

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
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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. 23.


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)/(1x)^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*n35)^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^235*n+20 \\ Charles R Greathouse IV, Oct 29 2011
(Magma) [((32*n35)^2 +55)/64: n in [1..40]]; // G. C. Greubel, Nov 09 2019
(Sage) [((32*n35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019
(GAP) List([1..40], n> ((32*n35)^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



