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

1,2

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).

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

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 *)

Prog

(Magma) [16*n^2-31*n+16: n in [1..50]]; // 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
(GAP) List([1..50], n-> ((32*n-31)^2+63)/64); # G. C. Greubel, Nov 09 2019

Crossrefs

Sequence in context: A237616 A044156 A044537 * A063523 A045234 A158056

Adjacent sequences: A143856 A143857 A143858 * A143860 A143861 A143862

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Status

approved