A266577 Square array read by descending antidiagonals: T(n,k) = ((2^(n+1) + 1)^(k-1) + 1)/2.

1, 3, 1, 13, 5, 1, 63, 41, 9, 1, 313, 365, 145, 17, 1, 1563, 3281, 2457, 545, 33, 1, 7813, 29525, 41761, 17969, 2113, 65, 1, 39063, 265721, 709929, 592961, 137313, 8321, 129, 1, 195313, 2391485, 12068785, 19567697, 8925313, 1073345, 33025, 257, 1, 976563, 21523361, 205169337, 645733985, 580145313, 138461441, 8487297, 131585, 513, 1

Offset

1,2

Comments

The matrix M in the definition of A292625 is given by this sequence, also, for each natural number m and each natural number c, ((2^(m+1)+1)^c-1)*(the product of any (m+1) not necessarily distinct terms of the m-th row) is palindromic in base (2^(m+1)+1), see the MathOverflow link. - Ahmad J. Masad, Apr 19 2023

Links

Amiram Eldar, Table of n, a(n) for n = 1..1275

Ahmad J. Masad, Conjecture on palindromic numbers, MathOverflow, Apr 2018.

Example

The array begins:
  1   3  13  63 313
  1   5  41 365
  1   9 145
  1  17
  1
Example of the result concerning palindromic numbers:
Take m=2, c=4, 2^(m+1) + 1 = 2^3 + 1 = 9, we choose 3 not necessarily distinct terms from the second row. Let them be 41, 365, 365; then we get 41*365*365*(9^4 - 1) = 35832196000 = 112435534211_9, which is a palindromic number in base 9.

Mathematica

T[n_, k_] := ((2^(n + 1) + 1)^(k - 1) + 1)/2; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Sep 14 2022 *)

Prog

(PARI) tabl(n) = matrix(n, n, i, j, ((2^(i+1)+1)^(j-1)+1)/2); \\ Michel Marcus, Jan 02 2016

Crossrefs

Cf. A034478.

Sequence in context: A322384 A360088 A113139 * A143411 A096773 A118384

Adjacent sequences: A266574 A266575 A266576 * A266578 A266579 A266580

Keyword

nonn,tabl

Author

Ahmad J. Masad, Jan 01 2016

Extensions

a(31) corrected by Georg Fischer, Nov 07 2021

Status

approved