%I #51 Mar 11 2022 20:43:56

%S 0,6,10,28,36,66,78,120,136,190,210,276,300,378,406,496,528,630,666,

%T 780,820,946,990,1128,1176,1326,1378,1540,1596,1770,1830,2016,2080,

%U 2278,2346,2556,2628,2850,2926,3160,3240,3486,3570,3828,3916,4186,4278,4560

%N Even triangular numbers.

%C Even numbers of the form n*(n+1)/2.

%C Even generalized hexagonal numbers. - _Omar E. Pol_, Apr 24 2016

%H Vincenzo Librandi, <a href="/A014494/b014494.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = (2*n+1)*(2*n+1-(-1)^n)/2. - _Ant King_, Nov 18 2010

%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - _Ant King_, Nov 18 2010

%F G.f.: -2*x*(3*x^2+2*x+3)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

%F a(n) = A000217(A014601(n)). - _Reinhard Zumkeller_, Oct 04 2004

%F a(n) = A014493(n+1)-(2n+1)*(-1)^n. - _R. J. Mathar_, Sep 15 2009

%F a(n) = A193867(n+1) - 1. - _Omar E. Pol_, Aug 17 2011

%F Sum_{n>=1} 1/a(n) = 2 - Pi/2. - _Robert Bilinski_, Jan 20 2021

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)-2. - _Amiram Eldar_, Mar 06 2022

%t Table[2Ceiling[n/2]*(2n + 1), {n, 0, 47}] (* _Robert G. Wilson v_, Nov 05 2004 *)

%t 1/2 (2#+1)(2#+1-(-1)^#) &/@Range[0,47] (* _Ant King_, Nov 18 2010 *)

%t Select[1/2 #(#+1) &/@Range[0,95],EvenQ] (* _Ant King_, Nov 18 2010 *)

%o (Magma) [1/2*(2*n+1)*(2*n+1-(-1)^n): n in [0..50]]; // _Vincenzo Librandi_, Aug 18 2011

%o (PARI) a(n)=(2*n+1)*(2*n+1-(-1)^n)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Python)

%o def A014494(n): return (2*n+1)*(n+n%2) # _Chai Wah Wu_, Mar 11 2022

%Y Cf. A000217, A000796, A014493, A056575, A074378, A193867.

%Y Cf. similar sequences listed in A299645.

%K nonn,easy

%O 0,2

%A _Mohammad K. Azarian_

%E More terms from _Erich Friedman_