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.
%I A348853 #45 Mar 22 2025 07:01:58
%S A348853 1,1,1,4,1,6,4,1,9,6,4,12,1,14,9,6,17,4,19,12,1,22,14,9,25,6,27,17,4,
%T A348853 30,19,12,33,1,35,22,14,38,9,40,25,6,43,27,17,46,4,48,30,19,51,12,53,
%U A348853 33,1,56,35,22,59,14,61,38,9,64,40,25,67,6,69,43,27,72
%N A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.
%C A348853 Terms are odd Zeckendorfs A003622 and the fixed points are where n is odd already so that a(n) = n iff n is in A003622.
%C A348853 A139764(n) is the least significant "10..00" part of n so Zeckendorf multiplication n = A101646(a(n), A139764(n)).
%C A348853 The equivalent delete least significant 0's in binary is A000265 so that conversion to Fibbinary (A003714) and back gives a(n) = A022290(A000265(A003714(n))).
%C A348853 a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00.
%C A348853 a(n) = 4 iff n is a Lucas number >= 4 (A000032) since they are Zeckendorf 10100..00 which reduces to 101.
%C A348853 In the Wythoff array A035513, a(n) is the term in column 0 of the row containing n, and hence the formula below using row number A019586 to select which of the odds (column 0) is a(n).
%H A348853 Kevin Ryde, <a href="/A348853/b348853.txt">Table of n, a(n) for n = 1..10000</a>
%H A348853 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>.
%F A348853 a(n) = n if A003849(n)=1, otherwise a(n) = a(A005206(n)) = a(A319433(n)).
%F A348853 a(n) = A003622(A019586(n) + 1).
%F A348853 Sum_{k=1..n} a(k) ~ n^2/(2*phi), where phi is the golden ratio (A001622). - _Amiram Eldar_, Feb 17 2024
%e A348853 n = 81 = Zeckendorf 101001000.
%e A348853 a(n) = 19 = Zeckendorf 101001.
%o A348853 (PARI) my(phi=quadgen(5)); a(n) = my(q,r); while([q,r]=divrem(n+2,phi); r<1, n=q-1); n;
%Y A348853 Cf. A189920 (Zeckendorf digits), A003622 (odds), A003849 (final digit), A005206, A319433 (shift down).
%Y A348853 Cf. A000045 (Fibonacci), A000032 (Lucas).
%Y A348853 Cf. A035513 (Wythoff array), A019586 (row number).
%Y A348853 Cf. A003714 (Fibbinary), A022290 (its inverse).
%Y A348853 In other bases: A000265 (binary), A004151 (decimal).
%Y A348853 Cf. A001622, A101646, A139764.
%K A348853 base,easy,nonn
%O A348853 1,4
%A A348853 _Kevin Ryde_, Nov 14 2021