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Search: id:A033991
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| 0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335, 3570, 3813, 4064, 4323, 4590, 4865, 5148, 5439, 5738, 6045, 6360, 6683, 7014, 7353, 7700, 8055
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Write 0,1,2,... in clockwise spiral; sequence gives numbers on negative x axis.
This sequence is the number of expressions x generated for a given modulus n in finite arithmetic. For example, n=1 (modulus 1) generates 3 expressions: 0+0=0(mod 1), 0-0=0(mod 1), 0*0=0(mod 1). By subtracting n from 4n^2, we eliminate the counting of those expressions that would include division by zero, which would be, of course, undefined. - David Quentin Dauthier (d_dauthier(AT)yahoo.com), Nov 04 2007
If A=[A033991] n(4n-1) (for n>0, 3,14,33,60,95,...,); Y=[A157330] 64*n-8 (56,120,184,...,); X=[A157331] 128*n^2-32*n+1 (97,449,1057,...,) ; , we have for all terms, Pell's equation X^2-A*Y^2=1. Example: 97^2-3*56^2=1; 449^2-14*120^2=1; 1057^2-33*184^2=1; 1921^2-60*248^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 27 2009]
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REFERENCES
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S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
Emilio Apricena, A version of the Ulam spiral
Vincenzo Librandi, X^2-AY^2=1 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 27 2009]
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FORMULA
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G.f.: x(3+5x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n)=A014635/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 16 2007
a(n)= A000326(n)+A005476(n), example:95=60+35, etc... a(n)=A049452(n)-A001105(n), example:33=51-18, etc. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007
a(n)=8*n+a(n-1)-13 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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EXAMPLE
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16 17 18 19 ...
15 4 5 6 ...
14 3 0 7 ...
13 2 1 8 ...
For n=2, a(2)=8*2+0-13=3; n=3, a(3)=8*3+3-13=14; n=4, a(4)=8*4+14-13=33 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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MAPLE
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[seq(binomial(4*n, 2)/2, n=0..45)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 16 2007
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MATHEMATICA
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(Mathematica 6.0) Manipulate[4 x^2 - x, {x, 1, 100, 1}] - David Quentin Dauthier (d_dauthier(AT)yahoo.com), Nov 04 2007
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PROGRAM
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(PARI) a(n)=4*n^2-n.
(Other) sage: [lucas_number1(3, 2*n, n) for n in xrange(0, 46)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 20 2009]
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CROSSREFS
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Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Cf. A016742.
a(n)=A007742(-n)=A074378(2n-1)=A014848(2n).
Cf. A014635.
Cf. A157330, A157331 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 27 2009]
Sequence in context: A071396 A032525 A130697 this_sequence A155154 A081269 A140064
Adjacent sequences: A033988 A033989 A033990 this_sequence A033992 A033993 A033994
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KEYWORD
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nonn,easy,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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