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A033996 8 times triangular numbers: a(n) = 4*n*(n+1). 63
0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Write 0, 1, 2, ... in clockwise spiral; sequence gives numbers on one of 4 diagonals.

Also, least m > n such that T(m)*T(n) is a square and more precisely that of A055112(n). {T(n) = A000217(n)} - Lekraj Beedassy, May 14 2004

Or, product of n-th even number and n-th even nonprime. - Juri-Stepan Gerasimov, Jul 26 2009

Also sequence found by reading the line from 0, in the direction 0, 8, ... and the same line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Axis perpendicular to A195146 in the same spiral. - Omar E. Pol, Sep 18 2011

Number of diagonals with length sqrt(5) in an (n+1) X (n+1) square grid. Every 1 X 2 rectangle has two such diagonals. - Wesley Ivan Hurt, Mar 25 2015

Imagine a board made of squares (like a chessboard), one of whose squares is completely surrounded by square-shaped layers made of adjacent squares. a(n) is the total number of squares in the first to n-th layer. a(1) = 8 because there are 8 neighbors to the unit square; adding them gives a 3 X 3 square. a(2) = 24 = 8 + 16 because we need 16 more squares in the next layer to get a 5 X 5 square: a(n) = (2*n+1)^2 - 1 counting the (2n+1) X (2n+1) square minus the central square. - R. J. Cano, Sep 26 2015

The three platonic solids (the simplex, hypercube, and cross-polytope) with unit side length in n dimensions all have rational volume if and only if n appears in this sequence, after 0. - Brian T Kuhns, Feb 26 2016

The number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016

The square-root of a(n), n>0, has continued fraction [2n; {1,4n}] with whole number part 2n and  period 1,4n. - Ron Knott, May 11 2017

Numbers k such that k+1 is a square and k is a multiple of 4. - Bruno Berselli, Sep 28 2017

a(n) is the number of vertices of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch, May 13 2018

REFERENCES

M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of  certain networks, J. of Optoelectronics and Advanced Materials , 18, No. 9-10, 2016, 884-892.

Stuart M. Ellerstein, J. Recreational Math. 29 (3) 188, 1998.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Eric Weisstein's World of Mathematics, Knight Graph

Eric Weisstein's World of Mathematics, Hamiltonian Path

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 4*n^2 + 4*n = (2*n+1)^2 - 1.

G.f.: 8*x/(1-x)^3.

a(n) = A016754(n) - 1 = 2*A046092(n) = 4*A002378(n). - Lekraj Beedassy, May 25 2004

a(n) = A049598(n) - A046092(n); a(n) = A124080(n) - A002378(n). - Zerinvary Lajos, Mar 06 2007

a(n) = 8*A000217(n). - Omar E. Pol, Dec 12 2008

a(n) = A005843(n) * A163300(n). - Juri-Stepan Gerasimov, Jul 26 2009

a(n) = a(n-1)+8*n (with a(0)=0). - Vincenzo Librandi, Nov 17 2010

For n>0, a(n) = A058031(n+1) - A062938(n-1). - Charlie Marion, Apr 11 2013

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Wesley Ivan Hurt, Mar 25 2015

a(n) = A000578(n+1) - A152618(n). - Bui Quang Tuan, Apr 01 2015

a(n) - a(n-1) = A008590(n), n>0. - Altug Alkan, Sep 26 2015

From Ilya Gutkovskiy, May 19 2016: (Start)

E.g.f.: 4*x*(2 + x)*exp(x).

Sum_{n>=1} 1/a(n) = 1/4. (End)

Product_{n>=1} a(n)/A016754(n) = Pi/4. - Daniel Suteu, Dec 25 2016

a(n) = A056220(n) + A056220(n+1). - Bruce J. Nicholson, May 29 2017

EXAMPLE

Spiral with 0, 8, 24, 48, ... along lower right diagonal:

.

  36--37--38--39--40--41--42

   |                       |

  35  16--17--18--19--20  43

   |   |               |   |

  34  15   4---5---6  21  44

   |   |   |       |   |   |

  33  14   3   0   7  22  45

   |   |   |   | \ |   |   |

  32  13   2---1   8  23  46

   |   |           | \ |   |

  31  12--11--10---9  24  47

   |                   | \ |

  30--29--28--27--26--25  48

                            \

[Reformatted by Jon E. Schoenfield, Dec 25 2016]

MAPLE

seq(8*binomial(n+1, 2), n=0..46); # Zerinvary Lajos, Nov 24 2006

[seq((2*n+1)^2-1, n=0..46)];

MATHEMATICA

s = 0; lst = {s}; Do[s += n++ + 8; AppendTo[lst, s], {n, 0, 7!, 8}]; lst (*  Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

Table[(2n - 1)^2 - 1, {n, 50}] (* Alonso del Arte, Mar 31 2013 *)

PROG

(PARI) nsqm1(n) = { forstep(x=1, n, 2, y = x*x-1; print1(y, ", ") ) }

(MAGMA) [ 4*n*(n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

CROSSREFS

Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A124080, A008590 (first differences).

Sequences from spirals: A001107, A002939, A002943, A007742, A033951, A033952, A033953, A033954, A033988, A033989, A033990, A033991, A033996. - Omar E. Pol, Dec 12 2008

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.

Sequence in context: A022763 A244370 A179682 * A146980 A028612 A068857

Adjacent sequences:  A033993 A033994 A033995 * A033997 A033998 A033999

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 11 1999

STATUS

approved

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Last modified August 17 09:30 EDT 2018. Contains 313814 sequences. (Running on oeis4.)