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

Author

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

GAP

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

Magma

[((32*n-35)^2 +55)/64: n in [1..40]]; // 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 *)

PARI

a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011

Sage

[((32*n-35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

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