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%I #107 Sep 08 2022 08:45:30
%S 1,2,7,16,29,46,67,92,121,154,191,232,277,326,379,436,497,562,631,704,
%T 781,862,947,1036,1129,1226,1327,1432,1541,1654,1771,1892,2017,2146,
%U 2279,2416,2557,2702,2851,3004,3161,3322,3487,3656,3829
%N a(n) = 2*n^2 - n + 1.
%C Maximum number of regions determined by n bent lines (angular sectors). See GKP Reference.
%C a(n)*Pi is the total length of half circle spiral after n rotations. It is formed as irregular spiral with two center points. At the 2nd stage, there are two alternatives: (1) select 2nd half circle radius, r2 = 2, the sequence will be A014105 or (2) select r2 = 0, the sequence will be A130883. See illustration in links. - _Kival Ngaokrajang_, Jan 19 2014
%C A128218(a(n)) = 2*n+1 and A128218(m) != 2*n+1 for m < a(n). - _Reinhard Zumkeller_, Jun 20 2015
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, 1994, pp7-8.
%H G. C. Greubel, <a href="/A130883/b130883.txt">Table of n, a(n) for n = 0..5000</a>
%H Dmitry Efimov, <a href="https://arxiv.org/abs/2101.09722">Hafnian of two-parameter matrices</a>, arXiv:2101.09722 [math.CO], 2021.
%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H Kival Ngaokrajang, <a href="/A130883/a130883_1.pdf">Illustration of irregular spirals (center points: 1, 2)</a>
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%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 - 3 for n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 23 2010
%F a(n) = A000124(2*n) - 2*n. - _Geoffrey Critzer_, Mar 30 2011
%F O.g.f.: (4*x^2-x+1)/(1-x)^3. - _Geoffrey Critzer_, Mar 30 2011
%F a(n) = 2*a(n-1) - a(n-2) + 4. - _Eric Werley_, Jun 27 2011
%F a(0)=1, a(1)=2, a(2)=7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jul 20 2011
%F a(n) = A128918(2*n). - _Reinhard Zumkeller_, Oct 27 2013
%F a(n) = 1 + A000384(n). - _Omar E. Pol_, Apr 27 2017
%F E.g.f.: (2*x^2 + x + 1)*exp(x). - _G. C. Greubel_, Jul 14 2017
%F a(n) = A152947(2*n+1). - _Franck Maminirina Ramaharo_, Jan 10 2018
%t a[n_]:=2*n^2-n+1; (* or *) Array[ -#*(1-#*2)+1&,5!,0] (* _Vladimir Joseph Stephan Orlovsky_, Dec 21 2008 *)
%t LinearRecurrence[{3,-3,1},{1,2,7},50] (* _Harvey P. Dale_, Jul 20 2011 *)
%o (Haskell)
%o a130883 = a128918 . (* 2) -- _Reinhard Zumkeller_, Oct 27 2013
%o (PARI) a(n)=2*n^2-n+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [2*n^2 - n + 1 : n in [0..50]]; // _Wesley Ivan Hurt_, Mar 25 2020
%o (Python)
%o def A130883(n): return n*(2*n - 1) + 1 # _Chai Wah Wu_, May 24 2022
%Y Cf. A000124, A084849, A128218, A128918.
%Y Cf. A270109. [_Bruno Berselli_, Mar 17 2016]
%K nonn,easy
%O 0,2
%A _Mohammad K. Azarian_, Jul 26 2007
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