A353448 Number w is in this sequence if every frame w X h, w >= h >= 3, contains more distinct distance quadrilaterals with corners interior to the 4 sides with concurrent diagonals, i.e., both ascending or both descending, than non-concurrent diagonals, or equivalently A353450(w,h) >= A353449(w,h).

3, 4, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 29, 31, 33, 34, 37, 41, 43, 45, 46, 49, 55, 57, 61, 67, 73, 79, 81, 85, 89, 91, 97, 109, 113, 121, 127, 133, 141, 145, 151, 157, 161, 169, 181, 193, 201, 205, 209, 211, 217, 221, 225, 241, 253, 261, 265, 271

Offset

1,1

Comments

It is conjectured that this sequence is a subsequence A160007 except for the small terms <= 46 (verified for all w < 661). The example section depicts the way this sequence matches the irregular pattern of A160007. Numbers w > 100 were computed by Hugo Pfoertner.

w = 617 is one of the rare occurrences of remainder 17 mod 60 in A160007, and it is also in this sequence. One might suspect that there would be less and less hits, but the time-consuming computation successfully countered the intuition.

Links

Rainer Rosenthal, Table of n, a(n) for n = 1..96

Hugo Pfoertner, Comparison to A225730, using Plot 2.

Example

.
w = 5 is in this sequence:
.
                        5 | . . . C .
   4 | . . C . .        4 | .       .    w = 5 is in this sequence because all
   3 | .       B        3 | .       B    quadrilaterals in (5,4) and (5,5)
   2 | D       .        2 | D       .    shown in the example section of A353450
   1 | . A . . .        1 | . A . . .    have concurrent diagonals.
   y /----------        y /----------
     x 1 2 3 4 5          x 1 2 3 4 5
.
w = 6 is *not* in this sequence:
.
   4 | . . . C . .         w = 6 is not in this sequence because of the single
   3 | D         .         quadrilateral in (6,4) shown in the example section
   2 | .         B         of A353449. Diagonal AC is rising while diagonal DB
   1 | . A . . . .         is falling (non-concurrent diagonals).
   y /------------         There is no (6,4) quadrilateral with all distances
     x 1 2 3 4 5 6         distinct and with concurrent diagonals!
.
            123456789012345678901234567890123456789012345678901234567890
    1 -  60   ::: x  :: x:::: x x : x   x x ::  x   x x ::  x     x x
   61 - 120 x     x   . x     x x   x   x x     x   . . .   x   x .
  121 - 180 x     x     x     . x   x     x     x   x .     x     . .
  181 - 240 x     .     x   . . x   x   x x     x   x . x   .     .
  241 - 300 x     .     x     . x   x     x .   .   x .     x     .
  301 - 360 x   . . .   x     . x   x     .     x   . .     . . . .
  361 - 420 x   . . .   .     x .   x     x .   x   x .     x     . .
  421 - 480 x     .     x     . x   .   . .     x   . x .   x     . .
  481 - 540 x     .   . .   . . .   x     .     .   x .     x   . .
  541 - 600 x   . .   . .     . x   .     . .   x   . .     x     .
  601 - 660 x     . .   x   x . .   x     x     .   x . .   x .   .
.
Legend:
  "x" marks numbers w belonging to this sequence and to A160007.
  ":" marks numbers w belonging to this sequence only.
  "." marks numbers w belonging to A160007 only.

Crossrefs

Cf. A160007, A353532 ("all tetrapods"), A353449 ("unisense"), A353450 ("contrasense").

Cf. A225730 (has many terms in common when 1 is added, see also comparison plot).

Sequence in context: A370862 A072556 A047365 * A048342 A159560 A288427

Adjacent sequences: A353445 A353446 A353447 * A353449 A353450 A353451

Keyword

nonn

Author

Rainer Rosenthal, May 22 2022

Status

approved