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Search: id:A074378
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| A074378 |
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Even triangular numbers halved. |
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+0
17
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| 0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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sum_{n>=0} q^a(n) = (prod_{n>0}(1-q^n))(sum_{n>=0} A035294(n)q^n).
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]
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]
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LINKS
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Neville Holmes, More Gemometric Integer Sequences
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FORMULA
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n(n+1)/4 where n(n+1)/2 is even.
G.f.: x(3+2x+3x^2)/((1-x)(1-x^2)^2).
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]
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MATHEMATICA
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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]
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PROGRAM
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(PARI) a(n)=(2*n+1)*(n-n\2)
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CROSSREFS
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Cf. A011848, A014493, A074377.
A007742(n)=a(2n), A033991(n)=a(2n-1).
Cf. A011848, A014493, A074377, A033991, A007742, A035294.
Cf. A010709, A047522 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 14 2009]
Sequence in context: A062698 A128341 A028942 this_sequence A026645 A026667 A104208
Adjacent sequences: A074375 A074376 A074377 this_sequence A074379 A074380 A074381
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KEYWORD
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easy,nonn
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AUTHOR
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Neville Holmes (neville.holmes(AT)utas.edu.au), Sep 04 2002
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