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A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

%I A163253 #9 Jan 05 2025 19:51:39
%S A163253 1,4,2,9,5,3,16,10,7,6,25,17,13,11,8,36,26,21,18,14,12,49,37,31,27,22,
%T A163253 19,15,64,50,43,38,32,28,23,20,81,65,57,51,44,39,33,29,24,100,82,73,
%U A163253 66,58,52,45,40,34,30,121,101,91,83,74,67,59,53,46,41,35
%N A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.
%C A163253 A permutation of the natural numbers.
%C A163253 Row 1 consists of the squares.
%C A163253 Beginning with row 5, the columns obey a 3rd-order recurrence:
%C A163253 c(n)=c(n-1)+c(n-2)-c(n-3)+1; thus disregarding row 1, the nonsquares are partitioned by this recurrence.
%C A163253 Except for initial terms, the first ten rows match A000290, A002522, A002061, A059100, A014209, A117950, A027688, A087475, A027689, A117951, and the first column, A035106.
%H A163253 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-1/Kimberling.pdf">Doubly interspersed sequences, double interspersions and fractal sequences</a>, The Fibonacci Quarterly 48 (2010) 13-20.
%F A163253 Let S(n,k) denote the k-th term in the n-th row. Three cases:
%F A163253 S(1,k)=k^2;
%F A163253 if n is even, then S(n,k)=k^2+(n-2)k+(n^2-2*n+4)/4;
%F A163253 if n>=3 is odd, then S(n,k)=k^2+(n-2)k+(n^2-2*n+1)/4.
%e A163253 Corner:
%e A163253 1....4....9...16...25
%e A163253 2....5...10...17...26
%e A163253 3....7...13...21...31
%e A163253 6...11...18...27...38
%e A163253 The double interspersion A161179 begins thus:
%e A163253 1....4....7...12...17...24
%e A163253 2....3....8...11...18...23
%e A163253 5....6...13...16...25...30
%e A163253 9...10...19...22...33...38
%e A163253 Expel the even-numbered columns, leaving
%e A163253 1....7...17...
%e A163253 2....8...18...
%e A163253 5...13...25...
%e A163253 9...19...33...
%e A163253 Then replace each of those numbers by its rank when all the numbers are jointly ranked.
%Y A163253 Cf. A000037, A000290, A161179, A163254, A163255, A163256, A163257, A163258.
%K A163253 nonn,tabl
%O A163253 1,2
%A A163253 _Clark Kimberling_, Jul 23 2009
%E A163253 Edited and augmented by _Clark Kimberling_, Jul 24 2009