A143859 Ulam's spiral (WNW spoke).

1, 18, 67, 148, 261, 406, 583, 792, 1033, 1306, 1611, 1948, 2317, 2718, 3151, 3616, 4113, 4642, 5203, 5796, 6421, 7078, 7767, 8488, 9241, 10026, 10843, 11692, 12573, 13486, 14431, 15408, 16417, 17458, 18531, 19636, 20773, 21942, 23143, 24376

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Keywords

• nonn
• easy

Comments

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 67, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Links

• Harvey P. Dale, 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-31)^2+63)/64); # G. C. Greubel, Nov 09 2019

Magma

[16*n^2-31*n+16: n in [1..50]]; // Vincenzo Librandi, Nov 08 2014

Maple

seq( ((32*n-31)^2+63)/64, n=1..50); # G. C. Greubel, Nov 09 2019

Mathematica

f[n_]:= 16n^2 -31n +16; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 03 2008 *)
LinearRecurrence[{3,-3,1},{1,18,67},40] (* Harvey P. Dale, Mar 24 2012 *)
CoefficientList[Series[(1+15x+16x^2)/(1-x)^3, {x,0,40}], x] (* Vincenzo Librandi, Nov 08 2014 *)

PARI

vector(50, n, 16*n^2-31*n+16) \\ Michel Marcus, Nov 08 2014

Sage

[((32*n-31)^2+63)/64 for n in (1..50)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = 16*n^2 - 31*n + 16. - R. J. Mathar, Sep 08 2008

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=18, a(3)=67. - Harvey P. Dale, Mar 24 2012

G.f.: x*(1 + 15*x + 16*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012

E.g.f.: -16 + (16 - 15*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019