A293703 -id:A293703 - cp's OEIS frontend

A293701 a(n) is the length of the longest palindromic subsequence in the first n terms of A293700.

1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 47, 49

Offset: 1

V.J. Pohjola, Oct 16 2017

If a(n + 1) > a(n) for some n, then A293700(n + 1) is in the longest palindrome. So to find a(n + 1) it suffices to check if A293700 is in the palindrome, which must be at least of length a(n). - David A. Corneth, Nov 25 2017

At points where a(n) = n, the whole sequence is a palindrome. For example at n=9105, the length of the longest palindrome a(9105) is 9105 (see A294923).

			For n = 1, roots = 1, 4; first differences = 3; longest palindrome = 3; a(n) = 1.
For n = 2, roots = 1, 4, 23; first differences = 3, 19; longest palindrome = 3; a(n) = 1.
For n = 3, roots = 1, 4, 23, 26; first differences = 3, 19, 3; longest palindrome = 3, 19, 3; a(n) = 3.
For n = 33, roots = 1, 4, 23, 26, 45, 48, 67, 70, 89, 92, 111, 114, 133, 136, 155, 158, 177, 180, 183, 199, 202, 205, 221, 224, 227, 243, 246, 249, 265, 268, 271, 290, 293, 312; first differences = 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 19; longest palindrome = 19, 3, 19, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 19, 3, 19; a(n) = 20.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed with the given Scheme-program from the b-file of A293700 provided by Robert Israel)
V.J. Pohjola, Line plot of n, a(n) for n=1...100
V.J. Pohjola, Line plot of n, a(n) for n=1...1400
V.J. Pohjola, Line plot of n, a(n) for n=1...10000

Cf. A293698, A293699, A293700, A293702, A293703, A293704, A293705, A293706, A293751.

rootsp = Flatten[Position[Table[Floor[Tan[i]], {i, 1, 10^4}], 1]];
lrootsp = Length[rootsp];
difp = Differences[rootsp];
ldp = Length[difp];
nmax = 200; palsp = {}; lenpalsp = {0};
Do[diffip = difp[[1 ;; n]]; lendiffip = Length[diffip];
  pmax = n - Last[lenpalsp];
  t = Table[difp[[p ;; n]], {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]]]], {n, 1, nmax}];
Drop[lenpalsp,1](*a(n)=Drop[lenpalsp,1][[n]]*)

firstsols(n) = {my(res = vector(n), i = 0, pi = [Pi, Pi], sols = [atan(1), atan(2)]); while(1, for(j = ceil(sols[1]), floor(sols[2]), i++; if(i>n, return(res)); res[i] = j); sols+=[Pi(), Pi()])} \\ A293698
diff(v) = vector(#v-1,i,v[i+1]-v[i]);
first(n) = {my(res = vector(n), m = 0, check = diff(firstsols(n+1))); for(i=1, n, for(j = 1, i - m, if(ispalindrome(check, j, i), m = i - j + 1; next(1))); res[i] = m); res}
ispalindrome(v, {llim = 1}, {ulim = #v}) = {for(i=0, (ulim - llim) \ 2, if(v[llim + i]!=v[ulim - i], return(0))); 1} \\ David A. Corneth, Nov 25 2017

;; This uses memoization-macro definec and assumes also that A293700 is available:
(definec (A293701 n) (if (= 1 n) n (let outloop ((k n)) (cond ((<= k (A293701 (- n 1))) (A293701 (- n 1))) (else (let inloop ((i n)) (let ((low-ind (+ 1 (- n k) (- n i)))) (cond ((< i low-ind) (max k (A293701 (- n 1)))) ((not (= (A293700 i) (A293700 low-ind))) (outloop (- k 1))) (else (inloop (- i 1)))))))))))
;; Antti Karttunen, Nov 25 2017

A293702 a(n) is the length of the longest palindromic subsequence in the first n terms of A293751.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 11, 11, 12, 14, 16, 18, 20, 22, 24, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 63, 63, 63, 63, 63, 63, 63

Offset: 1

V.J. Pohjola, Oct 16 2017

nonn

			For n = 1, Roots = 18, 21; First differences = 3; Longest palindrome = 3; a(n) = 1.
For n = 2, Roots = 18, 21, 40; First differences = 3, 19; Longest palindrome = 3; a(n) = 1.
For n = 3, Roots = 18, 21, 40, 43; First differences = 3, 19, 3; Longest palindrome = 3, 19, 3; a(n) = 3.
For n = 20, Roots = 18, 21, 40, 43, 62, 65, 84, 87, 90, 106, 109, 112, 128, 131,134, 150, 153, 156, 172, 175; First differences = 3, 19, 3, 19, 3, 19, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3; Longest palindrome = 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3, 16, 3, 3; a(n) = 14.

