A033951 Write 1,2,... in a clockwise spiral; sequence gives numbers on positive x axis.

1, 8, 23, 46, 77, 116, 163, 218, 281, 352, 431, 518, 613, 716, 827, 946, 1073, 1208, 1351, 1502, 1661, 1828, 2003, 2186, 2377, 2576, 2783, 2998, 3221, 3452, 3691, 3938, 4193, 4456, 4727, 5006, 5293, 5588, 5891, 6202, 6521, 6848, 7183, 7526, 7877, 8236, 8603, 8978

Offset: 0

Author

Olivier Gorin (gorin(AT)roazhon.inra.fr)

Keywords

• nonn
• easy
• nice

Comments

Ulam's spiral (S spoke of A054552). - Robert G. Wilson v, Oct 31 2011

a(n) is the first term in a sum of 2*n + 1 consecutive integers that equals (2*n + 1)^3. - Patrick J. McNab, Dec 24 2016

Examples

			Spiral begins:
.
  65--66--67--68--69--70--71--72--73
   |                               |
  64  37--38--39--40--41--42--43  74
   |   |                       |   |
  63  36  17--18--19--20--21  44  75
   |   |   |               |   |   |
  62  35  16   5---6---7  22  45  76
   |   |   |   |       |   |   |   |
  61  34  15   4   1   8  23  46  77
   |   |   |   |   |   |   |   |
  60  33  14   3---2   9  24  47
   |   |   |           |   |   |
  59  32  13--12--11--10  25  48
   |   |                   |   |
  58  31--30--29--28--27--26  49
   |                           |
  57--56--55--54--53--52--51--50
From _Aaron David Fairbanks_, Mar 06 2025: (Start)
Illustration of initial terms:
                                            o o o o
                        o o o             o o o o o o
          o o         o o o o o         o o o o o o o o
  o     o o o o     o o o o o o o     o o o o o o o o o o
          o o         o o o o o         o o o o o o o o
                        o o o             o o o o o o
                                            o o o o
  1        8              23                   46
(End)
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.

Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

Cf. A014848, A132774.

Programs

Maple

A033951:=n->4*n^2 + 3*n + 1: seq(A033951(n), n=0..100); # Wesley Ivan Hurt, Feb 11 2017

Mathematica

lst={};Do[p=4*n^2+3*n+1;AppendTo[lst, p], {n, 1, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,8,23},60] (* Harvey P. Dale, Feb 07 2015 *)
CoefficientList[Series[(1 + 5 x + 2 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Feb 12 2017 *)

PARI

a(n)=4*n^2+3*n+1

Python

[4*n**2 + 3*n + 1 for n in range(46)] # Michael S. Branicky, Jan 08 2021

Formula

a(n) = 4*n^2 + 3*n + 1.

G.f.: (1 + 5*x + 2*x^2)/(1-x)^3.

A014848(2n+1) = a(n).

Equals A132774 * [1, 2, 3, ...]; = binomial transform of [1, 7, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 28 2007

a(n) = A016754(n) - n. - Reinhard Zumkeller, May 17 2009

a(n) = a(n-1) + 8*n-1 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010

a(0)=1, a(1)=8, a(2)=23, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 07 2015

E.g.f.: exp(x)*(1 + 7*x + 4*x^2). - Stefano Spezia, Apr 24 2024

Extensions

Extended (with formula) by Erich Friedman