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A036286 Periodic vertical binary vectors of Fibonacci numbers, topmost bits being most significant.

%I A036286 #9 Jul 08 2025 21:03:26
%S A036286 3,6,90,202474,802914372650,124876754670311211270396330,
%T A036286 2261740218128437766312179308277308483058208661638110890,
%U A036286 7527129205899945471753233641719262207829849606092782843679109711117799287001392666047916596823438974998183293610
%N A036286 Periodic vertical binary vectors of Fibonacci numbers, topmost bits being most significant.
%H A036286 A. Karttunen, <a href="/A036286/b036286.txt">Table of n, a(n) for n = 0..10</a>
%H A036286 A. Karttunen, <a href="/A036284/a036284.c.txt">C program for computing this sequence</a>
%F A036286 a(n) = Sum_{k=0..A007283(n)-1} ([A000045((A007283(n)-1)-k)/(2^n)] mod 2) * 2^k, where [] stands for floor function.
%e A036286 When Fibonacci numbers are written in binary (see A004685), under each other as:
%e A036286 0000000 (0)
%e A036286 0000001 (1)
%e A036286 0000001 (1)
%e A036286 0000010 (2)
%e A036286 0000011 (3)
%e A036286 0000101 (5)
%e A036286 0001000 (8)
%e A036286 0001101 (13)
%e A036286 0010101 (21)
%e A036286 0100010 (34)
%e A036286 0110111 (55)
%e A036286 1011001 (89)
%e A036286 it can be seen that the bits in the n-th column from right repeat after the period of A007283(n): 3, 6, 12, 24, ... (see also A001175). This sequence is formed from those bits: 011, binary for 3, thus a(0) = 3. 000110, binary for 6, thus a(1) = 6, 000001011010, binary for 90, thus a(2) = 90. Cf. A036284.
%Y A036286 See comments at A036284. a(n)/A036287(n) can be interpreted as fractions.
%K A036286 nonn,base
%O A036286 0,1
%A A036286 _Antti Karttunen_, Nov 01 1998
%E A036286 Entry revised Dec 29 2007