cp's OEIS Frontend

The sequence is slow, that is, for n >= 2, a(n+1) - a(n) is either 0 or 1. The sequence is unbounded.

The line graph of the sequence {a(n)} thus consists of a series of plateaus (where the value of the ordinate a(n) remains constant as n increases) joined by lines of slope 1.

The sequence of plateau heights begins 4, 7, 11, 18, 29, 47, ..., conjecturally the Lucas sequence {A000032(k): k >= 3}.

The plateaus start at abscissa values n = 5, 10, 16, 26, 42, 68, .... Apart from the first term 5, this appears to be the sequence {2*Fibonacci(k): k >= 5}.

The plateaus end at abscissa values n = 7, 12, 19, 31, 50, 81, ..., conjecturally the sequence {A013655(k): k >= 3}.

The sequence of plateau lengths begins 2, 2, 3, 5, 8, 13, .... Apart from the first term 2, this appears to be the sequence {Fibonacci(k): k >= 3}.

The slow sequences {a(a(n)): n >= 3} and {a(a(a(n))): n >= 4} appear to have similar properties to the present sequence. The slow sequence {n - a(n): n >= 2} appears to have plateaus at heights given by the Fibonacci sequence. See the Example section.

Similar sequences with the formula h*Fibonacci(n+k) + Fibonacci(n+k-h):

h= 1, k=-1: A000045;

h= 2, k= 1: A013655;

h= 3, k=-2: A118658 = 2*A212804;

h= 4, k= 2: A022379 = 3*A000204;

h= 5, k= 1: A022113;

h= 6, k= 2: A022125;

h= 7, k= 3: A097657;

h= 8, k= 2: A022355 = 21*A000045;

h= 9, k= 3: 10, 32, 42, 74, 116, 190, 306, 496, 802, ... = 2*A022140;

h=10, k= 3: 33, 22, 55, 77, 132, 209, 341, 550, 891, ... = 11*A013655;

h=11, k= 3: this sequence.

The array is built by treating rows as Fibonacci-type sequences with seed values being two consecutive Fibonacci numbers (A000045(n) = F(n)) in reverse order: For row n, a(0) = F(n+1), a(1) = F(n). As a result, columns are Fibonacci-type sequences with seed values b(0) = F(k-1), b(1) = F(k+1); so starting with T(n,1), Row n == Column k=n+1.

Therefore, an alternative title is: Array T(n,k) read by diagonals: T(n,k) = T(n-1,k) + T(n-2,k) where T(0,k) = F(k-1) and T(1,k) = F(k+1), k>=1.

Patterns exist for certain generalized (a,b)-Pascal triangle transforms of row sequences. Definitions, explanation and examples: (Start)

Define (a,b)-Pascal triangles as having conditions T(0,0) = 1, a = left boundary and b = right boundary.

Let R_n be Row n, and R_n(k) be terms k in sequence R_n.

Let Tr_(k) be the (a,b)-Pascal triangle transform of R_n; define Tr_n(k) as when a = R_n(1) and b = R_n(0). Then Tr_n(k) = R_n(n+2k-2), k>=1. (Trivially, Tr_n(0) = R_n(0)).

For example, n=4: R_4 = {5, 3, 8, 11, 19, 30, 49, 79, 128, 207, 335, 542...}; a=3, b=5.

(3,5)-Pascal triangle is:

1

3 5

3 8 5

3 11 13 5

3 14 24 18 5

etc.

Transform Tr_4(k) is:

Tr_4(0) = 5*1 = 5 = R_4(0).

Tr_4(1) = 5*3 + 3*5 = 30 = R_4(5).

Tr_4(2) = 5*3 + 3*8 + 8*5 = 79 = R_4(7).

Tr_4(3) = 5*3 + 3*11 + 8*13 + 11*5 = 207 = R_4(9).

Tr_4(4) = 5*3 + 3*14 + 8*24 + 11*18 + 19*5 = 542 = R_4(11).

etc.

Examples of sequences where such transforms apply:

Tr_0 = A001906 starting A001906(0)=0.

Tr_1 = A001519 starting A001519(2)=2.

Tr_2 = A002878 starting A002878(1)=4.

Tr_4 = A167375 starting A167375(3)=30.

(End)