|
%I #95 Sep 16 2022 10:28:20
%S 0,5,18,39,68,105,150,203,264,333,410,495,588,689,798,915,1040,1173,
%T 1314,1463,1620,1785,1958,2139,2328,2525,2730,2943,3164,3393,3630,
%U 3875,4128,4389,4658,4935,5220,5513,5814,6123,6440,6765,7098,7439,7788,8145
%N a(n) = n*(4*n+1).
%C Write 0,1,2,... in a clockwise spiral; sequence gives the numbers that fall on the positive y-axis. (See Example section.)
%C Central terms of the triangle in A126890. - _Reinhard Zumkeller_, Dec 30 2006
%C a(n)*Pi is the total length of 4 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A004770. The spiral length ratio rounded down [floor(L(n)/L(1))] is A047497. See illustration in links. - _Kival Ngaokrajang_, Dec 27 2013
%C For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n; {4, 4n}]. For n=1, this collapses to [2, {4}]. - _Magus K. Chu_, Sep 15 2022
%D S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%H Vincenzo Librandi, <a href="/A007742/b007742.txt">Table of n, a(n) for n = 0..10000</a>
%H Emilio Apricena, <a href="/A035608/a035608.png">A version of the Ulam spiral</a>
%H Kival Ngaokrajang, <a href="/A007742/a007742.pdf">Illustration of 4 points circle center spiral</a>
%H Leo Tavares, <a href="/A007742/a007742.jpg">Illustration: Triangular Layers</a>
%H G. Thimm, <a href="/A007741/a007741.pdf">Emails to N. J. A. Sloane, Sep. 1994</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(5+3*x)/(1-x)^3. - _Michael Somos_, Mar 03 2003
%F a(n) = A033991(-n) = A074378(2*n).
%F a(n) = floor((n + 1/4)^2). - _Reinhard Zumkeller_, Feb 20 2010
%F a(n) = A110654(n) + A173511(n) = A002943(n) - n. - _Reinhard Zumkeller_, Feb 20 2010
%F a(n) = 8*n + a(n-1) - 3. - _Vincenzo Librandi_, Nov 21 2010
%F Sum_{n>=1} 1/a(n) = Sum_{k>=0} (-1)^k*zeta(2+k)/4^(k+1) = 0.349762131... . - _R. J. Mathar_, Jul 10 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=18. - _Philippe Deléham_, Mar 26 2013
%F a(n) = A118729(8n+4). - _Philippe Deléham_, Mar 26 2013
%F a(n) = A000217(3*n) - A000217(n). - _Bruno Berselli_, Sep 21 2016
%F E.g.f.: (4*x^2 + 5*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F From _Amiram Eldar_, Jul 03 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 4 - Pi/2 - 3*log(2).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(2) + log(2) + sqrt(2)*log(1 + sqrt(2)) - 4. (End)
%F a(n) = A081266(n) - A000217(n). - _Leo Tavares_, Mar 25 2022
%e Part of the spiral:
%e .
%e 64--65--66--67--68
%e |
%e 63 36--37--38--39--40--41--42
%e | | |
%e 62 35 16--17--18--19--20 43
%e | | | | |
%e 61 34 15 4---5---6 21 44
%e | | | | | | |
%e 60 33 14 3 0 7 22 45
%e | | | | | | | |
%e 59 32 13 2---1 8 23 46
%e | | | | | |
%e 58 31 12--11--10---9 24 47
%e | | | |
%e 57 30--29--28--27--26--25 48
%e | |
%e 56--55--54--53--52--51--50--49
%t LinearRecurrence[{3,-3,1},{0,5,18},50] (* _Vincenzo Librandi_, Jan 29 2012 *)
%t Table[n(4n+1),{n,0,50}] (* _Harvey P. Dale_, Aug 10 2017 *)
%o (PARI) a(n)=4*n^2+n
%o (Magma) I:=[0, 5, 18]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jan 29 2012
%Y Cf. A033991, A074378.
%Y Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%Y Cf. index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.
%Y Cf. A081266.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
|
