f.: x*(1 + 4*x + 4*x^2 + 4*x^3 + x^4 + 3*x^5 + 4*x^6 + 4*x^7 + x^8 + 3*x^9 + x^10 + 2*x^11) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)).

%F a(n) = a(n-1) + a(n-11) - a(n-12) for n>12.

%F (End)

%o (PARI) isok(n) = ((n%4)==1) || ((n%32)==14) || ((n%32)==26) || ((n%32)==30);

%o (PARI) Vec(x*(1 + 4*x + 4*x^2 + 4*x^3 + x^4 + 3*x^5 + 4*x^6 + 4*x^7 + x^8 + 3*x^9 + x^10 + 2*x^11) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)) + O(x^40)) \\ _Colin Barker_, May 20 2018

%Y Cf. A016777 (another such sequence), A210380.

%K nonn,easy

%O 1,2

%A _Michel Marcus_, May 20 2018