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%I #44 Sep 08 2022 08:45:38
%S 1,14,59,136,245,386,559,764,1001,1270,1571,1904,2269,2666,3095,3556,
%T 4049,4574,5131,5720,6341,6994,7679,8396,9145,9926,10739,11584,12461,
%U 13370,14311,15284,16289,17326,18395,19496,20629,21794,22991,24220
%N Ulam's spiral (NNE spoke).
%C 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.
%C 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
%D Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.
%H G. C. Greubel, <a href="/A143861/b143861.txt">Table of n, a(n) for n = 1..1000</a>
%H Martin Gardner, <a href="https://www.jstor.org/stable/24936050">Mathematical Recreations: The Remarkable Lore of the Prime Number</a>, Scientific American 210 3: 120 - 128.
%H Hermetic Systems, <a href="http://www.hermetic.ch/pns/pns.htm">Prime Number Spiral</a>
%H OEIS wiki, <a href="https://oeis.org/wiki/Ulam%27s_spiral">Ulam spiral</a>
%H Ivars Peterson's MathTrek, <a href="https://www.sciencenews.org/article/prime-spirals">Prime Spirals</a>, Science News, May 3 2002.
%H Robert Sacks, <a href="http://www.numberspiral.com/index.html">Number Spiral</a>
%H Scientific American, <a href="/A143861/a143861.jpg">Cover page of the March 1964</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_spiral">Ulam spiral</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Boxing_the_compass">Boxing the compass</a>
%H Robert G. Wilson v, <a href="/A143861/a143861.txt">Ulam's spiral</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 16*n^2 - 35*n + 20. - _R. J. Mathar_, Sep 08 2008
%F G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - _Colin Barker_, Aug 03 2012
%F E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - _G. C. Greubel_, Nov 09 2019
%p seq( ((32*n-35)^2 +55)/64, n=1..40); # _G. C. Greubel_, Nov 09 2019
%t (* From _Robert G. Wilson v_, Oct 29 2011 *)
%t f[n_]:= 16n^2 -35n +20; Array[f, 40]
%t LinearRecurrence[{3,-3,1}, {1,14,59}, 40]
%t FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
%t ((32*Range[40] -35)^2 +55)/64 (* _G. C. Greubel_, Nov 09 2019 *)
%o (PARI) a(n)=16*n^2-35*n+20 \\ _Charles R Greathouse IV_, Oct 29 2011
%o (Magma) [((32*n-35)^2 +55)/64: n in [1..40]]; // _G. C. Greubel_, Nov 09 2019
%o (Sage) [((32*n-35)^2 +55)/64 for n in (1..40)] # _G. C. Greubel_, Nov 09 2019
%o (GAP) List([1..40], n-> ((32*n-35)^2 +55)/64); # _G. C. Greubel_, Nov 09 2019
%Y Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.
%K nonn,easy
%O 1,2
%A _Vladimir Joseph Stephan Orlovsky_, Sep 03 2008
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