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A229083 Numbers k such that the distance between the k-th triangular number and the nearest square is at most 1.

%I A229083 #25 Jul 14 2021 01:55:50
%S A229083 1,2,4,5,8,15,25,32,49,90,148,189,288,527,865,1104,1681,3074,5044,
%T A229083 6437,9800,17919,29401,37520,57121,104442,171364,218685,332928,608735,
%U A229083 998785,1274592,1940449,3547970,5821348,7428869,11309768,20679087,33929305,43298624,65918161
%N A229083 Numbers k such that the distance between the k-th triangular number and the nearest square is at most 1.
%C A229083 The k-th triangular number (A000217) is a square, or a square plus or minus one.
%C A229083 Union of A006451 (k-th triangular number is a square minus one), A072221 (k-th triangular number is a square plus one), and A001108 (k-th triangular number is square). Also, union of A229131 and A001108.
%F A229083 G.f.: (x^7 - 2*x^6 + x^5 - 3*x^4 + x^3 + 2*x^2 + x + 1)/((1-2*x^2+x^4)*(1-2*x^2-x^4)*(1-x)) (conjectured).
%e A229083 A000217(4) = 10 and 10 - 3^2 = 1 so 4 is in the sequence.
%e A229083 A000217(5) = 15 and 4^2 - 15 = 1 so 5 is in the sequence.
%e A229083 A000217(8) = 36 = 6^2 so 8 is in sequence.
%o A229083 (PARI) for(n=1,10^8,for(i=-1,1,f=0;if(issquare(n*(n+1)/2+i),f=1;break));if(f,print1(n,",")))
%Y A229083 Cf. A000217, A001108, A006451, A072221, A229118, A229131.
%K A229083 nonn
%O A229083 1,2
%A A229083 _Ralf Stephan_, Sep 13 2013