A248808 Irregular triangle read by rows: row n gives golden ratio base representation of Fibonacci number F_n.

0, 0, 1, -2, 2, -2, 3, -1, -4, 4, 0, -4, 5, 1, -3, -6, 6, 2, -2, -6, 7, 3, -1, -5, -8, 8, 4, 0, -4, -8, 9, 5, 1, -3, -7, -10, 10, 6, 2, -2, -6, -10, 11, 7, 3, -1, -5, -9, -12, 12, 8, 4, 0, -4, -8, -12, 13, 9, 5, 1, -3, -7, -11, -14, 14, 10, 6, 2, -2, -6, -10, -14

Offset

1,4

Comments

Each row can be viewed as either a sequence of powers of phi (the golden ratio) or as a sequence of indices of combinatorial Fibonacci numbers (Fibonacci numbers offset so that f(0)=1,f(1)=1) (see Benjamin & Quinn).

From Wolfdieter Lang, Oct 16 2014: (Start)

With F(n) = A000045(n) (extended to negative arguments by F(-n) = (-1)^(n+1)*F(n)) one has (proof by induction, using the F recurrence and phi^n = phi^(n-1) + phi^(n-2))

F(2*k) = sum(phi^(2*(n-1) - 4*m), m = 0..k-1), k >= 1 and F(2*k+1) = sum(phi^(2*n-1 - 4*m), m = 0..k-1) + phi^(-2*n), k >= 0.

This shows, using phi^n = F(n-1) + F(n)*phi for integer n, the identity sum(F(2*(k-1) - 4*m, m=0..k-1) = 0 (and similar ones).

The row sums [0, 0, -1, 0, -2, 0, -3, 0, ...] (offset 1) have o.g.f. -x^3/(1-x^2)^2. The alternating row sums [0, 0, 3, 4, 0, 0, 7, 8, 0, 0, 11, 12, ...] have o.g.f. x^3*(3 - 2*x + x^2)/(1 - x + x^2 -x^3)^2.

(End)

This representation of the Fibonacci numbers can be found in the paper of Frougny and Sakarovitch. - Michel Dekking, Feb 10 2020

See also A105424. - Michel Dekking, Feb 10 2020

References

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that Really Count, Dolciani Mathematical Expositions (MAA), (2003).

Links

Michel Marcus, Rows n = 1..100 of triangle, flattened

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110.

Christiane Frougny and Jacques Sakarovitch, Automatic conversion from Fibonacci representation to representation in base phi and a generalization, (1998).

Dale Gerdemann, Golden Ratio Base Digit Patterns for Columns of the Fibonomial Triangle video (2014).

Dale Gerdemann, Fibonacci numbers in Golden Ratio Base, Seqfan post, Oct 12 2014.

Formula

T(2*k, m) = 2*(k-1)-4*m, m = 0, ..., k-1 for k >= 1, and T(2*k+1, m) = 2*k-1-4*m, m = 0, ..., k-1, and T(2*k+1, k) = -2*k for k >= 0. See a comment above for the proof. - Wolfdieter Lang, Oct 16 2014

Example

There are two rows of length one, two rows of length 2, etc. For example, the 6th row, 4,0,-4, is the sequence of powers of phi (1+sqrt(5))/2) used to represent the number 8.
[0] 1
[0] 1
[1, -2] 2
[2, -2] 3
[3, -1, -4] 5
[4,  0, -4] 8
[5,  1, -3, -6] 13
[6,  2, -2, -6] 21
[7,  3, -1, -5,  -8] 34
[8,  4,  0, -4,  -8] 55
[9,  5,  1, -3,  -7, -10] 89
[10, 6,  2, -2,  -6, -10] 144
[11, 7,  3, -1,  -5,  -9,  -12] 233
[12, 8,  4,  0,  -4,  -8,  -12] 377

Mathematica

T[n_, m_] := If[n == 2*m+1, -2*m, n-4*m-2];
Table[T[n, m], {n, 1, 15}, {m, 0, Floor[(n - 1)/2]}] // Flatten (* Jean-François Alcover, Jul 13 2016, after Wolfdieter Lang *)

Prog

(PARI) T(n, m) = if (n == 2*m+1, -2*m, n-4*m-2);
tabf(nn) = for (n=1, nn, for (k=0, (n-1)\2, print1(T(n, k), ", ")); ); \\ Michel Marcus, May 28 2019

Crossrefs

Cf. A000045.

Sequence in context: A256855 A273943 A256071 * A233206 A014843 A116987

Adjacent sequences: A248805 A248806 A248807 * A248809 A248810 A248811

Keyword

sign,tabf,nice

Author

Dale Gerdemann, Oct 14 2014

Extensions

Rows 12 to 14 added by Wolfdieter Lang, Oct 23 2014

More terms from Michel Marcus, May 28 2019

Status

approved