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A143861 Ulam's spiral (NNE spoke). 2
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.

Gardner, M. "Mathematical Recreations: The Remarkable Lore of the Prime Number" Scientific American 210 3: 120 - 128.

LINKS

Table of n, a(n) for n=1..40.

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) = 16n^2 - 35n + 20. - R. J. Mathar, Sep 08 2008

G.f.: x*(1+11*x+20*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012

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, Oct 29 2011 *)

a = 2; lst = {1}; Do[ AppendTo[ lst, 16n^2 - a]; a += 3, {n, 39}]; lst (* Vladimir Joseph Stephan Orlovsky *)

PROG

(PARI) a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011

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 21 13:27 EDT 2017. Contains 292300 sequences.