OFFSET
0,3
COMMENTS
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..10000
Rob Burns, Chung-Graham and Zeckendorf representations, arXiv:2502.18870 [math.NT], 2025.
Hung Viet Chu, Aney Manish Kanji, and Zachary Louis Vasseur, Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers, arXiv:2501.03231 [math.GM], 2024.
F. R. K. Chung and R. L. Graham, On irregularities of distribution of real sequences, Proc. Nat. Acad. Sci. USA, Vol. 78, No. 7 (1981), p. 4001.
F. R. K. Chung and R. L. Graham, On irregularities of distribution, Finite and Infinite Sets, Colloquia Mathematica Societatis János Bolyai, Vol. 37, North-Holland, 1984, pp. 181-222; alternative link.
EXAMPLE
a(3) = 10 since 3 = 1 * 3 + 0 * 1 = 3 * Fibonacci(2*2) + 0 * Fibonacci(2*1).
a(10) = 102 since 10 = 1 * 8 + 0 * 3 + 2 * 1 = 1 * Fibonacci(2*3) + 0 * Fibonacci(2*2) + 2 * Fibonacci(2*1).
MATHEMATICA
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]
PROG
(PARI) m = 20; fvec = vector(m, i, fibonacci(2*i)); f(n) = if(n <= m, fvec[n], fibonacci(2*n));
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; }
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Amiram Eldar, Feb 28 2025
STATUS
approved