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Search: id:A007742
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| 0, 5, 18, 39, 68, 105, 150, 203, 264, 333, 410, 495, 588, 689, 798, 915, 1040, 1173, 1314, 1463, 1620, 1785, 1958, 2139, 2328, 2525, 2730, 2943, 3164, 3393, 3630, 3875, 4128, 4389, 4658, 4935, 5220, 5513, 5814, 6123, 6440, 6765, 7098, 7439, 7788, 8145
(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 positive y axis.
Central terms of the triangle in A126890. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006
If A=[A007742] n(4n+1) (for n>0, 5,18,39,68,...,); Y=[A157336] 64*n+8 (72,136,200,...,); X=[A157337] 128*n^2+32*n+1 (161,577,1249,...,) ; , we have for all terms, Pell's equation X^2-A*Y^2=1. Example: 161^2-5*72^2=1; 577^2-18*136^2=1; 1249^2-39*200^2=1; 2177^2-68*264^2=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(5+3x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n)=8*n+a(n-1)-11 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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EXAMPLE
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Part of the spiral:
16 17 18 19 ...
15 4 5 6 ...
14 3 0 7 ...
13 2 1 8 ...
For n=2, a(2)=8*2+0-11=5; n=3, a(3)=8*3+5-11=18; n=4, a(4)=8*4+18-11=39 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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MATHEMATICA
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s=0; lst={s}; Do[s+=n++ +5; AppendTo[lst, s], {n, 0, 7!, 8}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 16 2008]
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PROGRAM
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(PARI) a(n)=4*n^2+n
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CROSSREFS
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a(n)=A033991(-n)=A074378(2n).
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Cf. A157336, A157337 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 27 2009]
Sequence in context: A038346 A065007 A031428 this_sequence A000338 A056640 A160969
Adjacent sequences: A007739 A007740 A007741 this_sequence A007743 A007744 A007745
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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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