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A381579 The Chung-Graham representation of n: representation of n in base of even-indexed Fibonacci numbers.

%I A381579 #9 Feb 28 2025 12:08:01
%S A381579 0,1,2,10,11,12,20,21,100,101,102,110,111,112,120,121,200,201,202,210,
%T A381579 211,1000,1001,1002,1010,1011,1012,1020,1021,1100,1101,1102,1110,1111,
%U A381579 1112,1120,1121,1200,1201,1202,1210,1211,2000,2001,2002,2010,2011,2012,2020,2021
%N A381579 The Chung-Graham representation of n: representation of n in base of even-indexed Fibonacci numbers.
%C A381579 Chung and Graham (1981, 1984) proved that every nonnegative integer n can be uniquely represented as a sum n = Sum_{i>=1} d_i * Fibonacci(2*i), where each d_i is in {0, 1, 2}, and if d_i = d_j = 2 with i < j, then for some k, i < k < j, we have d_k = 0.
%H A381579 Amiram Eldar, <a href="/A381579/b381579.txt">Table of n, a(n) for n = 0..10000</a>
%H A381579 Rob Burns, <a href="https://arxiv.org/abs/2502.18870">Chung-Graham and Zeckendorf representations</a>, arXiv:2502.18870 [math.NT], 2025.
%H A381579 Hung Viet Chu, Aney Manish Kanji, and Zachary Louis Vasseur, <a href="https://arxiv.org/abs/2501.03231">Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers</a>, arXiv:2501.03231 [math.GM], 2024.
%H A381579 F. R. K. Chung and R. L. Graham, <a href="https://doi.org/10.1073/pnas.78.7.4001">On irregularities of distribution of real sequences</a>, Proc. Nat. Acad. Sci. USA, Vol. 78, No. 7 (1981), p. 4001.
%H A381579 F. R. K. Chung and R. L. Graham, <a href="https://doi.org/10.1016/B978-0-444-86893-0.50016-4">On irregularities of distribution</a>, Finite and Infinite Sets, Colloquia Mathematica Societatis János Bolyai, Vol. 37, North-Holland, 1984, pp. 181-222; <a href="https://mathweb.ucsd.edu/~ronspubs/81_12_irregularities.pdf">alternative link</a>.
%e A381579 a(3) = 10 since 3 = 1 * 3 + 0 * 1 = 3 * Fibonacci(2*2) + 0 * Fibonacci(2*1).
%e A381579 a(10) = 102 since 10 = 1 * 8 + 0 * 3 + 2 * 1 = 1 * Fibonacci(2*3) + 0 * Fibonacci(2*2) + 2 * Fibonacci(2*1).
%t A381579 f[n_] := f[n] = Fibonacci[2*n]; a[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2*10^(k-1); m -= 2*f[k], s += 10^(k-1); m -= f[k]]]; s]; Array[a, 50, 0]
%o A381579 (PARI) m = 20; fvec = vector(m, i, fibonacci(2*i)); f(n) = if(n <= m, fvec[n], fibonacci(2*n));
%o A381579 a(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2*10^(k-1); m -= 2*f(k), s += 10^(k-1); m -= f(k))); s;}
%Y A381579 Cf. A000045, A001906, A291711 (sum of digits).
%Y A381579 Similar sequences: A014417, A104326.
%K A381579 nonn,easy,base
%O A381579 0,3
%A A381579 _Amiram Eldar_, Feb 28 2025