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a(n) = (4*n+1)*(4*n+5).
5

%I #54 Oct 08 2023 04:44:33

%S 5,45,117,221,357,525,725,957,1221,1517,1845,2205,2597,3021,3477,3965,

%T 4485,5037,5621,6237,6885,7565,8277,9021,9797,10605,11445,12317,13221,

%U 14157,15125,16125,17157,18221

%N a(n) = (4*n+1)*(4*n+5).

%C Bisection of A078371. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004

%C a(n) is the smallest number not in the sequence such that Sum_{k=0..n} 1/a(k) has a denominator 4*n+5. - _Derek Orr_, Jun 21 2015

%C a(n) is the number of 2 X 2 matrices with all elements in {-n,..,0,..,n} with permanent = determinant^n except for a(0), where a(0)=0, but A003185(0) = 5. - _Indranil Ghosh_, Jan 04 2017

%H Indranil Ghosh, <a href="/A003185/b003185.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 1 = Sum_{n>=0} 4/a(n). Sum_{k=0..n} 4/a(k) = 4(n+1)/[4(n+1)+1]. Integral_{x=0..1} 1/(1 + x^4) = Sum_{n>=0} 4/a(2*n) = Sum_{n>=0} (-1)^n/(4n+1). - _Gary W. Adamson_, Jun 18 2003

%F 1 = 1/5 + Sum_{n>=1} 16/a(n); with partial sums (4n+1)/(4n+5). - _Gary W. Adamson_, Jun 18 2003

%F From _R. J. Mathar_, Apr 04 2008: (Start)

%F O.g.f.: (-5-30*x+3*x^2)/(-1+x)^3.

%F a(3*n) = A001513(2*n).

%F Conjecture: a(n+1)-a(n) = A063164(n+2). (End)

%F a(n) = 32*n + a(n-1) + 8 (with a(0)=5). - _Vincenzo Librandi_, Nov 12 2010

%F a(0)=5, a(1)=45, a(2)=117, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jan 27 2013

%F Sum_{n>=0} (-1)^n/a(n) = (log(2*sqrt(2)+3) + Pi)/(8*sqrt(2)) - 1/4. - _Amiram Eldar_, Oct 08 2023

%t Table[(4n+1)(4n+5),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{5,45,117},40] (* _Harvey P. Dale_, Jan 27 2013 *)

%o (PARI) a(n) = (4*n+1)*(4*n+5); \\ _Michel Marcus_, Jan 17 2023

%o (Python) a= lambda n: (4*n+1)*(4*n+5) # _Indranil Ghosh_, Jan 04 2017

%Y Cf. A001513, A003185, A063164, A078371.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_