OFFSET
1,2
COMMENTS
-A293751 are the negative roots of floor(tan(k))=1.
Each increment of n increases the length of the sequence of the first differences by two, whereby the length of the palindrome increases by 0, 1 or 2.
LINKS
V.J. Pohjola, Table of n, a(n) for n = 1..3001
V.J. Pohjola, Line plot for n=1...20
V.J. Pohjola, Line plot for n=1...200
V.J. Pohjola, Line plot for n=1...3000
EXAMPLE
For n = 1, the roots are -18, 1; the first differences are 19; the longest palindrome is 19; so a(n) = 1.
For n = 2, the roots are -21, -18, 1, 4; the first differences are 3, 19, 3; the longest palindrome is 3, 19, 3; so a(n) = 3.
For n = 8, the roots are -87, -84, -65, -62, -43, -40, -21, -18, 1, 4, 23, 26, 45, 48, 67, 70; the first differences are 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3; the longest palindrome is 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3; so a(n) = 15.
For n = 9, the roots are -90, -87, -84, -65, -62, -43, -40, -21, -18, 1, 4, 23, 26, 45, 48, 67, 70, 89; first differences are 16, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19; the longest palindrome is 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3; so a(n) = 15.
MATHEMATICA
rootsA = {}; Do[
If[Floor[Tan[i]] == 1, AppendTo[rootsA, i]], {i, -10^5, 10^5}]
lenN = Length[Select[rootsA, # < 0 &]]
r = 200; roots = rootsA[[lenN - r ;; lenN + r + 1]]
diff = Differences[roots]
center = (Length[diff] + 1)/2; kmax = (Length[diff] + 1)/2 -
1; pals = {}; lenpals = {}; lenpal = 1;
Do[diffk = diff[[center - k ;; center + k]];
lendiffk = Length[diffk]; w = 3;
lenpal = lenpal + 2; (Label[alku]; w = w - 1;
pmax = lendiffk - lenpal - (w - 1);
t = Table[diffk[[p ;; lenpal + w + p - 1]], {p, 1, pmax}];
s = Select[t, # == Reverse[#] &]; If[s != {}, Goto[end], Goto[alku]];
Label[end]); AppendTo[pals, First[s]];
AppendTo[lenpals, Length[Flatten[First[s]]]];
lenpal = Length[Flatten[First[s]]], {k, 0, kmax}]
lenpals (*a[n]=lenpals[[n]]*)
CROSSREFS
KEYWORD
nonn
AUTHOR
V.J. Pohjola, Oct 20 2017
STATUS
approved