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 A293705 a(n) is the shift of the longest palindromic subsequence in the first n terms of A293699. 10
 0, -1, 0, -1, 0, -1, 0, -1, -2, -3, -4, -5, -6, 6, 5, 7, 6, 5, 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, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 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. When shift is a positive number, it indicates the number of steps that the palindrome has moved to the right from its symmetric position. LINKS V.J. Pohjola, Table of n, a(n) for n = 1..10000 V.J. Pohjola, Line plot for n=1...10000 V.J. Pohjola, Line plot for n=1...100 EXAMPLE For n = 1, 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 = 14, differences = 3, 19, 3, 19, 3, 19, 3, 3, 16, 3, 3, 16, 3, 3; longest palindrome = 3, 3, 16, 3, 3, 16, 3, 3; a(14) = 6 - 0 = 6. MATHEMATICA rootsn = Flatten[Position[Table[Floor[Tan[-i]], {i, 1, 10^4}], 1]]; difn = Differences[rootsn]; ldn = Length[difn]; kmax = 500; palsn = {}; lenpalsn = {0}; shiftn = {}; posn = {}; Do[diffin = difn[[1 ;; k]]; lendiffin = Length[diffin];   pmax = k - Last[lenpalsn];   t = Table[difn[[p ;; k]], {p, 1, pmax}];   sn = Flatten[Select[t, # == Reverse[#] &]];   If[sn == {},    AppendTo[palsn, Last[palsn]] && AppendTo[lenpalsn, Last[lenpalsn]],    AppendTo[palsn, sn] && AppendTo[lenpalsn, Length[Flatten[sn]]]];   AppendTo[posn, Position[t, Last[palsn]]]; pp = Last[Flatten[posn]] - 1;   qq = lendiffin - (pp + Last[lenpalsn]);   AppendTo[shiftn, pp - qq], {k, 1, kmax}]; shiftn (*a(n)=shiftn[[n]]*) CROSSREFS Cf. A293698, A293751, A293700, A293702, A293704, A293699, A293701, A293706, A293703. Sequence in context: A307311 A272081 A317582 * A036055 A034151 A101272 Adjacent sequences:  A293702 A293703 A293704 * A293706 A293707 A293708 KEYWORD sign AUTHOR V.J. Pohjola, Oct 21 2017 STATUS approved

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Last modified January 22 23:50 EST 2022. Contains 350504 sequences. (Running on oeis4.)