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

1, 2, 7, 16, 29, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561

Offset: 0

Author

Mohammad K. Azarian, Jul 26 2007

Keywords

• nonn
• easy

Comments

Maximum number of regions determined by n bent lines (or angular sectors). See Concrete Mathematics reference.

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

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

A128218(a(n)) = 2*n+1 and A128218(m) != 2*n+1 for m < a(n). - Reinhard Zumkeller, Jun 20 2015

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
• Guo-Niu Han, Enumeration of Standard Puzzles
• Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• Kival Ngaokrajang, Illustration of irregular spirals (center points: 1, 2)
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
• N. J. A. Sloane, Illustration for a(3) = 16
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000124, A000384, A014105, A084849, A128218, A128918, A152947, A270109.

See also A117625.

A row of the array in A386478.

Programs

Haskell

a130883 = a128918 . (* 2)  -- Reinhard Zumkeller, Oct 27 2013

Magma

[2*n^2 - n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Mar 25 2020

Mathematica

a[n_]:=2*n^2-n+1; (* or *) Array[ -#*(1-#*2)+1&,5!,0] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{3,-3,1},{1,2,7},50] (* Harvey P. Dale, Jul 20 2011 *)

PARI

a(n)=2*n^2-n+1 \\ Charles R Greathouse IV, Sep 24 2015

Python

def A130883(n): return n*(2*n - 1) + 1 # Chai Wah Wu, May 24 2022

Formula

a(n) = a(n-1) + 4*n - 3 for n > 0, a(0)=1. - Vincenzo Librandi, Nov 23 2010

a(n) = A000124(2*n) - 2*n. - Geoffrey Critzer, Mar 30 2011

O.g.f.: (4*x^2-x+1)/(1-x)^3. - Geoffrey Critzer, Mar 30 2011

a(n) = 2*a(n-1) - a(n-2) + 4. - Eric Werley, Jun 27 2011

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

a(n) = A128918(2*n). - Reinhard Zumkeller, Oct 27 2013

a(n) = 1 + A000384(n). - Omar E. Pol, Apr 27 2017

E.g.f.: (2*x^2 + x + 1)*exp(x). - G. C. Greubel, Jul 14 2017

a(n) = A152947(2*n+1). - Franck Maminirina Ramaharo, Jan 10 2018