A014494 Even triangular numbers.

0, 6, 10, 28, 36, 66, 78, 120, 136, 190, 210, 276, 300, 378, 406, 496, 528, 630, 666, 780, 820, 946, 990, 1128, 1176, 1326, 1378, 1540, 1596, 1770, 1830, 2016, 2080, 2278, 2346, 2556, 2628, 2850, 2926, 3160, 3240, 3486, 3570, 3828, 3916, 4186, 4278, 4560

Offset

0,2

Comments

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

Even generalized hexagonal numbers. - Omar E. Pol, Apr 24 2016

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = (2*n+1)*(2*n+1-(-1)^n)/2. - Ant King, Nov 18 2010

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). - Ant King, Nov 18 2010

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

a(n) = A000217(A014601(n)). - Reinhard Zumkeller, Oct 04 2004

a(n) = A014493(n+1)-(2n+1)*(-1)^n. - R. J. Mathar, Sep 15 2009

a(n) = A193867(n+1) - 1. - Omar E. Pol, Aug 17 2011

Sum_{n>=1} 1/a(n) = 2 - Pi/2. - Robert Bilinski, Jan 20 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)-2. - Amiram Eldar, Mar 06 2022

Mathematica

Table[2Ceiling[n/2]*(2n + 1), {n, 0, 47}] (* Robert G. Wilson v, Nov 05 2004 *)
1/2 (2#+1)(2#+1-(-1)^#) &/@Range[0, 47] (* Ant King, Nov 18 2010 *)
Select[1/2 #(#+1) &/@Range[0, 95], EvenQ] (* Ant King, Nov 18 2010 *)

Prog

(Magma) [1/2*(2*n+1)*(2*n+1-(-1)^n): n in [0..50]]; // Vincenzo Librandi, Aug 18 2011
(PARI) a(n)=(2*n+1)*(2*n+1-(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015
(Python)
def A014494(n): return (2*n+1)*(n+n%2) # Chai Wah Wu, Mar 11 2022

Crossrefs

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

Cf. similar sequences listed in A299645.

Sequence in context: A184387 A295185 A225845 * A318894 A129545 A097578

Adjacent sequences: A014491 A014492 A014493 * A014495 A014496 A014497

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Erich Friedman

Status

approved