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A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

%I A163257 #9 Jan 05 2025 19:51:39
%S A163257 1,5,2,11,6,3,19,12,8,4,29,20,15,10,7,41,30,24,18,14,9,55,42,35,28,23,
%T A163257 17,13,71,56,48,40,34,27,22,16,89,72,63,54,47,39,33,26,21,109,90,80,
%U A163257 70,62,53,46,38,32,25,131,110,99,88,79,69,61,52,45,37,31,155,132,120,108
%N A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.
%C A163257 A permutation of the natural numbers.
%C A163257 Beginning at row 6, the columns obey a 3rd-order recurrence:
%C A163257 c(n)=c(n-1)+c(n-2)-c(n-3)+1.
%C A163257 Except for initial terms, the first seven rows are A028387, A002378, A005563, A028552, A008865, A014209, A028873, and the first column, A004652.
%H A163257 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 A163257 Let S(n,k) denote the k-th term in the n-th row. Four cases:
%F A163257 S(1,k)=k^2+k-1
%F A163257 S(2,k)=k^2+k
%F A163257 if n>1 is odd, then S(n,k)=k^2+(n-1)k+(n-1)(n-3)/4
%F A163257 if n>2 is even, then S(n,k)= k^2+(n-1)k+n(n-4)/4.
%e A163257 Corner:
%e A163257 1....5...11...19
%e A163257 2....6...12...20
%e A163257 3....8...15...24
%e A163257 4...10...18...28
%e A163257 The double interspersion A161179 begins thus:
%e A163257 1....4....7...12...17...24
%e A163257 2....3....8...11...18...23
%e A163257 5....6...13...16...25...30
%e A163257 9...10...19...22...33...38
%e A163257 Expel the odd-numbered columns and then swap rows 1 and 2, leaving
%e A163257 3....11...23...39
%e A163257 4....12...24...40
%e A163257 6....16...30...48
%e A163257 10...22...38...58
%e A163257 Then replace each of those numbers by its rank when all the numbers are jointly ranked.
%Y A163257 Cf. A163253, A163254, A163255, A163256, A163258.
%K A163257 nonn,tabl
%O A163257 1,2
%A A163257 _Clark Kimberling_, Jul 24 2009