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 A143861 Ulam's spiral (NNE spoke). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 24 16:22 EDT 2023. Contains 365579 sequences. (Running on oeis4.)