A206917 Sum of binary palindromes in the half-open interval [2^(n-1), 2^n).

0, 1, 3, 12, 24, 96, 192, 768, 1536, 6144, 12288, 49152, 98304, 393216, 786432, 3145728, 6291456, 25165824, 50331648, 201326592, 402653184, 1610612736, 3221225472, 12884901888, 25769803776, 103079215104, 206158430208, 824633720832, 1649267441664, 6597069766656

Offset

0,3

Comments

First differences of A206918.

Links

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (0,8).

Index to sequences related to palindromes.

Formula

a(n) = (3/8)*2^(n+floor((n+1)/2)).

From Colin Barker, May 31 2013: (Start)

a(n) = 8*a(n-2) for n > 3.

G.f.: -x*(4*x^2+3*x+1) / (8*x^2-1). (End)

From Amiram Eldar, Feb 20 2026: (Start)

Sum_{n>=1} 1/a(n) = 31/21.

Sum_{n>=1} (-1)^(n+1)/a(n) = 5/7. (End)

Example

a(0) = 0, since there is no binary palindrome 2^(-1) <= p < 2^0.
a(3) = 12, since 2^2 <= p < 2^3 for p = 5 and p = 7.

Mathematica

LinearRecurrence[{0, 8}, {0, 1, 3, 12}, 30] (* Amiram Eldar, Feb 20 2026 *)

Crossrefs

Cf. A006995, A016116, A052995, A206918.

Sequence in context: A028725 A047166 A076506 * A320334 A009776 A331241

Adjacent sequences: A206914 A206915 A206916 * A206918 A206919 A206920

Keyword

nonn,base,easy

Author

Hieronymus Fischer, Feb 18 2012

Extensions

More terms from Colin Barker, May 31 2013

Status

approved