This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143861 #45 Feb 16 2025 08:33:08
%S A143861 1,14,59,136,245,386,559,764,1001,1270,1571,1904,2269,2666,3095,3556,
%T A143861 4049,4574,5131,5720,6341,6994,7679,8396,9145,9926,10739,11584,12461,
%U A143861 13370,14311,15284,16289,17326,18395,19496,20629,21794,22991,24220
%N A143861 Ulam's spiral (NNE spoke).
%C A143861 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 A143861 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 A143861 Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.
%H A143861 G. C. Greubel, <a href="/A143861/b143861.txt">Table of n, a(n) for n = 1..1000</a>
%H A143861 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 A143861 Hermetic Systems, <a href="http://www.hermetic.ch/pns/pns.htm">Prime Number Spiral</a>
%H A143861 OEIS wiki, <a href="https://oeis.org/wiki/Ulam%27s_spiral">Ulam spiral</a>
%H A143861 Ivars Peterson's MathTrek, <a href="https://www.sciencenews.org/article/prime-spirals">Prime Spirals</a>, Science News, May 3 2002.
%H A143861 Robert Sacks, <a href="http://www.numberspiral.com/index.html">Number Spiral</a>
%H A143861 Scientific American, <a href="/A143861/a143861.jpg">Cover page of the March 1964</a>
%H A143861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral</a>
%H A143861 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ulam_spiral">Ulam spiral</a>
%H A143861 Wikipedia, <a href="http://en.wikipedia.org/wiki/Boxing_the_compass">Boxing the compass</a>
%H A143861 Robert G. Wilson v, <a href="/A143861/a143861.txt">Ulam's spiral</a>
%H A143861 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A143861 a(n) = 16*n^2 - 35*n + 20. - _R. J. Mathar_, Sep 08 2008
%F A143861 G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - _Colin Barker_, Aug 03 2012
%F A143861 E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - _G. C. Greubel_, Nov 09 2019
%p A143861 seq( ((32*n-35)^2 +55)/64, n=1..40); # _G. C. Greubel_, Nov 09 2019
%t A143861 (* From _Robert G. Wilson v_, Oct 29 2011 *)
%t A143861 f[n_]:= 16n^2 -35n +20; Array[f, 40]
%t A143861 LinearRecurrence[{3,-3,1}, {1,14,59}, 40]
%t A143861 FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
%t A143861 ((32*Range[40] -35)^2 +55)/64 (* _G. C. Greubel_, Nov 09 2019 *)
%o A143861 (PARI) a(n)=16*n^2-35*n+20 \\ _Charles R Greathouse IV_, Oct 29 2011
%o A143861 (Magma) [((32*n-35)^2 +55)/64: n in [1..40]]; // _G. C. Greubel_, Nov 09 2019
%o A143861 (Sage) [((32*n-35)^2 +55)/64 for n in (1..40)] # _G. C. Greubel_, Nov 09 2019
%o A143861 (GAP) List([1..40], n-> ((32*n-35)^2 +55)/64); # _G. C. Greubel_, Nov 09 2019
%Y A143861 Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.
%K A143861 nonn,easy
%O A143861 1,2
%A A143861 _Vladimir Joseph Stephan Orlovsky_, Sep 03 2008