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A307188 Positions of 0's in the square spiral in A275609.

%I A307188 #31 Jan 22 2022 23:41:08
%S A307188 0,9,11,14,16,18,20,22,30,47,49,51,53,59,61,63,66,68,70,72,74,76,88,
%T A307188 90,92,100,117,119,121,123,125,127,137,139,141,147,149,151,153,155,
%U A307188 158,160,162,178,180,182,184,186,194,196,198,210,219,221,223,225,227
%N A307188 Positions of 0's in the square spiral in A275609.
%C A307188 This sequence and A307189-A307192 together partition the nonnegative numbers.
%C A307188 Comment from _N. J. A. Sloane_, Mar 31 2019: (Start)
%C A307188 Both Rémy Sigrist's illustration and mine show the same block structure. A block B(r,s) will mean an array of r X s 0's separated by rows and columns of blanks. Here is B(3,5), with blanks represented by dots:
%C A307188 ......
%C A307188 0.0.0.
%C A307188 ......
%C A307188 0.0.0.
%C A307188 ......
%C A307188 0.0.0.
%C A307188 The dots are always to the right of or above the 0's.
%C A307188 The block B(r,s) has size 2r X 2s.
%C A307188 The block containing the center point is a B(3,5).
%C A307188 Looking at the blocks along the x-axis, moving from West to East, the successive r-values are:
%C A307188 ... 23, 19, 15, 11, 7, (3), 5, 9, 13, 17, 21, ...
%C A307188 Looking at the blocks along the y-axis, moving from South to North, the successive s-values are:
%C A307188 ... 24, 20, 16, 12, 8, (5), 6, 10, 14, 18, 22, ...
%C A307188 Except at the central block B(3,5), the differences between adjacent r- and s-values always have magnitude 4.
%C A307188 Furthermore, as we move North, each block is displaced by one step to the East from the block below it. As we move East, each block is displaced by one step to the South from the block to the West of it.
%C A307188 These displacements can be seen in both illustrations. In Sigrist's illustration, notice how the blocks gradually shift to the East along the y-xis, and to the South along along the x-axis.
%C A307188 (End)
%H A307188 Alois P. Heinz, <a href="/A307188/b307188.txt">Table of n, a(n) for n = 1..20000</a> (first 1240 terms from N. J. A. Sloane)
%H A307188 Rémy Sigrist, <a href="/A307188/a307188.png">Colored scatterplot of (x,y) such that A275609(x,y) = 0 and -500 <= x <= 500 and -500 <= y <= 500</a> (where the color is function of (x mod 2) + 2 * (y mod 2))
%H A307188 N. J. A. Sloane, <a href="/A307188/a307188.txt">Central portion of spiral in A275609 showing just the 0's</a> [Entries 1,2,3,4 in the spiral have been replaced by dots. The central 0 has been changed to an X. The illustration shows cells (x,y) with -35 <= x <= 35, -33 <= y <= 36.]
%Y A307188 Cf. A275609, A307189-A307192.
%K A307188 nonn
%O A307188 1,2
%A A307188 _N. J. A. Sloane_, Mar 28 2019