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A074378
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Even triangular numbers halved.
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35
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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
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OFFSET
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0,2
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COMMENTS
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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, Sep 01 2008]
a(n) is the set of all n such that 16n+1 is a perfect square. [From Gary Detlefs, Feb 21 2010]
Also integers of the form sum(k=0..n, k/2). [Arkadiusz Wesolowski, Feb 07 2012]
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LINKS
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Table of n, a(n) for n=0..49.
Neville Holmes, More Gemometric Integer Sequences
Index to sequences with linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
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FORMULA
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n(n+1)/4 where n(n+1)/2 is even.
G.f.: x*(3+2*x+3*x^2)/((1-x)*(1-x^2)^2).
a(n) = (2*n+1)*floor((n+1)/2); a(2*k) = k(4*k+1); a(2*k+1) = (k+1)(4*k+3). [From Benoit Jubin, Feb 05 2009]
a(2*n) = A007742(n), a(2*n-1) = A033991(n). [Arkadiusz Wesolowski, Jul 20 2012]
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MATHEMATICA
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lst = {}; Do[a = 4*n^2; x = {a - n, a + n}; AppendTo[lst, x], {n, 0, 25}]; Rest@Flatten[lst] (* Arkadiusz Wesolowski, Jul 20 2012 *)
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PROG
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(PARI) a(n)=(2*n+1)*(n-n\2)
(MAGMA) f:=func<n | n*(4*n+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..25]]; // Bruno Berselli, Nov 13 2012
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CROSSREFS
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Cf. A011848, A014493, A074377, A035294.
Cf. A010709, A047522 [From Vincenzo Librandi, Feb 14 2009]
Sequence in context: A177007 A028942 A179213 * A185301 A179304 A026645
Adjacent sequences: A074375 A074376 A074377 * 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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STATUS
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approved
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