

A033996


8 times triangular numbers: a(n) = 4*n*(n+1).


82



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
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OFFSET

0,2


COMMENTS

Write 0, 1, 2, ... in a clockwise spiral; sequence gives numbers on one of 4 diagonals.
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 squareshaped layers made of adjacent squares. a(n) is the total number of squares in the first to nth 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 crosspolytope) 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 the nth stage of growth of the twodimensional cellular automaton defined by "Rule 645", based on the 5celled 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 periodic part {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
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
a(n2) is the number of ways to place 3 adjacent marks in a diagonal, horizontal, or vertical row on an n X n tictactoe grid.  Matej Veselovac, May 28 2021


REFERENCES

Stuart M. Ellerstein, J. Recreational Math. 29 (3) 188, 1998.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. AddisonWesley, Reading, MA, 2nd ed., 1994, p. 99.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.


LINKS



FORMULA

a(n) = 4*n^2 + 4*n = (2*n+1)^2  1.
G.f.: 8*x/(1x)^3.
E.g.f.: 4*x*(2 + x)*exp(x).
Sum_{n>=1} 1/a(n) = 1/4. (End)
sqrt(a(n)+1)  sqrt(a(n)) = (sqrt(n+1)  sqrt(n))^2.  Seiichi Manyama, Dec 23 2018
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
Product_{n>=1} (1  1/a(n)) = (4/Pi)*cos(Pi/sqrt(2)).
Product_{n>=1} (1 + 1/a(n)) = 4/Pi (A088538). (End)


EXAMPLE

Spiral with 0, 8, 24, 48, ... along lower right diagonal:
.
36373839404142
 
35 1617181920 43
   
34 15 456 21 44
     
33 14 3 0 7 22 45
    \   
32 13 21 8 23 46
   \  
31 1211109 24 47
  \ 
302928272625 48
\


MAPLE

[seq((2*n+1)^21, n=0..46)];


MATHEMATICA



PROG

(PARI) nsqm1(n) = { forstep(x=1, n, 2, y = x*x1; print1(y, ", ") ) }


CROSSREFS

Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A088538, A124080, A008590 (first differences), A130809 (partial sums).
Sequences from spirals: A001107, A002939, A002943, A007742, A033951, A033952, A033953, A033954, A033988, A033989, A033990, A033991, A033996.  Omar E. Pol, Dec 12 2008


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



