%I A028560
%S A028560 0,7,16,27,40,55,72,91,112,135,160,187,216,247,280,315,352,391,432,475,
%T A028560 520,567,616,667,720,775,832,891,952,1015,1080,1147,1216,1287,1360,
%U A028560 1435,1512,1591,1672,1755,1840,1927,2016,2107,2200,2295,2392
%N A028560 n(n + 6), also numbers such that 9(9 + n) is a perfect square.
%C A028560 Sequence allows us to find X values of the equation: X + (X + 3)^2 + 
               (X + 6)^3 = Y^2. To prove that X = n^2 + 6n: Y^2 = X + (X + 3)^2 
               + (X + 6)^3 = X^3 + 19*X^2 + 115X + 225 = (X + 9)(X^2 + 10X + 25) 
               = (X + 9)*(X + 5)^2 it means: (X + 9) must be a perfect square, so 
               X = k^2 - 9 with k>=3. we can put: k = n + 3, which gives: X = n^2 
               + 6n and Y = (n + 3)(n^2 + 6n + 5). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               Nov 12 2007
%C A028560 Number of units of a(n) belongs to a periodic sequence: 0, 7, 6, 7, 0, 
               5, 2, 1, 2, 5. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 
               04 2009]
%C A028560 Apart the first term, a(n)=2*n+a(n-1)+5 (with a(1)=7) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%H A028560 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A028560 P. De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic 
               Quasipronics of the form n(n+x)</a>
%F A028560 a(n) = (n+3)^2 -3^2 = n*(n+6*n), n>=0.
%F A028560 G.f.: x*(7-5*x)/(1-x)^3.
%p A028560 a:=n->sum(n,j=7..n): seq(a(n), n=6..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 12 2007
%p A028560 a:=n->sum(1+sum(1, k=1..n),k=6..n):seq(a(n), n=5...51); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 11 2008
%t A028560 Table[n(n + 6), {n, 0, 65}] or Select[ Range[0, 5000], IntegerQ[ Sqrt[9(9 
               + #)]] & ]
%o A028560 (Other) SAGE: [lucas_number2(2,n,4-n) for n in xrange(2,49)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 19 2009]
%Y A028560 a(n-3), n>=4, third column (used for the Paschen series of the hydrogen 
               atom) of triangle A120070.
%Y A028560 Cf. A005563.
%Y A028560 Sequence in context: A017245 A052221 A119461 this_sequence A133694 A024627 
               A140511
%Y A028560 Adjacent sequences: A028557 A028558 A028559 this_sequence A028561 A028562 
               A028563
%K A028560 nonn,new
%O A028560 0,2
%A A028560 Patrick De Geest (pdg(AT)worldofnumbers.com)
%E A028560 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2002