A093573 - OEIS

A093573

Triangle read by rows: row n gives positions where n occurs in the Golay-Rudin-Shapiro related sequence A020986.

%I #40 Jan 28 2020 12:07:52

%S 0,1,3,2,4,6,5,7,13,15,8,12,14,16,26,9,11,17,19,25,27,10,18,20,22,24,

%T 28,30,21,23,29,31,53,55,61,63,32,50,52,54,56,60,62,64,106,33,35,49,

%U 51,57,59,65,67,105,107,34,36,38,48,58,66,68,70,104,108,110,37,39,45,47,69,71,77,79,101,103,109,111

%N Triangle read by rows: row n gives positions where n occurs in the Golay-Rudin-Shapiro related sequence A020986.

%C Each positive integer n occurs n times, so the n-th row has length n.

%H Reinhard Zumkeller, <a href="/A093573/b093573.txt">Rows n = 1..150 of triangle, flattened</a>

%H John Brillhart, Patrick Morton, <a href="http://projecteuclid.org/euclid.ijm/1256048841">Über Summen von Rudin-Shapiroschen Koeffizienten</a>, (German) Illinois J. Math. 22 (1978), no. 1, 126--148. MR0476686 (57 #16245). - _N. J. A. Sloane_, Jun 06 2012

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rudin-ShapiroSequence.html">Rudin-Shapiro Sequence</a>

%e A020986(n) for n = 0, 1, ... is 1, 2, 3, 2, 3, 4, 3, 4, 5, 6, ..., so the positions of 1, 2, 3, 4, ... are 0; 1, 3; 2, 4, 6; 5, 7, 13, 15; ...

%e From _Seiichi Manyama_, Apr 23 2017: (Start)

%e Triangle begins:

%e 0,

%e 1, 3,

%e 2, 4, 6,

%e 5, 7, 13, 15,

%e 8, 12, 14, 16, 26,

%e 9, 11, 17, 19, 25, 27,

%e 10, 18, 20, 22, 24, 28, 30,

%e 21, 23, 29, 31, 53, 55, 61, 63,

%e 32, 50, 52, 54, 56, 60, 62, 64, 106,

%e 33, 35, 49, 51, 57, 59, 65, 67, 105, 107,

%e 34, 36, 38, 48, 58, 66, 68, 70, 104, 108, 110,

%e ... (End)

%t With[{n = 16}, TakeWhile[#, Length@ #2 == #1 & @@ # &][[All, -1]] &@ Transpose@ {Keys@ #, Lookup[#, Keys@ #]} &[PositionIndex@ Accumulate@ Array[1 - 2 Mod[Length[FixedPointList[BitAnd[#, # - 1] &, BitAnd[#, Quotient[#, 2]]]], 2] &, n^2, 0] - 1]] // Flatten (* _Michael De Vlieger_, Jan 25 2020 *)

%o (Haskell)

%o a093573 n k = a093573_row n !! (k-1)

%o a093573_row n = take n $ elemIndices n a020986_list

%o a093573_tabl = map a093573_row [1..]

%o -- _Reinhard Zumkeller_, Jun 06 2012

%Y Column k=1 gives A212591. Diagonal k=n gives A020991.

%Y Cf. A020985, A020986.

%K nonn,tabl

%O 1,3

%A _Eric W. Weisstein_, Apr 01 2004

%E Offset corrected by _Reinhard Zumkeller_, Jun 06 2012