%I A144912
%S A144912 0,2,2,1,0,4,1,2,2,6,1,0,0,4,8,3,2,2,2,6,10,2,4,2,0,4,8,
%T A144912 12,0,4,0,2,2,6,10,14,0,2,2,4,0,4,8,12,16,2,0,4,2,2,2,6,
%U A144912 10,14,18,0,2,0,0,6,0,4,8,12,16,20
%V A144912 0,2,-2,-1,0,-4,1,2,-2,-6,1,0,0,-4,-8,3,2,2,-2,-6,-10,-2,4,-2,0,-4,-8,
%W A144912 -12,0,-4,0,2,-2,-6,-10,-14,0,-2,2,-4,0,-4,-8,-12,-16,2,0,4,-2,2,-2,-6,
%X A144912 -10,-14,-18,0,-2,0,0,-6,0,-4,-8,-12,-16,-20
%N A144912 Unreduced numerators of digital mean, dm_num(b, n), with rows n in {2, 
               3, 4, ...} and columns b in {2, 3, 4, ..., n}.
%C A144912 The unreduced numerator of dm(b, n) is sigma(i in [1, d]: d_i * 2 - (b 
               - 1)), where d is the number of digits in the base b representation 
               of n and d_i the individual digits. The corresponding denominator 
               is 2 * d, giving a value in (-(b - 1) / 2, (b - 1) / 2] for n > 0.
%C A144912 dm_num(b, n) = d(b - 1) iff all the digits in n are b - 1.
%C A144912 dm_num(b, n) = -2(b - 2) for b = n, because n in base n is 10, giving 
               dm_num(n, n) = 2 - n + 1 + 0 - n + 1 = 4 - 2 * n = -2(n - 2).
%C A144912 dm_num(b, n) = 0 for odd b and n having all digits equal to (b - 1) / 
               2, as well as for many other (b, n).
%C A144912 Defining m = ceil((n + 1) / 2):
%C A144912 dm_num(b, n) = dm_num(b - 1, n) - 4 for b in [m + 1, n].
%C A144912 dm_num(m, n) = 0 for even n and 2 for odd n.
%C A144912 dm_num(m - 1, n) = 6 - n for even n > 4 and 9 - n for odd n > 5, producing 
               a sequence of first differences {+2, -4, +2, -4, ...}.
%C A144912 Triangular patterns become clearly visible for large n, defined by additive 
               periodicities along rational slopes. Zeros along the triangle borders 
               correspond to ones in the Redheffer matrix until odd values become 
               dominant. The line along m is the border between the two largest 
               triangles. This pattern is masked by aliasing effects for small bases, 
               notably including base 10, due to the thinness of the triangles which 
               dominate at small b. Odd values may represent "artifacts" caused 
               by "interference".
%H A144912 Reikku Kulon, <a href="b144912.txt">Rows of triangle for b in [2, 141]</
               a>
%H A144912 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RedhefferMatrix.html"> Redheffer Matrix</a>
%H A144912 Reikku Kulon, <a href="a144912.c">C99 source to produce the triangle</
               a>
%H A144912 Reikku Kulon, <a href="a144912.png">Triangle as 2048x2048 PNG image</
               a> (zero is white, primes are black and darker greys indicate fewer 
               prime factors)
%H A144912 Reikku Kulon, <a href="a144912_shadow.png">Triangle as 2048x2048 PNG 
               image, extended to b in [2, 2 * n + 1]</a>
%H A144912 Reikku Kulon, <a href="a144912_primes.png">Prime band as 16384x256 PNG 
               image</a> (note the curves coincident with new strips of primes, 
               as well as the second band which appears at 4096 and corresponds 
               to the 637/638 gap in A031443)
%H A144912 Reikku Kulon, <a href="a144912_primes_64k.png">Prime band as 16384x256 
               PNG image, starting from n = 57344</a>
%Y A144912 Cf. A002321, A083058, A031443, A144777, A144798, A144799, A144800, A144801, 
               A144812
%Y A144912 Sequence in context: A155161 A065177 A064044 this_sequence A145337 A071464 
               A071510
%Y A144912 Adjacent sequences: A144909 A144910 A144911 this_sequence A144913 A144914 
               A144915
%K A144912 base,easy,frac,sign,tabl
%O A144912 2,2
%A A144912 Reikku Kulon (reikku(AT)gmail.com), Sep 25 2008, Oct 03 2008