A343560 a(n) = (n-1)*(4*n+1).

0, 9, 26, 51, 84, 125, 174, 231, 296, 369, 450, 539, 636, 741, 854, 975, 1104, 1241, 1386, 1539, 1700, 1869, 2046, 2231, 2424, 2625, 2834, 3051, 3276, 3509, 3750, 3999, 4256, 4521, 4794, 5075, 5364, 5661, 5966, 6279, 6600, 6929, 7266, 7611, 7964, 8325, 8694, 9071

Offset

1,2

Comments

A polynomial curve. However, write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers parallel to the negative y-axis (see Example section). This sequence only encounters composite numbers as it expands to infinity.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = A164754(n+1) + 1 = A001107(n+1), for n >= 2.

G.f.: x^2*(9-x)/(1-x)^3. - R. J. Mathar, Sep 15 2021, edited by M. F. Hasler, Apr 14 2026

Sum_{n>=2} 1/a(n) = 24/25 -3*log(2)/5 -Pi/10. - R. J. Mathar, May 30 2022

From Elmo R. Oliveira, Apr 14 2026: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

E.g.f.: 1 + exp(x)*(4*x^2 + x - 1). (End)

Sum_{n>=2} (-1)^n/a(n) = (32 - 5*sqrt(2)*Pi + 10*log(2) - 10*sqrt(2)*log(1+sqrt(2)))/50. - Amiram Eldar, Apr 22 2026

Example

On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence, not including "0", will lie parallel to the negative y-axis, located within the fourth quadrant, as seen in the example below:
  99  64--65--66--67--68--69--70--71--72
   |   |                               |
  98  63  36--37--38--39--40--41--42  73
   |   |   |                       |   |
  97  62  35  16--17--18--19--20  43  74
   |   |   |   |               |   |   |
  96  61  34  15   4---5---6  21  44  75
   |   |   |   |   |       |   |   |   |
  95  60  33  14   3  *0*  7  22  45  76
   |   |   |   |   |   |   |   |   |   |
  94  59  32  13   2---1   8  23  46  77
   |   |   |   |           |   |   |   |
  93  58  31  12--11--10--*9* 24  47  78
   |   |   |                   |   |   |
  92  57  30--29--28--27-*26*-25  48  79
   |   |                           |   |
  91  56--55--54--53--52-*51*-50--49  80
   |                                   |
  90--89--88--87--86--85-*84*-83--82--81

Maple

A343560 := n -> 4*n^2 - 3*n - 1;
seq(A343560(n), n = 1 .. 50);

Mathematica

A343560[n_] := (4*n + 1)*(n - 1); Array[A343560, 100] (* or *)
LinearRecurrence[{3, -3, 1}, {0, 9, 26}, 100] (* Paolo Xausa, Aug 27 2025 *)

Prog

(C) int a(int n) { return 4*n*n-3*n-1; }
(PARI) apply( {A343560(n)=(n-1)*(4*n+1)}, [1..55]) \\ M. F. Hasler, Apr 14 2026

Crossrefs

Cf. A001107, A164754, A180863.

Sequence in context: A075395 A352775 A085367 * A330395 A081267 A052153

Adjacent sequences: A343557 A343558 A343559 * A343561 A343562 A343563

Keyword

nonn,easy

Author

Zachary Dove, Apr 19 2021

Status

approved