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A143861
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Ulam's spiral (NNE spoke).
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1
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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; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pgs 2&3.
Gardner, M. "Mathematical Recreations: The Remarkable Lore of the Prime Number" Scientific American 210 3: 120 - 128.
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LINKS
| Hermetic Systems, Prime Number Spiral
OEIS wiki, Ulam spiral
Ivars Peterson's MathTrek, Prime Spirals
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
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FORMULA
| a(n)=16n^2-35n+20. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 08 2008]
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MATHEMATICA
| f[n_] := 16n^2 -35n +20; Array[f, 40] (* Or *)LinearRecurrence[{3, -3, 1}, {1, 14, 59}, 40] (* Or *)FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* Or *)
a = 1; lst = {a}; Do[a += n + 12; AppendTo[ lst, a], {n, 1, 1240, 32}]; lst (* Robert G. Wilson v, 29 Oct 2011 *)
a = 2; lst = {1}; Do[ AppendTo[ lst, 16n^2 - a]; a += 3, {n, 39}]; lst (* Vladimir Orlovsky *)
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PROG
| (PARI) a(n)=16*n^2-35*n+20 \\ Charles R Greathouse, Oct 29 2011
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CROSSREFS
| Cf. A054556, A073337, A054554, A143856, A054552, A143855, A016754, A143839, A033951, A143838, A054569, A143854, A054567, A143859, A053755, A143860, A033638.
Sequence in context: A063537 A084195 A033856 * A100174 A120371 A062022
Adjacent sequences: A143858 A143859 A143860 * A143862 A143863 A143864
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KEYWORD
| nonn,easy
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AUTHOR
| Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 03 2008
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