A137707 Secondary Wythoff Array read by descending antidiagonals.

1, 3, 2, 5, 4, 7, 9, 6, 13, 8, 15, 10, 21, 14, 11, 25, 16, 35, 22, 19, 12, 41, 26, 57, 36, 31, 20, 17, 67, 42, 93, 58, 51, 32, 29, 18, 109, 68, 151, 94, 83, 52, 47, 30, 23, 177, 110, 245, 152, 135, 84, 77, 48, 39, 24, 286, 178, 397, 246, 220, 136, 125, 78, 63, 40, 27

Offset

1,2

Comments

(1) Delete the odd numbered rows and get twice the Wythoff array, A035513.

(2) Subtract 1 from the even numbered rows and get the odd numbered rows.

(3) As a sequence, this is a permutation of the positive integers.

(4) The array is a dispersion and an interspersion.

(5) Let c = ordered union of odd numbered columns and let d = ordered union of even numbered columns; then c and d are the unique solutions of the complementary equation d(n)=c(c(n))+2 and also of the complementary equation d(n)=c(n)+2*Floor[(n+2)/2].

(6) c=A137708, d=A137709.

Links

Table of n, a(n) for n=1..66.

Formula

Odd numbered rows: r(n)=r(n-1)+r(n-2)+1, Even numbered rows: r(n)=r(n-1)+r(n-2).

Example

Corner:
    1    3    5     9    15    25    41    67    109   177
    2    4    6    10    16    26    42    68    110   178
    7   13   21    35    57    93   151   245    397   643
    8   14   22    36    58    94   152   246    398   644
   11   19   31    51    83   135   219   355    575   931
   12   20   32    52    84   136   220   356    576   932
   17   29   47    77   125   203   329   533    863  1397
   18   30   48    78   126   204   330   534    864  1398
   23   39   63   103   167   271   439   711   1151  1863
   24   40   64   104   168   272   440   712   1152  1864
   27   45   73   119   193   313   507   821   1329  2151
   28   46   74   120   194   314   508   822   1330  2152
   33   55   89   145   235   381   617   999   1617  2617

Mathematica

w[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
t = Table[2 w[n, k], {n, 1, 12}, {k, 1, 12}]; Grid[Riffle[t - 1, t]]
 (* Clark Kimberling, May 09 2025 *)

Crossrefs

Cf. A035513, A137708, A137709.

Sequence in context: A191709 A194077 A257334 * A164380 A143527 A284153

Adjacent sequences: A137704 A137705 A137706 * A137708 A137709 A137710

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 07 2008

Status

approved