A143861 Ulam's spiral (NNE spoke).

1, 14, 59, 136, 245, 386, 559, 764, 1001, 1270, 1571, 1904, 2269, 2666, 3095, 3556, 4049, 4574, 5131, 5720, 6341, 6994, 7679, 8396, 9145, 9926, 10739, 11584, 12461, 13370, 14311, 15284, 16289, 17326, 18395, 19496, 20629, 21794, 22991, 24220

Offset

1,2

Comments

Stanislaw M. Ulam was doodling during the presentation of a "long and very boring paper" at a scientific meeting in 1963. The spiral is its result. Note that conforming to trigonometric conventions, the spiral begins on the abscissa and rotates counterclockwise. Other spirals, orientations, direction of rotation and initial values exist, even in the OEIS.

Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 59, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

References

Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Martin Gardner, Mathematical Recreations: The Remarkable Lore of the Prime Number, Scientific American 210 3: 120 - 128.

Hermetic Systems, Prime Number Spiral

OEIS wiki, Ulam spiral

Ivars Peterson's MathTrek, Prime Spirals, Science News, May 3 2002.

Robert Sacks, Number Spiral

Scientific American, Cover page of the March 1964

Eric Weisstein's World of Mathematics, Prime Spiral

Wikipedia, Ulam spiral

Wikipedia, Boxing the compass

Robert G. Wilson v, Ulam's spiral

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 16*n^2 - 35*n + 20. - R. J. Mathar, Sep 08 2008

G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012

E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

Maple

seq( ((32*n-35)^2 +55)/64, n=1..40); # G. C. Greubel, Nov 09 2019

Mathematica

(* From Robert G. Wilson v, Oct 29 2011 *)
f[n_]:= 16n^2 -35n +20; Array[f, 40]
LinearRecurrence[{3, -3, 1}, {1, 14, 59}, 40]
FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
((32*Range[40] -35)^2 +55)/64 (* G. C. Greubel, Nov 09 2019 *)

Prog

(PARI) a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011
(Magma) [((32*n-35)^2 +55)/64: n in [1..40]]; // G. C. Greubel, Nov 09 2019
(Sage) [((32*n-35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019
(GAP) List([1..40], n-> ((32*n-35)^2 +55)/64); # G. C. Greubel, Nov 09 2019

Crossrefs

Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.

Sequence in context: A063537 A084195 A033856 * A100174 A120371 A062022

Adjacent sequences: A143858 A143859 A143860 * A143862 A143863 A143864

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Status

approved