A143856 Ulam's spiral (ENE spoke).

1, 12, 55, 130, 237, 376, 547, 750, 985, 1252, 1551, 1882, 2245, 2640, 3067, 3526, 4017, 4540, 5095, 5682, 6301, 6952, 7635, 8350, 9097, 9876, 10687, 11530, 12405, 13312, 14251, 15222, 16225, 17260, 18327, 19426, 20557, 21720, 22915, 24142

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Keywords

• nonn
• easy

Comments

Also sequence found by reading the segment (1, 12) together with the line from 12, in the direction 12, 55,..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Links

• Vincenzo Librandi, 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..40], n-> ((32*n-37)^2 +39)/64); # G. C. Greubel, Nov 09 2019

Magma

[16*n^2 -37*n +22: n in [1..40]]; // Vincenzo Librandi, Jul 10 2012

Maple

seq( ((32*n-37)^2 +39)/64, n=1..40); # G. C. Greubel, Nov 09 2019

Mathematica

f[n_]:= 16n^2 -37n +22; Array[f, 40] (* Robert G. Wilson v, Oct 31 2011 *)
Table[16n^2-37*n+22,{n,1,40}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{1,12,55},50] (* Harvey P. Dale, Sep 02 2024 *)

PARI

a(n)=16*n^2-37*n+22 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[((32*n-37)^2 +39)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = 16*n^2 - 37*n + 22. - R. J. Mathar, Sep 08 2008

G.f. x*(1 + 9*x + 22*x^2)/(1-x)^3. - R. J. Mathar, Oct 31 2011

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 10 2012

E.g.f.: -22 + (22 - 21*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019