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 A074378 Even triangular numbers halved. 32
 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; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Set of integers k such that k + (1 + 2 + 3 + 4 + ... + x) = 3*k, where x is sufficiently large. For example, 203 is a term because 203 + (1 + 2 + 3 + 4 + ... +28) = 609 and 609 = 3*203. - Gil Broussard, Sep 01 2008 Set of all m such that 16*m+1 is a perfect square. - Gary Detlefs, Feb 21 2010 Integers of the form Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Feb 07 2012 Numbers of the form h*(4*h + 1) for h = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018 Numbers whose distance to nearest square equals their distance to nearest oblong; that is, numbers k such that A053188(k) = A053615(k). - Lamine Ngom, Oct 27 2020 LINKS David A. Corneth, Table of n, a(n) for n = 0..9999 Neville Holmes, More Geometric Integer Sequences Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). FORMULA Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n). a(n) = 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). From Benoit Jubin, Feb 05 2009: (Start) 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). (End) a(2*n) = A007742(n), a(2*n-1) = A033991(n). - Arkadiusz Wesolowski, Jul 20 2012 a(n) = (4*n + 1 - (-1)^n)*(4*n + 3 - (-1)^n)/4^2. - Peter Bala, Jan 21 2019 a(n) = (2*n+1)*(n+1)*(1+(-1)^(n+1))/4 + (2*n+1)*(n)*(1+(-1)^n)/4. - Eric Simon Jacob, Jan 16 2020 From Amiram Eldar, Jul 03 2020: (Start) Sum_{n>=1} 1/a(n) = 4 - Pi (A153799). Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(2) - 4 (See A016687). (End) MAPLE a:=n->(2*n+1)*floor((n+1)/2): seq(a(n), n=0..50); # Muniru A Asiru, Feb 01 2019 MATHEMATICA 1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* Michael De Vlieger, Jun 01 2017, Version 10.4 *) PROG (PARI) a(n)=(2*n+1)*(n-n\2) (Magma) f:=func; [0] cat [f(n*m): m in [-1, 1], n in [1..25]]; // Bruno Berselli, Nov 13 2012 CROSSREFS Cf. A011848, A014493, A074377, A035294. Cf. A010709, A047522. [Vincenzo Librandi, Feb 14 2009] Cf. A266883 (numbers n such that 16*n-15 is a square). Cf. A016687, A153799. Cf. A053615, A053188. Sequence in context: A289622 A331996 A179213 * A185301 A179304 A026645 Adjacent sequences: A074375 A074376 A074377 * A074379 A074380 A074381 KEYWORD nonn,easy AUTHOR W. Neville Holmes, Sep 04 2002 STATUS approved

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Last modified June 5 16:06 EDT 2023. Contains 363137 sequences. (Running on oeis4.)