A143838 Ulam's spiral (SSW spoke).

1, 22, 75, 160, 277, 426, 607, 820, 1065, 1342, 1651, 1992, 2365, 2770, 3207, 3676, 4177, 4710, 5275, 5872, 6501, 7162, 7855, 8580, 9337, 10126, 10947, 11800, 12685, 13602, 14551, 15532, 16545, 17590, 18667, 19776, 20917, 22090, 23295, 24532

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 02 2008

Keywords

• nonn
• easy

Comments

Also sequence found by reading the segment (1, 22) together with the line from 22, in the direction 22, 75, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

GAP

List([1..50], n-> ((32*n-27)^2 +39)/64); # G. C. Greubel, Nov 09 2019

Magma

[16*n^2-27*n+12: n in [1..50]]; // Vincenzo Librandi, Nov 08 2014

Maple

seq( ((32*n-27)^2 +39)/64, n=1..50); # G. C. Greubel, Nov 09 2019

Mathematica

f[n_]:= 16n^2 -27n +12; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[(1+19x+12x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
((32*Range[50]-27)^2 +39)/64 (* G. C. Greubel, Nov 09 2019 *)
LinearRecurrence[{3,-3,1},{1,22,75},40] (* Harvey P. Dale, Sep 26 2020 *)

PARI

vector(50, n, 16*n^2-27*n+12) \\ Michel Marcus, Nov 08 2014

Sage

[((32*n-27)^2 +39)/64 for n in (1..50)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = 16*n^2 - 27*n + 12, n>0. - R. J. Mathar, Sep 04 2008

G.f.: x*(1 + 19*x + 12*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012

E.g.f.: -12 + (12 - 11*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019