A293704 a(n) is the shift of the longest palindromic subsequence in the first n terms of A293700.

0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, 18, 17, 16

Offset

1,19

Comments

Shift is the measure of the position of the palindromic subsequence within the corresponding sequence of first differences, defined as the number of terms being dropped from the left end of the sequence of first differences minus those dropped from its right end. Thus, when shift is negative, the palindrome has moved leftward from its symmetric position.

Links

V.J. Pohjola, Table of n, a(n) for n = 1..10000

V. J. Pohjola, Line plot drawn for n=1...150

V. J. Pohjola, Line plot drawn for n=1...2500

V. J. Pohjola, Line plot drawn for n=1...10000

Example

For n = 1, first differences = 3; longest palindrome = 3; a(1) = 0 - 0 = 0.
For n = 2, differences = 3, 19; longest palindrome = 3; a(2) = 0 - 1 = -1.
For n = 22, differences = 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 3, 16, 3, 3, 16; longest palindrome = 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3; a(22) = 0 - 5 = -5.

Mathematica

rootsp = Flatten[Position[Table[Floor[Tan[i]], {i, 1, 10^4}], 1]];
lrootsp = Length[rootsp];
difp = Differences[rootsp];
ldp = Length[difp];
kmax = 500; palsp = {}; lenpalsp = {0}; shiftp = {}; posp = {};
Do[diffip = difp[[1 ;; k]]; lendiffip = Length[diffip];
  pmax = k - Last[lenpalsp];
  t = Table[difp[[p ;; k]], {p, 1, pmax}];
  sp = Flatten[Select[t, # == Reverse[#] &]];
  If[sp == {},
   AppendTo[palsp, Last[palsp]] && AppendTo[lenpalsp, Last[lenpalsp]],
   AppendTo[palsp, sp] && AppendTo[lenpalsp, Length[Flatten[sp]]]];
  AppendTo[posp, Position[t, Last[palsp]]]; pp = Last[Flatten[posp]] - 1;
  qq = lendiffip - (pp + Last[lenpalsp]);
  AppendTo[shiftp, pp - qq], {k, 1, kmax}];
lenpalsp;
shiftp (*a(n)=shiftp[[n]]*)

Crossrefs

Cf. A293698, A293751, A293700, A293701, A293705, A293699, A293702, A293706, A293703.

Sequence in context: A318891 A318890 A124196 * A322021 A295888 A295880

Adjacent sequences: A293701 A293702 A293703 * A293705 A293706 A293707

Keyword

sign

Author

V.J. Pohjola, Oct 21 2017

Status

approved