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A161179 A double interspersion, R(n,k), by antidiagonals. 3
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, 21, 27, 40, 32, 30, 33, 29, 34, 28, 35, 49, 39, 41, 38, 42, 37, 43, 36, 44, 60, 50, 48, 51, 47, 52, 46, 53, 45, 54, 71, 59, 61, 58, 62, 57, 63, 56, 64, 55, 65, 84, 72, 70, 73, 69, 74 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

(1) Every positive integer occurs exactly once; thus as a sequence, A161179

is a permutation of the positive integers.

(2) The double interspersion property for every pair of rows

s=(s(1),s(2),s(3),...) and t=(t(1),t(2),t(3),...) can be stated for

s(1)<t(1) as follows: there exist indices i and j such that

s(i)<s(i+1)<t(j)<t(j+1)<s(i+2)<s(i+3)<t(j+2)<t(j+3)<...

(For a [single] interspersion, see A035513, the Wythoff array.)

LINKS

Table of n, a(n) for n=1..72.

FORMULA

If k is odd and n=1: R(n,k)=(1/2)(k^2+2k-1)

if k is odd and n>1: R(n,k)=(1/2)(n^2+(2k-1)n+k^2-2k-1)

if k is even and n=1: R(n,k)=(1/2)(k^2+2k)

if k is even and n>1: R(n,k)=(1/2)(n^2+(2k-3)n+k^2-2k)

To derive the above, use the easily discerned patterns of

the difference array of R. Quite a different construction

of R takes place in two steps: first, form an array U from the

golden mean, tau, and the Fibonacci numbers A000045, as follows:

if n=1 then U(n,k)=F(k+1)

if n>1, then U(n,k)=F(n+k-1)+F(k+2)-tau*F(k+1).

Then replace each U(n,k) by its rank when all the numbers U(n,k)

are arranged in increasing order. The resulting array is R. (The

rows of U are doubly interspersed and satisfy the Fibonacci

recurrence; no array of rational integers has both of those properties.)

EXAMPLE

Northwest corner:

1....4....7...12...17...24...31...40...49...60

2....3....8...11...18...23...32...39...50...59

5....6...13...16...25...30...41...48...61...70

9...10...19...22...33...38...51...58...73...82

Rows 2 and 4 are doubly interspersed because

3<8<9<10<11<18<19<22<23<32<33<38<...

CROSSREFS

A000045, A035513.

Sequence in context: A182364 A021238 A222622 * A105727 A126091 A026189

Adjacent sequences:  A161176 A161177 A161178 * A161180 A161181 A161182

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 05 2009

STATUS

approved

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Last modified June 28 09:49 EDT 2017. Contains 288813 sequences.