cp's OEIS Frontend

A309616 a(n) is the number of ways to represent 2*n in the decibinary system.

%I A309616 #38 Mar 08 2024 23:11:24
%S A309616 1,2,4,6,10,13,18,22,30,36,45,52,64,72,84,93,110,122,140,154,177,192,
%T A309616 214,230,258,277,304,324,356,376,405,426,464,490,528,557,604,634,678,
%U A309616 710,765,802,854,892,952,989,1042,1080,1146,1190,1253,1300,1374,1420,1486,1533,1612,1664
%N A309616 a(n) is the number of ways to represent 2*n in the decibinary system.
%C A309616 It appears that a(n) is the number of decibinary numbers that can be constructed to represent the decimal numbers 2n-2 and 2n-1. To make this more clear let's consider n = 5: a(5) = 10 means that there are 10 decibinary numbers that represent the decimal numbers 2*5 - 2 = 8 and 2*5 - 1 = 9.
%C A309616 Furthermore, a(n) is the number of k such that A028897(k)=2*n.
%H A309616 HackerRank, <a href="https://www.hackerrank.com/challenges/decibinary-numbers/problem">Decibinary Numbers</a>.
%F A309616 a(1) = 1. a(n) = a(n-1) + a(ceiling(n/2)) if 1 < n <= 5.
%F A309616 Conjecture: a(n) = a(n-1) + a(ceiling(n/2)) - a(ceiling((n-5)/2)) if n > 5.
%F A309616 I think this sequence is closely related to the 10th binary partition function. The only difference is that every second number is omitted. At the moment, the 10th binary partition function is not in the OEIS. However, my experiments strongly suggest that the 10th binary partition function would indeed look like 1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 13, 13, ...
%e A309616 a(1) = 1.
%e A309616 a(2) = a(2-1) + a(ceiling(2/2)) = a(1) + a(1) = 1 + 1 = 2.
%e A309616 a(3) = a(3-1) + a(ceiling(3/2)) = a(2) + a(2) = 2 + 2 = 4.
%e A309616 a(4) = a(4-1) + a(ceiling(4/2)) = a(3) + a(2) = 4 + 2 = 6.
%e A309616 a(5) = a(5-1) + a(ceiling(5/2)) = a(4) + a(3) = 6 + 4 = 10.
%e A309616 a(6) = a(6-1) + a(ceiling(6/2)) - a(ceiling((6-5)/2)) = a(5) + a(3) - a(1) = 10 + 4 - 1 = 13.
%e A309616 a(7) = a(7-1) + a(ceiling(7/2)) - a(ceiling((7-5)/2)) = a(6) + a(4) - a(1) = 13 + 6 - 1 = 18.
%e A309616 a(8) = a(8-1) + a(ceiling(8/2)) - a(ceiling((8-5)/2)) = a(7) + a(4) - a(2) = 18 + 6 - 2 = 22.
%e A309616 a(9) = a(9-1) + a(ceiling(9/2)) - a(ceiling((9-5)/2)) = a(8) + a(5) - a(2) = 22 + 10 - 2 = 30.
%e A309616 a(10) = a(10-1) + a(ceiling(10/2)) - a(ceiling((10-5)/2)) = a(9) + a(5) - a(3) = 30 + 10 - 4 = 36.
%t A309616 Nest[Append[#1, #1[[-1]] + #1[[Ceiling[#2/2] ]] - If[#2 > 5, #1[[Ceiling[(#2 - 5)/2] ]], 0 ]] & @@ {#, Length@ # + 1} &, {1}, 57] (* _Michael De Vlieger_, Sep 29 2019 *)
%o A309616 (C++) int a(int n) {
%o A309616     std::vector<int> seq;
%o A309616     int a = 1;
%o A309616     seq.push_back(a);
%o A309616     for (int i = 1; i < n; i++) {
%o A309616         a += seq.at(i / 2);
%o A309616         a -= (i >= 5) ? seq.at((i - 5) / 2) : 0;
%o A309616         seq.push_back(a);
%o A309616     }
%o A309616     return seq.back();
%o A309616 }
%Y A309616 Cf. A007728: superseeker found that the deltas of the sequence a(n+1) - a(n) match transformations of the original query.
%Y A309616 Cf. A028897.
%K A309616 nonn,base
%O A309616 0,2
%A A309616 _Jonas Hollm_, Aug 10 2019
%E A309616 Name corrected by _Rémy Sigrist_, Oct 15 2019