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%I #213 Feb 21 2023 02:14:49
%S 0,8,24,48,80,120,168,224,288,360,440,528,624,728,840,960,1088,1224,
%T 1368,1520,1680,1848,2024,2208,2400,2600,2808,3024,3248,3480,3720,
%U 3968,4224,4488,4760,5040,5328,5624,5928,6240,6560,6888,7224,7568,7920,8280
%N 8 times triangular numbers: a(n) = 4*n*(n+1).
%C Write 0, 1, 2, ... in a clockwise spiral; sequence gives numbers on one of 4 diagonals.
%C 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
%C 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
%C 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
%C 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
%C 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
%C The number of active (ON,black) cells in the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 19 2016
%C The square root of a(n), n>0, has continued fraction [2n; {1,4n}] with whole number part 2n and periodic part {1,4n}. - _Ron Knott_, May 11 2017
%C Numbers k such that k+1 is a square and k is a multiple of 4. - _Bruno Berselli_, Sep 28 2017
%C 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
%C a(n) is the number of vertices in conjoined n X n octagons which are arranged into a square array, a.k.a. truncated square tiling. - _Donghwi Park_, Dec 20 2020
%C a(n-2) is the number of ways to place 3 adjacent marks in a diagonal, horizontal, or vertical row on an n X n tic-tac-toe grid. - _Matej Veselovac_, May 28 2021
%D Stuart M. Ellerstein, J. Recreational Math. 29 (3) 188, 1998.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%D Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H Ivan Panchenko, <a href="/A033996/b033996.txt">Table of n, a(n) for n = 0..1000</a>
%H M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, <a href="https://joam.inoe.ro/articles/computing-topological-indices-of-certain-networks/">Computing topological indices of certain networks</a>, J. of Optoelectronics and Advanced Materials, 18, No. 9-10, 2016, 884-892.
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H Leo Tavares, <a href="/A033996/a033996.jpg">Illustration: Centroid Diamonds</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>.
%H Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>.
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>.
%F a(n) = 4*n^2 + 4*n = (2*n+1)^2 - 1.
%F G.f.: 8*x/(1-x)^3.
%F a(n) = A016754(n) - 1 = 2*A046092(n) = 4*A002378(n). - _Lekraj Beedassy_, May 25 2004
%F a(n) = A049598(n) - A046092(n); a(n) = A124080(n) - A002378(n). - _Zerinvary Lajos_, Mar 06 2007
%F a(n) = 8*A000217(n). - _Omar E. Pol_, Dec 12 2008
%F a(n) = A005843(n) * A163300(n). - _Juri-Stepan Gerasimov_, Jul 26 2009
%F a(n) = a(n-1) + 8*n (with a(0)=0). - _Vincenzo Librandi_, Nov 17 2010
%F For n > 0, a(n) = A058031(n+1) - A062938(n-1). - _Charlie Marion_, Apr 11 2013
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Wesley Ivan Hurt_, Mar 25 2015
%F a(n) = A000578(n+1) - A152618(n). - _Bui Quang Tuan_, Apr 01 2015
%F a(n) - a(n-1) = A008590(n), n > 0. - _Altug Alkan_, Sep 26 2015
%F From _Ilya Gutkovskiy_, May 19 2016: (Start)
%F E.g.f.: 4*x*(2 + x)*exp(x).
%F Sum_{n>=1} 1/a(n) = 1/4. (End)
%F Product_{n>=1} a(n)/A016754(n) = Pi/4. - _Daniel Suteu_, Dec 25 2016
%F a(n) = A056220(n) + A056220(n+1). - _Bruce J. Nicholson_, May 29 2017
%F sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^2. - _Seiichi Manyama_, Dec 23 2018
%F a(n)*a(n+k) + 4*k^2 = m^2 where m = (a(n) + a(n+k))/2 - 2*k^2; for k=1, m = 4*n^2 + 8*n + 2 = A060626(n). - _Ezhilarasu Velayutham_, May 22 2019
%F Sum_{n>=1} (-1)^n/a(n) = 1/4 - log(2)/2. - _Vaclav Kotesovec_, Dec 21 2020
%F From _Amiram Eldar_, Feb 21 2023: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = -(4/Pi)*cos(Pi/sqrt(2)).
%F Product_{n>=1} (1 + 1/a(n)) = 4/Pi (A088538). (End)
%e Spiral with 0, 8, 24, 48, ... along lower right diagonal:
%e .
%e 36--37--38--39--40--41--42
%e | |
%e 35 16--17--18--19--20 43
%e | | | |
%e 34 15 4---5---6 21 44
%e | | | | | |
%e 33 14 3 0 7 22 45
%e | | | | \ | | |
%e 32 13 2---1 8 23 46
%e | | | \ | |
%e 31 12--11--10---9 24 47
%e | | \ |
%e 30--29--28--27--26--25 48
%e \
%e [Reformatted by _Jon E. Schoenfield_, Dec 25 2016]
%p seq(8*binomial(n+1, 2), n=0..46); # _Zerinvary Lajos_, Nov 24 2006
%p [seq((2*n+1)^2-1, n=0..46)];
%t Table[(2n - 1)^2 - 1, {n, 50}] (* _Alonso del Arte_, Mar 31 2013 *)
%o (PARI) nsqm1(n) = { forstep(x=1,n,2, y = x*x-1; print1(y, ", ") ) }
%o (Magma) [ 4*n*(n+1) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%Y Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A088538, A124080, A008590 (first differences), A130809 (partial sums).
%Y Sequences from spirals: A001107, A002939, A002943, A007742, A033951, A033952, A033953, A033954, A033988, A033989, A033990, A033991, A033996. - _Omar E. Pol_, Dec 12 2008
%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.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 11 1999
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