login
a(n) = 2*n^2 + 2*n - 1.
29

%I #90 Feb 29 2024 18:12:08

%S -1,3,11,23,39,59,83,111,143,179,219,263,311,363,419,479,543,611,683,

%T 759,839,923,1011,1103,1199,1299,1403,1511,1623,1739,1859,1983,2111,

%U 2243,2379,2519,2663,2811,2963,3119,3279,3443,3611,3783,3959,4139,4323,4511,4703,4899,5099

%N a(n) = 2*n^2 + 2*n - 1.

%C Essentially the same as A132209.

%C From _Vincenzo Librandi_, Nov 25 2010: (Start)

%C Numbers k such that 2*k + 3 is a square.

%C First diagonal of A144562. (End)

%C The terms a(n) give the values for c of indefinite binary quadratic forms [a, b, c] = [2, 4n+2, a(n)] of discriminant D = 12, where a and c can be switched. The positive numbers represented by these forms are given in A084917. - _Klaus Purath_, Aug 31 2023

%H Reinhard Zumkeller, <a href="/A142463/b142463.txt">Table of n, a(n) for n = 0..10000</a>

%H Leo Tavares, <a href="/A142463/a142463.jpg">Illustration: Hexagonic Diamonds</a>.

%H Leo Tavares, <a href="/A142463/a142463_1.jpg">Illustration: Hexagonic Rectangles</a>.

%H Leo Tavares, <a href="/A142463/a142463_2.jpg">Illustration: Hexagonic Crosses</a>.

%H Leo Tavares, <a href="/A142463/a142463_3.jpg">Illustration: Hexagonic Columns</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + 4*n.

%F From _Paul Barry_, Nov 03 2009: (Start)

%F G.f.: (1 - 6*x + x^2)/(1-x)^3.

%F a(n) = 4*C(n+1,2) - 1. (End)

%F a(n) = -A188653(2*n+1). - _Reinhard Zumkeller_, Apr 13 2011

%F a(n) = 3*( Sum_{k=1..n} k^5 )/( Sum_{k=1..n} k^3 ), n > 0. - _Gary Detlefs_, Oct 18 2011

%F a(n) = (A005408(n)^2 - 3)/2. - _Zhandos Mambetaliyev_, Feb 11 2017

%F E.g.f.: (-1 + 4*x + 2*x^2)*exp(x). - _G. C. Greubel_, Mar 01 2021

%F From _Leo Tavares_, Nov 22 2021: (Start)

%F a(n) = 2*A005563(n) - A005408(n). See Hexagonic Diamonds illustration.

%F a(n) = A016945(n-1) + A001105(n-1). See Hexagonic Rectangles illustration.

%F a(n) = A004767(n-1) + A046092(n-1). See Hexagonic Crosses illustration.

%F a(n) = A002378(n) + A028387(n-1). See Hexagonic Columns illustration. (End)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Wesley Ivan Hurt_, Dec 03 2021

%F Sum_{n>=0} 1/a(n) = tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - _Amiram Eldar_, Sep 16 2022

%p A142463:= n-> 2*n^2 +2*n -1; seq(A142463(n), n=0..50); # _G. C. Greubel_, Mar 01 2021

%t Array[ -#*(2-#*2)-1&,5!,1] (* _Vladimir Joseph Stephan Orlovsky_, Dec 21 2008 *)

%t Table[2n^2+2n-1,{n,0,50}] (* _Harvey P. Dale_, Feb 29 2024 *)

%o (Magma) [2*n^2+2*n-1: n in [0..100]]

%o (PARI) a(n)=2*n^2+2*n-1 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (Sage) [2*n^2 +2*n -1 for n in (0..50)] # _G. C. Greubel_, Mar 01 2021

%Y Cf. A000096, A005408, A132209, A144562, A188653.

%Y Cf. A005563, A016945, A001105, A000290, A004767, A046092, A002378, A028387.

%K sign,easy

%O 0,2

%A _Roger L. Bagula_, Sep 19 2008

%E Edited by the Associate Editors of the OEIS, Sep 02 2009