A368290 a(n) is the length of the longest palindromic subsequence at symmetrically-spaced indices ending at a(n-1); a(1)=1.

1, 1, 2, 1, 3, 1, 3, 3, 2, 5, 1, 7, 1, 6, 1, 6, 3, 5, 5, 2, 4, 1, 7, 4, 2, 9, 1, 11, 1, 6, 5, 4, 5, 4, 3, 7, 3, 9, 5, 6, 7, 7, 7, 5, 9, 6, 5, 7, 5, 5, 8, 1, 11, 6, 7, 7, 9, 10, 1, 9, 9, 6, 9, 6, 11, 7, 13, 1, 12, 1, 14, 1, 16, 1, 17, 1, 19, 1, 14, 7, 9, 7, 11

Offset

1,3

Comments

A set of indices is symmetric if, listed in increasing or decreasing order, its first differences are a palindromic sequence.

A new value is always followed by 1.

An alternate definition: a(n) is the largest number of coincidences between the subsequence enclosed by m..n-1 and its reverse, where a(n-1)=a(m), maximized over m.

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Neal Gersh Tolunsky, Ordinal transform of first 30000 terms.

Neal Gersh Tolunsky, Graph of first 100000 terms.

Example

a(10)=5 because we find the following length-5 palindromic subsequence at symmetric indices ending at i=a(n-1)=a(9)=2:
  S: 1,1,2,1,3,1,3,3,2
  P:     2,  3,1,3,  2
a(14)=6 because of the following length-6 palindromic subsequence:
  S: 1,1,2,1,3,1,3,3,2,5,1,7,1
  P:   1,  1,    3,3,    1,  1

Prog

(PARI) { for (n = 1, #a = vector(83, n, 1), for (k = 1, n-1, if (a[k] == a[n-1], a[n] = max(a[n], sum (i = k, n-1, a[i] == a[n-1+k-i]); ); ); ); print1 (a[n]", "); ); } \\ Rémy Sigrist, Dec 20 2023

Crossrefs

Cf. A308659, A362881.

Sequence in context: A308659 A145652 A373745 * A111248 A100714 A339364

Adjacent sequences: A368287 A368288 A368289 * A368291 A368292 A368293

Keyword

nonn

Author

Neal Gersh Tolunsky, Dec 19 2023

Extensions

More terms from Rémy Sigrist, Dec 20 2023

Status

approved