A074378 Even triangular numbers halved.

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

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

The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5)) = 1 + q^3 + q^5 + q^14 + q^18 + .... - Peter Bala, Dec 30 2024

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)

a(n) = A014494(n)/2 = A274757(n)/3 = A266883(n) - 1. - Hugo Pfoertner, Dec 31 2024

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 *)
Select[Accumulate[Range[0, 100]], EvenQ]/2 (* Harvey P. Dale, Feb 15 2025 *)

Prog

(PARI) a(n)=(2*n+1)*(n-n\2)
(Magma) f:=func<i | i*(4*i+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

Crossrefs

Cf. A011848, A014493, A014494, A074377, A035294, A274757.

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