A143860 Ulam's spiral (NNW spoke).

1, 16, 63, 142, 253, 396, 571, 778, 1017, 1288, 1591, 1926, 2293, 2692, 3123, 3586, 4081, 4608, 5167, 5758, 6381, 7036, 7723, 8442, 9193, 9976, 10791, 11638, 12517, 13428, 14371, 15346, 16353, 17392, 18463, 19566, 20701, 21868, 23067, 24298

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Keywords

• nonn
• easy

Comments

Also, except for the first term, sequence found by reading the line from 16, in the direction 16, 63,... 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-33)^2 +63)/64); # G. C. Greubel, Nov 09 2019

Magma

[16*n^2-33*n+18: n in [1..40]]; // Vincenzo Librandi, Jul 10 2012

Maple

seq( ((32*n-33)^2 +63)/64, n=1..40); # G. C. Greubel, Nov 09 2019

Mathematica

f[n_]:= 16n^2 -33n +18; Array[f, 40] (* Robert G. Wilson v, Oct 31 2011 *)
((32*Range[50]-33)^2 +63)/64 (* G. C. Greubel, Nov 09 2019 *)

PARI

a(n)=16*n^2-33*n+18 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[((32*n-33)^2 +63)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = 16*n^2 - 33*n + 18. - R. J. Mathar, Sep 08 2008

G.f. x*(1 + 13*x + 18*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.: -18 + (18 - 17*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019