cp's OEIS Frontend

A161179 A double interspersion, R(n,k), by antidiagonals.

%I A161179 #2 Mar 30 2012 18:57:11
%S A161179 1,4,2,7,3,5,12,8,6,9,17,11,13,10,14,24,18,16,19,15,20,31,23,25,22,26,
%T A161179 21,27,40,32,30,33,29,34,28,35,49,39,41,38,42,37,43,36,44,60,50,48,51,
%U A161179 47,52,46,53,45,54,71,59,61,58,62,57,63,56,64,55,65,84,72,70,73,69,74
%N A161179 A double interspersion, R(n,k), by antidiagonals.
%C A161179 (1) Every positive integer occurs exactly once; thus as a sequence, A161179
%C A161179 is a permutation of the positive integers.
%C A161179 (2) The double interspersion property for every pair of rows
%C A161179 s=(s(1),s(2),s(3),...) and t=(t(1),t(2),t(3),...) can be stated for
%C A161179 s(1)<t(1) as follows: there exist indices i and j such that
%C A161179 s(i)<s(i+1)<t(j)<t(j+1)<s(i+2)<s(i+3)<t(j+2)<t(j+3)<...
%C A161179 (For a [single] interspersion, see A035513, the Wythoff array.)
%F A161179 If k is odd and n=1: R(n,k)=(1/2)(k^2+2k-1)
%F A161179 if k is odd and n>1: R(n,k)=(1/2)(n^2+(2k-1)n+k^2-2k-1)
%F A161179 if k is even and n=1: R(n,k)=(1/2)(k^2+2k)
%F A161179 if k is even and n>1: R(n,k)=(1/2)(n^2+(2k-3)n+k^2-2k)
%F A161179 To derive the above, use the easily discerned patterns of
%F A161179 the difference array of R. Quite a different construction
%F A161179 of R takes place in two steps: first, form an array U from the
%F A161179 golden mean, tau, and the Fibonacci numbers A000045, as follows:
%F A161179 if n=1 then U(n,k)=F(k+1)
%F A161179 if n>1, then U(n,k)=F(n+k-1)+F(k+2)-tau*F(k+1).
%F A161179 Then replace each U(n,k) by its rank when all the numbers U(n,k)
%F A161179 are arranged in increasing order. The resulting array is R. (The
%F A161179 rows of U are doubly interspersed and satisfy the Fibonacci
%F A161179 recurrence; no array of rational integers has both of those properties.)
%e A161179 Northwest corner:
%e A161179 1....4....7...12...17...24...31...40...49...60
%e A161179 2....3....8...11...18...23...32...39...50...59
%e A161179 5....6...13...16...25...30...41...48...61...70
%e A161179 9...10...19...22...33...38...51...58...73...82
%e A161179 Rows 2 and 4 are doubly interspersed because
%e A161179 3<8<9<10<11<18<19<22<23<32<33<38<...
%Y A161179 A000045, A035513.
%K A161179 nonn,tabl
%O A161179 1,2
%A A161179 _Clark Kimberling_, Jun 05 2009