A261076 The infinite trunk of Zeckendorf (Fibonacci) beanstalk, with reversed subsections.

0, 1, 2, 4, 7, 5, 12, 9, 20, 17, 14, 33, 29, 27, 24, 22, 54, 50, 47, 45, 42, 40, 37, 35, 88, 83, 79, 76, 74, 70, 67, 63, 61, 58, 56, 143, 138, 134, 130, 126, 123, 121, 117, 113, 110, 108, 104, 101, 97, 95, 92, 90, 232, 226, 221, 217, 213, 209, 205, 201, 198, 193, 189, 185, 181, 178, 176, 172, 168, 165, 163, 159, 156, 152, 150, 147, 145

Offset

0,3

Comments

This can be viewed as an irregular table: after the initial zero on row 0, start each row n with k = F(n+2)-1 and subtract repeatedly the number of "1-fibits" (number of terms in Zeckendorf expansion of k) from k to get successive terms, until the number that has already been listed (which is always (F(n+1)-1) is encountered, which is not listed second time, but instead, the current row is finished and the next row starts with (F(n+3))-1, with the same process repeated. Here F(n) = the n-th Fibonacci number, A000045(n).

Links

Antti Karttunen, Table of n, a(n) for n = 0..11817; rows 0 .. 23 of the irregular table.

Indranil Ghosh, Python program to generate the sequence

Formula

For n <= 2, a(n) = n; for n >= 3, if A219641(a(n-1)) = F(k)-1 [i.e., one less than some Fibonacci number F(k)] then a(n) = F(k+2)-1, otherwise a(n) = A219641(a(n-1)).

As a composition:

a(n) = A219648(A261102(n)).

Example

As an irregular table, the sequence looks like:
0;
1;
2;
4;
7, 5;
12, 9;
20, 17, 14;
33, 29, 27, 24, 22;
54, 50, 47, 45, 42, 40, 37, 35;
...
After zero, each row n is A261091(n) elements long.

Prog

(Scheme, with memoization-macro definec)
(definec (A261076 n) (cond ((<= n 2) n) ((A219641 (A261076 (- n 1))) => (lambda (maybe_next) (if (= 1 (A007895 (+ 1 maybe_next))) (+ -1 (A000045 (+ 3 (A072649 (+ 1 maybe_next))))) maybe_next)))))

Crossrefs

Cf. A000045, A000071, A007895, A072649, A219641, A219648, A261091, A261102.

Cf. A218616 (analogous sequence for base-2).

Sequence in context: A182310 A244591 A299324 * A302991 A015791 A335818

Adjacent sequences: A261073 A261074 A261075 * A261077 A261078 A261079

Keyword

nonn,tabf

Author

Antti Karttunen, Aug 09 2015

Status

approved