A132774 -id:A132774 - cp's OEIS frontend

A132775 A007818 * A132774.

1, 3, 3, 5, 10, 5, 7, 21, 21, 7, 9, 36, 54, 36, 9, 11, 55, 110, 110, 55, 11, 13, 78, 195, 260, 195, 78, 13, 15, 105, 315, 525, 525, 315, 105, 15, 17, 136, 476, 952, 1190, 952, 476, 136, 17, 19, 171, 684, 1596, 2394, 2394, 1596, 684, 171, 19

Offset: 0

Gary W. Adamson, Aug 29 2007

Row sums = A014480: (1, 6, 20, 56, 144, 352, 832, ...).

			First few rows of the triangle:
   1;
   3,   3;
   5,  10,   5;
   7,  21,  21,   7;
   9,  36,  54,  36,   9;
  11,  55, 110, 110,  55,  11;
  13,  78, 195, 260, 195,  78,  13;
  15, 105, 315, 525, 525, 315, 105, 15;
  ...
Row 3 = (7, 21, 21, 7) = 7 * (1, 3, 3, 1).

Cf. A132774, A014480.

Binomial transform of A132774. T(n,k) = (2n+1) * A007318(n,k).

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

Original entry on oeis.org

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

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

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

			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)

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).

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.

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

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
```

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

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

Extended (with formula) by Erich Friedman

cp's OEIS Frontend

A132775 A007818 * A132774.

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

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