|
%I #56 Feb 05 2024 02:23:07
%S 3,10,19,30,43,58,75,94,115,138,163,190,219,250,283,318,355,394,435,
%T 478,523,570,619,670,723,778,835,894,955,1018,1083,1150,1219,1290,
%U 1363,1438,1515,1594,1675,1758,1843,1930,2019,2110,2203,2298,2395
%N a(n) = (n+3)^2 - 6.
%C a(n-3) = n^2 - 6, n>=0, with a(-3) = -6, a(-2) = -5, a(-1) = -2 gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 24 for b = 2*n. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - _Wolfdieter Lang_, Aug 16 2013
%H G. C. Greubel, <a href="/A028878/b028878.txt">Table of n, a(n) for n = 0..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</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) + 2*n + 5 (with a(0)=3). - _Vincenzo Librandi_, Aug 05 2010
%F From _Bruno Berselli_, Sep 02 2011: (Start)
%F G.f.: (x+1)*(3-2*x)/(1-x)^3.
%F a(n) = a(-n-6).
%F a(n) mod (n+1) = n-1. (End)
%F a(n) = A000290(n+3) - 6. - _Omar E. Pol_, Dec 12 2012
%F E.g.f.: (x^2 + 7*x + 3)*exp(x). - _G. C. Greubel_, Aug 19 2017
%F From _Amiram Eldar_, Nov 04 2020: (Start)
%F Sum_{n>=0} 1/a(n) = (47 - 5*sqrt(6)*Pi*cot(sqrt(6)*Pi))/60.
%F Sum_{n>=0} (-1)^n/a(n) = (-23 + 5*sqrt(6)*Pi*cosec(sqrt(6)*Pi))/60. (End)
%F From _Amiram Eldar_, Feb 05 2024: (Start)
%F Product_{n>=0} (1 - 1/a(n)) = (5*sqrt(2/21)/3)*sin(sqrt(7)*Pi)/sin(sqrt(6)*Pi).
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(15/2)*sin(sqrt(5)*Pi)/sin(sqrt(6)*Pi). (End)
%p A028878:=n->(n+3)^2-6: seq(A028878(n), n=0..100); # _Wesley Ivan Hurt_, Apr 28 2017
%t Table[(n + 3)^2 - 6, {n,0,50}] (* _G. C. Greubel_, Aug 19 2017 *)
%o (PARI) a(n)=(n+3)^2-6 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000290.
%K nonn,easy
%O 0,1
%A _Patrick De Geest_
%E Definition corrected by _Omar E. Pol_, Jul 27 2009
|
