A265026 First differences of A048701.

3, 6, 6, 18, 12, 6, 12, 66, 24, 12, 24, 6, 24, 12, 24, 258, 48, 24, 48, 12, 48, 24, 48, 6, 48, 24, 48, 12, 48, 24, 48, 1026, 96, 48, 96, 24, 96, 48, 96, 12, 96, 48, 96, 24, 96, 48, 96, 6, 96, 48, 96, 24, 96, 48, 96, 12, 96, 48, 96, 24, 96, 48

Offset

1,1

Comments

Comment from Altug Alkan, Dec 03 2015: (Start) Except for 3, all terms are divisible by 6 (cf. A048702, A265027).

Proof: Binary palindromes of even length (A048701) are odd for n > 0. So A048701(n) - A048701(n-1) is an even number for n > 1. Because the length is even and palindromic numbers are symmetric, for any digit “1” that is related with 2^n in its expansion which n is even, there are another digit “1” that is related with 2^m in its expansion which m is odd. 2^n+2^m is always divisible by 3 if n is even and m is odd. Therefore A048701(n) is divisible by 3, so A048701(n) - A048701(n-1) is divisible by 3 for n > 0. In conclusion, A048701(n) - A048701(n-1) is always divisible by 6 for n > 1. (End)

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..32767

Formula

a(n) = A048701(n) - A048701(n-1). - Altug Alkan, Dec 03 2015

Prog

(PARI) 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) - a048701(n-1))) \\ Altug Alkan, Dec 03 2015

Crossrefs

Cf. A048701, A048702, A265027.

Sequence in context: A036252 A103463 A370154 * A304335 A223048 A319446

Adjacent sequences: A265023 A265024 A265025 * A265027 A265028 A265029

Keyword

nonn

Author

N. J. A. Sloane, Nov 30 2015

Status

approved