V.J. Pohjola, Table of n, a(n) for n = 1..10000
V.J. Pohjola, Line plot for n=1…30
V.J. Pohjola, Line plot for n=1...10000

Cf. A293698, A293751, A293700, A293701, A293705, A293704, A293699, A293703, A293706.

rootsn = Flatten[Position[Table[Floor[Tan[-i]], {i, 1, 10^4}], 1]];
difn = Differences[rootsn];
imax = 100; palsn = {}; lenpalsn = {0};
Do[diffin = difn[[1 ;; i]]; lendiffin = Length[diffin];
  pmax = i - Last[lenpalsn];
  t = Table[difn[[p ;; i]], {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]]]], {i, 1, imax}];
Drop[lenpalsn, 1] (* a(n)=Drop[lenpalsn, 1][[n]] *)

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

Original entry on oeis.org

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

V.J. Pohjola, Oct 21 2017

sign

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.

			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.

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

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

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]]*)

A293705 a(n) is the shift of the longest palindromic subsequence in the first n terms of A293699.

Original entry on oeis.org

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

Offset: 1

V.J. Pohjola, Oct 21 2017

sign

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.

			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.

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

Cf. A293698, A293751, A293700, A293702, A293704, A293699, A293701, A293706, A293703.

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]]*)

A293706 a(n) is the shift of the longest palindromic subsequence within the first differences of the concatenation of the first n negative and positive roots of floor(tan(k)) = 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

V.J. Pohjola, Oct 23 2017

sign

Shift is the measure of the position of a palindromic subsequence within the corresponding sequence of first differences, being defined as the number of terms omitted from the left end of the sequence of first differences minus those omitted from its right end. Thus, when shift is, say, 10, the position of the palindrome is 10 steps to the right from the center of the first differences.

a(n) remains at value 10 from n=18 to 1183 after which it drops stepwise linearly to -1544.

			For n = 1, roots=-18,1; differences = 19; longest palindrome = 19; a(n) = 0.
For n = 2, roots=-21, -18, 1, 4; differences = 3,19,3; longest palindrome = 3,19,3  a(2) = 0.
For n = 9, roots=-106, -90, -87, -84, -65, -62, -43, -40, -21, -18, 1, 4, 23, 26, 45, 48, 67, 70, 89, 92; differences = 16, 3, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3; longest palindrome = 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3, 19, 3; a(9) = 2 - 0 = 2.

V.J. Pohjola, Table of n, a(n) for n = 1..3001
V.J. Pohjola, Line plot for n=1..30
V.J. Pohjola, Line plot for n=1..3000

Cf. A293698, A293751, A293700, A293703, A293704, A293699, A293701, A293705, A293702.

rootsA = {}; Do[
If[Floor[Tan[i]] == 1, AppendTo[rootsA, i]], {i, -10^4, 10^4}]
lenN = Length[Select[rootsA, # < 0 &]];
r = 1000; roots = rootsA[[lenN - r ;; lenN + r + 1]];
diff = Differences[roots];
center = Length[roots]/2;
pals = {}; lenpals = {}; lenpal = 1; pos = {}; shift = {};
Do[diffn = diff[[center - (n - 1) ;; center + (n - 1)]];
lendiffn = Length[diffn]; w = 3;
lenpal = lenpal + 2; (Label[alku]; w = w - 1;
  pmax = lendiffn - lenpal - (w - 1);
  t = Table[diffn[[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]]]];
AppendTo[pos, Flatten[Position[t, First[s]]]]; pp = Last[Flatten[pos]];
qq = lendiffn - (pp - 1 + Last[lenpals]);
AppendTo[shift, pp - 1 - qq], {n, 1, center}]
shift

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A293701 a(n) is the length of the longest palindromic subsequence in the first n terms of A293700.

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A293702 a(n) is the length of the longest palindromic subsequence in the first n terms of A293751.

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

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A293705 a(n) is the shift of the longest palindromic subsequence in the first n terms of A293699.

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A293706 a(n) is the shift of the longest palindromic subsequence within the first differences of the concatenation of the first n negative and positive roots of floor(tan(k)) = 1.

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