A003231 -id:A003231 - cp's OEIS frontend

A006133 Related to representations as sums of Fibonacci numbers.

3, 21, 32, 79, 144, 155, 173, 202, 220, 231, 401, 524, 542, 553, 600, 922, 987, 998, 1016, 1045, 1063, 1074, 1121, 1139, 1150, 1168, 1186, 1197, 1215, 1244, 1367, 1385, 1396, 1443, 1508, 1519, 1537, 1566, 1584, 1595, 1765, 2608, 2731, 2749, 2760, 2807, 3129

Offset: 1

N. J. A. Sloane

nonn

Common value of A003231(x) and A003234(x) for x in A006132 (see Table 1 p. 357 in Carlitz link). - Michel Marcus, Feb 02 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

A001950(n) = floor(n*(sqrt(5)+3)/2); \\ b
A003231(n) = floor(n*(sqrt(5)+5)/2); \\ c
iss(n) = A003231(A001950(n)) == A001950(A003231(n)) - 1;
lista(nn) = {v003231 = vector(nn, i, floor(i*(sqrt(5)+5)/2)); v003234 = select(n->iss(n), vector(5*nn, i, i)); for (n=1, nn, if (v003231[n] == v003234[n], print1(v003231[n], ", ")););}  \\ Michel Marcus, Feb 02 2014

a(n) = A003231(A006132(n)) = A003234(A006132(n)). - Michel Marcus, Feb 02 2014

More terms from Michel Marcus, Feb 02 2014

A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

Original entry on oeis.org

3, 5, 7, 10, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 34, 36, 39, 41, 43, 45, 47, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 97, 99, 101, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 126, 128, 130, 133, 135, 137, 139, 141, 144

Offset: 1

Michel Dekking, May 06 2019

nonn

Let b = A214970 be the sequence of the integers k for which the base phi representation includes 1, and let c be the sequence of integers k for which the base phi representation includes phi.
Note that a, b and c form a complementary triple (since consecutive digits 11 do not occur in a base phi representation).
Conjecture (Moses 2012/Baruchel 2018): b is the generalized Beatty sequence b(n) = floor(n*phi) + 2*n + 1.
Conjecture (Kimberling 2012): c = A054770 = A000201(n) + 2*n - 1 = floor(n*phi) + 2*n - 1.
One can prove that the Moses/Baruchel conjecture and the Kimberling conjecture are equivalent.
Conjecture: (a(n)) is a union of two generalized Beatty sequences v and w, given by v(n) = floor(n*phi) + 2*n = A003231(n), and w(n) = 3*floor(n*phi) + n + 1 = A190509(n) + 1.
This conjecture is compatible with the Moses/Baruchel/Kimberling conjecture.
These three conjectures are proved in my paper 'Base phi representations and golden mean beta-expansions'. - Michel Dekking, Jun 26 2019

			3 = phi^2 + phi^{-2}, 5 = phi^3 + phi^{-1} + phi^{-4}.

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
George Bergman, A Number System with an Irrational Base, Mathematics Magazine, Vol. 31, No. 2 (Nov. - Dec., 1957), pp. 98-110.
M. Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.

Cf. A214970, A054770, A000201.

cp's OEIS Frontend

A006133 Related to representations as sums of Fibonacci numbers.

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