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A130883 a(n) = 2*n^2 - n + 1.

%I A130883 #127 Jul 24 2025 17:57:05
%S A130883 1,2,7,16,29,46,67,92,121,154,191,232,277,326,379,436,497,562,631,704,
%T A130883 781,862,947,1036,1129,1226,1327,1432,1541,1654,1771,1892,2017,2146,
%U A130883 2279,2416,2557,2702,2851,3004,3161,3322,3487,3656,3829,4006,4187,4372,4561
%N A130883 a(n) = 2*n^2 - n + 1.
%C A130883 Maximum number of regions determined by n bent lines (or angular sectors). See Concrete Mathematics reference.
%C A130883 A "bent line" may also be regarded as a "long-legged letter V", meaning a letter V with both line segments extended to infinity.  See A117625 for the analogous sequence for a long-legged Z. - _N. J. A. Sloane_, Jun 18 2025
%C A130883 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 A130883 A128218(a(n)) = 2*n+1 and A128218(m) != 2*n+1 for m < a(n). - _Reinhard Zumkeller_, Jun 20 2015
%D A130883 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, 1994, pp. 7-8, and Problem 1.18, pages 19 and 500.
%H A130883 G. C. Greubel, <a href="/A130883/b130883.txt">Table of n, a(n) for n = 0..5000</a>
%H A130883 Dmitry Efimov, <a href="https://arxiv.org/abs/2101.09722">Hafnian of two-parameter matrices</a>, arXiv:2101.09722 [math.CO], 2021.
%H A130883 Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H A130883 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H A130883 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 A130883 Kival Ngaokrajang, <a href="/A130883/a130883_1.pdf">Illustration of irregular spirals (center points: 1, 2)</a>
%H A130883 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 A130883 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 A130883 N. J. A. Sloane, <a href="/A130883/a130883.jpg">Illustration for a(3) = 16</a>
%H A130883 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A130883 a(n) = a(n-1) + 4*n - 3 for n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 23 2010
%F A130883 a(n) = A000124(2*n) - 2*n. - _Geoffrey Critzer_, Mar 30 2011
%F A130883 O.g.f.: (4*x^2-x+1)/(1-x)^3. - _Geoffrey Critzer_, Mar 30 2011
%F A130883 a(n) = 2*a(n-1) - a(n-2) + 4. - _Eric Werley_, Jun 27 2011
%F A130883 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 A130883 a(n) = A128918(2*n). - _Reinhard Zumkeller_, Oct 27 2013
%F A130883 a(n) = 1 + A000384(n). - _Omar E. Pol_, Apr 27 2017
%F A130883 E.g.f.: (2*x^2 + x + 1)*exp(x). - _G. C. Greubel_, Jul 14 2017
%F A130883 a(n) = A152947(2*n+1). - _Franck Maminirina Ramaharo_, Jan 10 2018
%t A130883 a[n_]:=2*n^2-n+1; (* or *) Array[ -#*(1-#*2)+1&,5!,0] (* _Vladimir Joseph Stephan Orlovsky_, Dec 21 2008 *)
%t A130883 LinearRecurrence[{3,-3,1},{1,2,7},50] (* _Harvey P. Dale_, Jul 20 2011 *)
%o A130883 (Haskell)
%o A130883 a130883 = a128918 . (* 2)  -- _Reinhard Zumkeller_, Oct 27 2013
%o A130883 (PARI) a(n)=2*n^2-n+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A130883 (Magma) [2*n^2 - n + 1 : n in [0..50]]; // _Wesley Ivan Hurt_, Mar 25 2020
%o A130883 (Python)
%o A130883 def A130883(n): return n*(2*n - 1) + 1 # _Chai Wah Wu_, May 24 2022
%Y A130883 Cf. A000124, A000384, A014105, A084849, A128218, A128918, A152947, A270109.
%Y A130883 See also A117625.
%Y A130883 A row of the array in A386478.
%K A130883 nonn,easy
%O A130883 0,2
%A A130883 _Mohammad K. Azarian_, Jul 26 2007