This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014494 #63 Apr 05 2025 10:54:34
%S A014494 0,6,10,28,36,66,78,120,136,190,210,276,300,378,406,496,528,630,666,
%T A014494 780,820,946,990,1128,1176,1326,1378,1540,1596,1770,1830,2016,2080,
%U A014494 2278,2346,2556,2628,2850,2926,3160,3240,3486,3570,3828,3916,4186,4278,4560
%N A014494 Even triangular numbers.
%C A014494 Even numbers of the form n*(n+1)/2.
%C A014494 Even generalized hexagonal numbers. - _Omar E. Pol_, Apr 24 2016
%C A014494 The sequence terms occur as the exponents in the expansion of (1 - q^6) * Product_{n >= 1} (1 - q^(16*n-6))*(1 - q^(16*n))*(1 - q^(16*n+6)) = Sum_{n in Z} (-1)^n * q^(2*n*(4*n+1)) = 1 - q^6 - q^10 + q^28 + q^36 - q^66 - q^78 + + - - . - _Peter Bala_, Dec 23 2024
%H A014494 Vincenzo Librandi, <a href="/A014494/b014494.txt">Table of n, a(n) for n = 0..10000</a>
%H A014494 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A014494 From _Ant King_, Nov 18 2010: (Start)
%F A014494 a(n) = (2*n+1)*(2*n+1-(-1)^n)/2.
%F A014494 a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). (End)
%F A014494 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 A014494 a(n) = A000217(A014601(n)). - _Reinhard Zumkeller_, Oct 04 2004
%F A014494 a(n) = A014493(n+1)-(2n+1)*(-1)^n. - _R. J. Mathar_, Sep 15 2009
%F A014494 a(n) = A193867(n+1) - 1. - _Omar E. Pol_, Aug 17 2011
%F A014494 Sum_{n>=1} 1/a(n) = 2 - Pi/2. - _Robert Bilinski_, Jan 20 2021
%F A014494 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)-2. - _Amiram Eldar_, Mar 06 2022
%F A014494 E.g.f.: x*(5 + 2*x)*cosh(x) + (1 + x)*(1 + 2*x)*sinh(x). - _Stefano Spezia_, Dec 24 2024
%t A014494 Table[2Ceiling[n/2]*(2n + 1), {n, 0, 47}] (* _Robert G. Wilson v_, Nov 05 2004 *)
%t A014494 1/2 (2#+1)(2#+1-(-1)^#) &/@Range[0,47] (* _Ant King_, Nov 18 2010 *)
%t A014494 Select[1/2 #(#+1) &/@Range[0,95],EvenQ] (* _Ant King_, Nov 18 2010 *)
%o A014494 (Magma) [1/2*(2*n+1)*(2*n+1-(-1)^n): n in [0..50]]; // _Vincenzo Librandi_, Aug 18 2011
%o A014494 (PARI) a(n)=(2*n+1)*(2*n+1-(-1)^n)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A014494 (Python)
%o A014494 def A014494(n): return (2*n+1)*(n+n%2) # _Chai Wah Wu_, Mar 11 2022
%Y A014494 Cf. A000217, A000796, A014493, A056575, A074378, A193867.
%Y A014494 See also similar sequences listed in A299645.
%K A014494 nonn,easy
%O A014494 0,2
%A A014494 _Mohammad K. Azarian_
%E A014494 More terms from _Erich Friedman_