A007742 a(n) = n*(4*n+1).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Write 0,1,2,... in a clockwise spiral; sequence gives the numbers that fall on the positive y-axis. (See Example section.)

Central terms of the triangle in A126890. - Reinhard Zumkeller, Dec 30 2006

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

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

Examples

			Part of the spiral:
.
  64--65--66--67--68
   |
  63  36--37--38--39--40--41--42
   |   |                       |
  62  35  16--17--18--19--20  43
   |   |   |               |   |
  61  34  15   4---5---6  21  44
   |   |   |   |       |   |   |
  60  33  14   3   0   7  22  45
   |   |   |   |   |   |   |   |
  59  32  13   2---1   8  23  46
   |   |   |           |   |   |
  58  31  12--11--10---9  24  47
   |   |                   |   |
  57  30--29--28--27--26--25  48
   |                           |
  56--55--54--53--52--51--50--49
		

References

• 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Emilio Apricena, A version of the Ulam spiral
• Robert FERREOL, Illustration by pentagons
• Kival Ngaokrajang, Illustration of 4 points circle center spiral
• Leo Tavares, Illustration: Triangular Layers
• G. Thimm, Emails to N. J. A. Sloane, Sep. 1994
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A033991, A074378.

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

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.

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.

Cf. index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.

Cf. A081266.

Programs

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

Mathematica

LinearRecurrence[{3,-3,1},{0,5,18},50] (* Vincenzo Librandi, Jan 29 2012 *)
Table[n(4n+1),{n,0,50}] (* Harvey P. Dale, Aug 10 2017 *)

PARI

a(n)=4*n^2+n

Formula

G.f.: x*(5+3*x)/(1-x)^3. - Michael Somos, Mar 03 2003

a(n) = A033991(-n) = A074378(2*n).

a(n) = floor((n + 1/4)^2). - Reinhard Zumkeller, Feb 20 2010

a(n) = A110654(n) + A173511(n) = A002943(n) - n. - Reinhard Zumkeller, Feb 20 2010

a(n) = 8*n + a(n-1) - 3. - Vincenzo Librandi, Nov 21 2010

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

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

a(n) = A118729(8n+4). - Philippe Deléham, Mar 26 2013

a(n) = A000217(3*n) - A000217(n). - Bruno Berselli, Sep 21 2016

E.g.f.: (4*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017

From Amiram Eldar, Jul 03 2020: (Start)

Sum_{n>=1} 1/a(n) = 4 - Pi/2 - 3*log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(2) + log(2) + sqrt(2)*log(1 + sqrt(2)) - 4. (End)

a(n) = A081266(n) - A000217(n). - Leo Tavares, Mar 25 2022