Search: id:A074378
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%I A074378
%S A074378 0,3,5,14,18,33,39,60,68,95,105,138,150,189,203,248,264,315,333,390,
%T A074378 410,473,495,564,588,663,689,770,798,885,915,1008,1040,1139,1173,1278,
%U A074378 1314,1425,1463,1580,1620,1743,1785,1914,1958,2093,2139,2280,2328,2475
%N A074378 Even triangular numbers halved.
%C A074378 sum_{n>=0} q^a(n) = (prod_{n>0}(1-q^n))(sum_{n>=0} A035294(n)q^n).
%C A074378 a(n) is also the exact set of integers a(n) such that a(n)+1+2+3+4+...x=3a(n), 
               where x is sufficiently large. For example a(15)=203 because 203+(1+2+3+4+...+28)=609 
               and 609=3*203. [From Gil Broussard (gilbroussard(AT)bellsouth.net), 
               Sep 01 2008]
%C A074378 Except for the first term of [A047522] and the first term of [A074378], 
               if X=[A047522], Y=[A010709], A=[A074378], we have, for all other 
               terms, Pell's equation X^2-A*Y^2=1. Example 9^2-5*4^2=1; 15^2-14*4^2=1; 
               17^2-18*4^2=1 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 14 2009]
%H A074378 Neville Holmes, <a href="http://www.comp.utas.edu.au/users/nholmes/sqncs/
               gmtrc.htm#A074377">More Gemometric Integer Sequences</a>
%F A074378 n(n+1)/4 where n(n+1)/2 is even.
%F A074378 G.f.: x(3+2x+3x^2)/((1-x)(1-x^2)^2).
%F A074378 a(n) = (2n+1)*floor((n+1)/2); a(2k) = k(4k+1); a(2k+1) = (k+1)(4k+3). 
               [From Benoit Jubin (benoit_jubin(AT)yahoo.fr), Feb 05 2009]
%t A074378 lst={};s=0;Do[s+=n/2;If[Floor[s]==s,AppendTo[lst,s]],{n,0,7!}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 06 2009]
%o A074378 (PARI) a(n)=(2*n+1)*(n-n\2)
%Y A074378 Cf. A011848, A014493, A074377.
%Y A074378 A007742(n)=a(2n), A033991(n)=a(2n-1).
%Y A074378 Cf. A011848, A014493, A074377, A033991, A007742, A035294.
%Y A074378 Cf. A010709, A047522 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 14 2009]
%Y A074378 Sequence in context: A062698 A128341 A028942 this_sequence A026645 A026667 
               A104208
%Y A074378 Adjacent sequences: A074375 A074376 A074377 this_sequence A074379 A074380 
               A074381
%K A074378 easy,nonn
%O A074378 0,2
%A A074378 Neville Holmes (neville.holmes(AT)utas.edu.au), Sep 04 2002

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