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
KEYWORD
nonn,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, Sep 03 2008
STATUS
approved