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A308092 The sum of the first n terms of the sequence is the concatenation of the first n bits of the sequence read as binary, with a(1) = 1. 4

%I #34 Apr 10 2021 16:34:05

%S 1,2,3,7,14,28,56,112,224,448,896,1791,3583,7166,14332,28663,57326,

%T 114653,229306,458612,917223,1834446,3668892,7337785,14675570,

%U 29351140,58702279,117404558,234809116,469618232,939236465,1878472930,3756945860,7513891719

%N The sum of the first n terms of the sequence is the concatenation of the first n bits of the sequence read as binary, with a(1) = 1.

%C In binary, the sequence begins 1, 10, 11, 111, 1110, 11100, 111000, 1110000, 11100000, 111000000, 1110000000, 11011111111, 110111111111, 1101111111110, 11011111111100, ...

%C Conjecture: The number of 1's in the binary representation of each term is weakly increasing, i.e., A000120(a(n)) >= A000120(a(n-1)).

%C Proved by _Matthew Scroggs_; see link. - _Peter Kagey_, Jun 19 2019

%H Peter Kagey, <a href="/A308092/b308092.txt">Table of n, a(n) for n = 1..1000</a>

%H Matthew Scroggs, <a href="http://www.mscroggs.co.uk/blog/65">Number of 1s in A308092</a>

%F a(n) = c(n) - c(n-1) for n > 2, where c(n) is the concatenation of the first n bits of the sequence.

%e For n=5, 1 + 2 + 3 + 7 + 14 = 1_2 + 10_2 + 11_2 + 111_2 + 1110_2 = 11011_2, the first five bits of the sequence.

%t a[1]=1;a[2]=2;a[n_]:=a[n]=FromDigits[Flatten[IntegerDigits[#,2]&/@Table[a[k],{k,n-1}]][[;;n]],2]-Total@Table[a[m],{m,n-1}]

%t Table[a[l],{l,40}] (* _Giorgos Kalogeropoulos_, Mar 30 2021 *)

%o (Ruby)

%o def first_bits(n, seq); seq.map { |i| i.to_s(2) }.join[0...n].to_i(2) end

%o def next_term(n, seq); first_bits(n,seq) - first_bits(n-1,seq) end

%o def a308092_list(n)

%o (3..n).reduce([1,2]) { |accum, i| accum << next_term(i, accum) }

%o end

%o (Python)

%o def aupton(terms):

%o alst, bstr = [1, 2], "110"

%o for n in range(3, terms+1):

%o an = int(bstr[:n], 2) - int(bstr[:n-1], 2)

%o alst, bstr = alst + [an], bstr + bin(an)[2:]

%o return alst

%o print(aupton(34)) # _Michael S. Branicky_, Mar 30 2021

%Y Cf. A000120, A300000 (decimal analog).

%K nonn,base

%O 1,2

%A _Peter Kagey_, May 12 2019

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)