|
%I #71 Mar 26 2022 23:21:20
%S 1,15,45,91,153,231,325,435,561,703,861,1035,1225,1431,1653,1891,2145,
%T 2415,2701,3003,3321,3655,4005,4371,4753,5151,5565,5995,6441,6903,
%U 7381,7875,8385,8911,9453,10011,10585,11175,11781,12403,13041,13695,14365,15051
%N a(n) = (2*n+1)*(4*n+1).
%C Odd hexagonal numbers. Bisection of A000384. - _Omar E. Pol_, Apr 06 2008
%C Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the triangular numbers A000217. - _Omar E. Pol_, Sep 03 2011
%C a(n) is also the sum of natural numbers which can be placed in a center box and expanded ones on 4 arms on N, S, E, W or NE, NW, SW, SE directions. See illustration in links. - _Kival Ngaokrajang_, Jul 08 2014
%H G. C. Greubel, <a href="/A014634/b014634.txt">Table of n, a(n) for n = 0..5000</a>
%H Kival Ngaokrajang, <a href="/A014634/a014634.pdf">Illustration of initial terms</a>.
%H Leo Tavares, <a href="/A014634/a014634.jpg">Illustration: Dual Square Stars</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A157870(n)/2. - _Vladimir Joseph Stephan Orlovsky_, Mar 10 2009
%F a(n) = a(n-1) + 16*n-2 (with a(0)=1). - _Vincenzo Librandi_, Nov 20 2010
%F G.f.: (1+12*x+3*x^2)/(1-x)^3. - _Colin Barker_, Jan 08 2012
%F a(n) = A005408(n) * A016813(n). - _Omar E. Pol_, Nov 05 2013
%F a(n) = 2*A033954(n) + 1 = A194268(n) - n. - _Wesley Ivan Hurt_, Jul 14 2014
%F E.g.f.: (8*x^2 +14*x + 1)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F From _Amiram Eldar_, Feb 28 2022: (Start)
%F Sum_{n>=0} 1/a(n) = Pi/4 + log(2)/2.
%F Sum_{n>=0} (-1)^n/a(n) = Pi*(sqrt(2)-1)/4 + log(sqrt(2)+1)/sqrt(2). (End)
%F a(n) = A003154(n+1) + 2*A000290(n). - _Leo Tavares_, Mar 26 2022
%p A014634:=n->(2*n+1)*(4*n+1); seq(A014634(k), k=0..100); # _Wesley Ivan Hurt_, Nov 04 2013
%t lst={};Do[a=(2*n+1)*(4*n+1);AppendTo[lst,a],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 10 2009 *)
%t Table[(2 n + 1) (4 n + 1), {n, 0, 50}] (* _Wesley Ivan Hurt_, Jul 09 2014 *)
%t LinearRecurrence[{3,-3,1},{1,15,45},50] (* _Harvey P. Dale_, Aug 30 2021 *)
%o (Magma) [(2*n+1)*(4*n+1) : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 09 2014
%o (PARI) a(n)=(2*n+1)*(4*n+1) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000217, A000384, A005408, A016813, A033954, A157870, A194268.
%Y Cf. A003154, A000290.
%K nonn,easy
%O 0,2
%A _Mohammad K. Azarian_, _N. J. A. Sloane_
%E More terms from _Wesley Ivan Hurt_, Jul 09 2014
|
