A265026 -id:A265026 - cp's OEIS frontend

A048701 List of binary palindromes of even length (written in base 10).

0, 3, 9, 15, 33, 45, 51, 63, 129, 153, 165, 189, 195, 219, 231, 255, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843, 891, 903, 951, 975, 1023, 2049, 2145, 2193, 2289, 2313, 2409, 2457, 2553, 2565, 2661, 2709, 2805, 2829, 2925, 2973, 3069, 3075, 3171, 3219, 3315

Offset: 0

Antti Karttunen, Mar 07 1999

A178225(a(n)) = 1. - Reinhard Zumkeller, Oct 21 2011

a(n) is divisible by 3 and it is always an odd number for n > 0. Therefore a(n) is in A016945 for n > 0. - Altug Alkan, Dec 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

See also A048702 = this sequence divided by 3, A048700 = binary palindromes of odd length, A006995 = all binary palindromes, A048703 = quaternary (base 4) palindromes of even length.
For first differences see A265026, A265027.
Cf. A030308, A007088, A178225.

a048701 n = foldr (\d v -> 2 * v + d) 0 (reverse bs ++ bs) where
   bs = a030308_row (n)
-- Reinhard Zumkeller, Feb 19 2003, Oct 21 2011

Prepend[Select[Range@ 3315, Reverse@ # == # && EvenQ@ Length@ # &@ IntegerDigits[#, 2] &], 0] (* Michael De Vlieger, Dec 04 2015 *)

a048701(n) = my(f); f = length(binary(n)) - 1; 2^(f+1)*n + sum(i=0, f, bittest(n, i) * 2^(f-i)); \\ Altug Alkan, Dec 03 2015

def A048701(n):
    s = bin(n)[2:]
    return int(s+s[::-1],2) # Chai Wah Wu, Feb 26 2021

a(n) = (2^(floor_log_2(n)+1))*n + Sum_{i=0..floor_log_2(n)} '(bit_i(n, i)*(2^(floor_log_2(n)-i)))'.

Offset corrected by Reinhard Zumkeller, Oct 21 2011

Offset changed back to 0 by Andrey Zabolotskiy, Dec 26 2022

A265027 First differences of A048701 divided by 6.

Original entry on oeis.org

1, 1, 3, 2, 1, 2, 11, 4, 2, 4, 1, 4, 2, 4, 43, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 171, 16, 8, 16, 4, 16, 8, 16, 2, 16, 8, 16, 4, 16, 8, 16, 1, 16, 8, 16, 4, 16, 8, 16, 2, 16, 8, 16, 4, 16, 8, 16, 683, 32, 16, 32, 8, 32, 16, 32, 4, 32, 16, 32, 8, 32, 16, 32, 2, 32, 16, 32, 8, 32, 16, 32, 4, 32, 16, 32, 8

Offset: 2

N. J. A. Sloane, Nov 30 2015

nonn

Indices n such that a(n) = 1 are equal to row sums of Lucas triangle. In other words, a(A042950(n)) = 1. Additionally, a(A070875(n)) = 2 and a(A123760(n)) = 4. - Altug Alkan, Dec 04 2015

N. J. A. Sloane, Table of n, a(n) for n = 2..32767

Cf. A042950, A048701, A070875, A123760, A265026.

Differences@ Select[Range@ 12000, Reverse@ # == # && EvenQ@ Length@ # &@ IntegerDigits[#, 2] &]/6 (* Michael De Vlieger, Dec 04 2015 *)

a048701(n) = my(f); f = length(binary(n)) - 1; 2^(f+1)*n + sum(i=0, f, bittest(n, i) * 2^(f-i));
vector(100, n, (a048701(n+1) - a048701(n)) / 6) \\ Altug Alkan, Dec 03 2015

a(n) = A265026(n) / 6, for n > 1. - Altug Alkan, Dec 03 2015

cp's OEIS Frontend

A048701 List of binary palindromes of even length (written in base 10).

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