A261076 -id:A261076 - cp's OEIS frontend

A219648 The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf representation of a(n).

0, 1, 2, 4, 5, 7, 9, 12, 14, 17, 20, 22, 24, 27, 29, 33, 35, 37, 40, 42, 45, 47, 50, 54, 56, 58, 61, 63, 67, 70, 74, 76, 79, 83, 88, 90, 92, 95, 97, 101, 104, 108, 110, 113, 117, 121, 123, 126, 130, 134, 138, 143, 145, 147, 150, 152, 156, 159, 163, 165, 168

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "Zeckendorf beanstalk" from its root (zero).
There are many finite sequences such as 0,1,2; 0,1,2,4,5; etc. (see A219649) and as the length increases, so (necessarily) does the similarity to this infinite sequence.
There can be only one infinite trunk in "Zeckendorf beanstalk" as all paths downwards from numbers >= A000045(i) must pass through A000045(i)-1 (i.e. A000071(i)). This provides also a well-defined method to compute the sequence, for example, via a partially reversed version A261076.
See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.

Antti Karttunen, Table of n, a(n) for n = 0..11817

Cf. A000045, A000071, A007895, A014417, A219641, A219649, A261076, A261102. For all n, A219642(a(n)) = n and A219643(n) <= a(n) <= A219645(n). Cf. also A261083 & A261084.

Other similarly constructed sequences: A179016, A219666, A255056.

(define (A219648 n) (A261076 (A261102 n)))

a(n) = A261076(A261102(n)).

A261091 a(n) = number of steps required to reach F(n+1)-1 from F(n+2)-1 by repeatedly subtracting from a natural number the number of ones in its Zeckendorf representation. Here F(n) = the n-th Fibonacci number, F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, ...

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 5, 8, 11, 17, 25, 37, 56, 85, 130, 199, 305, 469, 723, 1118, 1733, 2693, 4193, 6539, 10211, 15962, 24974, 39103, 61262, 96030, 150608, 236338, 371101, 583118, 916978, 1443204, 2273434, 3584522, 5656786, 8934696, 14123156, 22340250

Offset: 0

Antti Karttunen, Aug 08 2015

nonn

Cf. A000045, A000071, A219641, A219642.
From a(1) onward the first differences of both A261081 and A261082.
Cf. A261090 (first differences of this sequence).
Cf. also A261102, A261076.

(define (A261091 n) (let ((end (- (A000045 (+ 1 n)) 1))) (let loop ((k (- (A000045 (+ 2 n)) 1)) (s 0)) (if (= k end) s (loop (A219641 k) (+ 1 s))))))

a(n) = A219642(A000071(n+2)) - A219642(A000071(n+1)). [By definition.]

a(n) = A219642(A000045(n+2)) - A219642(A000045(n+1)). [Equally.]

A261102 Simple self-inverse permutation of natural numbers: after zero, list each block of A261091(n) numbers in reverse order, from A261082(n) to A261081(n+1).

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 7, 6, 10, 9, 8, 15, 14, 13, 12, 11, 23, 22, 21, 20, 19, 18, 17, 16, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 113, 112, 111

Offset: 0

Antti Karttunen, Aug 09 2015

nonn

Maps between sequences A219648 and A261076.

Antti Karttunen, Table of n, a(n) for n = 0..7624
Index entries for sequences that are permutations of the natural numbers

Cf. A261081, A261082, A261091, A261101.
Cf. A219648, A261076.
Cf. also A218602 (analogous sequence for base-2).

cp's OEIS Frontend

A219648 The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf representation of a(n).

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A261091 a(n) = number of steps required to reach F(n+1)-1 from F(n+2)-1 by repeatedly subtracting from a natural number the number of ones in its Zeckendorf representation. Here F(n) = the n-th Fibonacci number, F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, ...

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Scheme

Formula

A261102 Simple self-inverse permutation of natural numbers: after zero, list each block of A261091(n) numbers in reverse order, from A261082(n) to A261081(n+1).

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