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A248808
Irregular triangle read by rows: row n gives golden ratio base representation of Fibonacci number F_n.
1
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
George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110.
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
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