This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(n) = rewrite_0to0_x1to1(fibbinary(j)) (where fibbinary(j) = A003714[ n ]) rewrite_0to0_x1to1 := proc(n) option remember; if(0 = n) then RETURN(n); else RETURN((2 * rewrite_0to0_x1to1(floor(n/(2^(1+(n mod 2)))))) + (n mod 2)); fi; end; fastfib := n -> round((((sqrt(5)+1)/2)^n)/sqrt(5)); fibinv_appr := n -> floor(log[ (sqrt(5)+1)/2 ](sqrt(5)*n)); fibinv := n -> (fibinv_appr(n) + floor(n/fastfib(1+fibinv_appr(n)))); fibbinary := proc(n) option remember; if(n <= 2) then RETURN(n); else RETURN((2^(fibinv(n)-2))+fibbinary_seq(n-fastfib(fibinv(n)))); fi; end; # second Maple program: b:= proc(n) is(n=0) end: a:= proc(n) option remember; local h; h:= iquo(a(n-1), 2)+1; while b(h) do h:= h*2 od; b(h):=true; h end: a(0):=0: seq(a(n), n=0..100); # Alois P. Heinz, Sep 22 2014
b[n_] := n==0; a[n_] := a[n] = Module[{h}, h = Quotient[a[n-1], 2] + 1; While[b[h], h = h*2]; b[h] = True; h]; a[0]=0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649 A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); } A007814(n) = valuation(n,2); A000265(n) = (n/2^valuation(n, 2)); A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n)))); A048679(n) = if(!n,n,A106151(2*A003714(n))); \\ Antti Karttunen, May 13 2018, after Reinhard Zumkeller's May 09 2005 formula.
from itertools import count, islice def A048679_gen(): # generator of terms return map(lambda n: int(bin(n)[2:].replace('01','1'),2),filter(lambda n:not (n<<1)&n,count(0))) A048679_list = list(islice(A048679_gen(),20)) # Chai Wah Wu, Mar 18 2024
def A048679(n): tlist, s = [1,2], 0 while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2]) for d in tlist[::-1]: if d <= n: s += 1 n -= d else: s <<= 1 return s # Chai Wah Wu, Apr 24 2025
rewrite_0to1_1to10_n_i_times := proc(n,i) local z,j; z := n; j := i; while(j > 0) do z := rewrite_0to1_1to10(z); j := j - 1; od; RETURN(z); end; rewrite_0to1_1to10 := proc(n) option remember; if(n < 2) then RETURN(n + 1); else RETURN(((2^(1+(n mod 2))) * rewrite_0to1_1to10(floor(n/2))) + (n mod 2) + 1); fi; end;
a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n-1]*2^Fibonacci[n-1] + a[n-2]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jul 27 2011 *)
def A005203(n): s = '0' for i in range(n): s = s.replace('0','a').replace('1','10').replace('a','1') return int(s,2) # Chai Wah Wu, Apr 24 2025
First levels of the tree: ...................1 ...................2 ...........3.................5 .......4.......7........8........13 .....6..10...11..18....12..20...21..34
a = {1, 2}; row = {a[[-1]]}; r = GoldenRatio; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)
11=1011 in binary, thus is rewritten as 100101 = 37 in decimal.
a048678 0 = 0
a048678 x = 2 * (b + 1) * a048678 x' + b
where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 31 2015
rewrite_0to0_1to01 := proc(n) option remember; if(n < 2) then RETURN(n); else RETURN(((2^(1+(n mod 2))) * rewrite_0to0_1to01(floor(n/2))) + (n mod 2)); fi; end;
f[n_] := FromDigits[ Flatten[IntegerDigits[n, 2] /. {1 -> {0, 1}}], 2]; Table[f@n, {n, 0, 60}] (* Robert G. Wilson v, Dec 11 2009 *)
a(n)=if(n<1,0,(3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2)
a(n) = if(n == 0, 0, my(A = -2); sum(i = 0, logint(n, 2), A++; if(bittest(n, i), 1 << (A++)))) \\ Mikhail Kurkov, Mar 14 2024
def a(n):
return 0 if n==0 else (3 - (-1)**n)*a(n//2) + (1 - (-1)**n)//2
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017
def A048678(n): return int(bin(n)[2:].replace('1','01'),2) # Chai Wah Wu, Mar 18 2024
88 = 55 + 21 + 8 + 3 + 1. 122 = 89 + 21 + 8 + 3 + 1. 135 = 89 + 34 + 8 + 3 + 1. 140 = 89 + 34 + 13 + 3 + 1. 142 = 89 + 34 + 13 + 5 + 1. 81 is not in the sequence because, although it is the sum of five Fibonacci numbers (81 = 5 + 8 + 13 + 21 + 34), its Zeckendorf representation only has three terms: 81 = 55 + 21 + 5.
a179245 n = a179245_list !! (n-1) a179245_list = filter ((== 5) . a007895) [1..] -- Reinhard Zumkeller, Mar 10 2013
with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0: m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: Q := {}: for i from fibonacci(11)-1 to 400 do if B(i) = 5 then Q := `union`(Q, {i}) else end if end do: Q;
zeck = DigitCount[Select[Range[3000], BitAnd[#, 2*#] == 0 &], 2, 1]; Position[zeck, 5] // Flatten (* Jean-François Alcover, Jan 30 2018 *)
A048680(n)=my(k=1,s);while(n,if(n%2,s+=fibonacci(k++)); k++; n>>=1); s [A048680(n)|n<-[1..100],hammingweight(n)==5] \\ Charles R Greathouse IV, Nov 17 2013
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