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A048702 Binary palindromes of even length divided by 3. 7
0, 1, 3, 5, 11, 15, 17, 21, 43, 51, 55, 63, 65, 73, 77, 85, 171, 187, 195, 211, 215, 231, 239, 255, 257, 273, 281, 297, 301, 317, 325, 341, 683, 715, 731, 763, 771, 803, 819, 851, 855, 887, 903, 935, 943, 975, 991 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let the length of A048701(n) in binary be 2k. Since it is a palindrome of even length, its digits come in pairs which are equal: one in the left half and the other in the right half. Thus, A048701(n) is a sum of numbers of the form d * 2^m * (2^(2k-2m-1) + 1). The number 2^(2k-2m-1) = 2 * 4^(k-m-1) is congruent to 2 (mod 3), so 2^(2k-2m-1) + 1 is divisible by 3. This means A048701(n) is divisible by 3, and therefore a(n) is an integer. - Michael B. Porter, Jun 18 2019

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = A048701(n)/3.

a(n) = 2^floor(log_2(n)) * Sum_{i=1..n} 1/(2^v_2(i)), for n >= 1, where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i (conjectured).

a(n) = n*2^floor(log_2(n)) - Sum_{i=1..floor(log_2(n))} 2^(floor(log_2(n)) - i)*floor(n/(2^i)) (conjectured).

MAPLE

# Two unproved formulas which are not based upon first generating a palindrome and then dividing by 3, recursive and more direct:

# Here d is 2^(the distance between the most and least significant 1-bit of n):

bper3_rec := proc(n) option remember; local d; if(0 = n) then RETURN(0); fi; d := 2^([ log2(n) ]-A007814[ n ]);

if(1 = d) then RETURN((2*bper3_rec(n-1))+d); else RETURN(bper3_rec(n-1)+d); fi; end;

# or more directly (after K. Atanassov's formula for partial sums of A007814):

bper3_direct := proc(n) local l, j; l := [ log2(n) ]; RETURN((2/3*((2^(2*l))-1))+1+ sum('(2^(l-j)*floor((n-(2^l)+2^j)/(2^(j+1))))', 'j'=0..l)); end;

# Can anybody find an even simpler closed form? See A005187 for inspiration.

MATHEMATICA

Join[{0}, Reap[For[k = 1, k < 3000, k += 2, bb = IntegerDigits[k, 2]; If[bb == Reverse[bb], If[EvenQ[Length[bb]], Sow[k/3]]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)

CROSSREFS

Cf. A048701, A048704 (base 4 palindromes of even length divided by 5), A044051 (binary palindromes plus one divided by 2: A006995(n)+1)/2), A000975.

Sequence in context: A086284 A136500 A024897 * A003546 A082421 A153798

Adjacent sequences:  A048699 A048700 A048701 * A048703 A048704 A048705

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Mar 07 1999

STATUS

approved

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Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)