A379147 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the integers m such that A184617(abs(m)) = A003714(n).

0, -1, 1, -2, 2, -4, 4, -5, -3, 3, 5, -8, 8, -9, -7, 7, 9, -10, -6, 6, 10, -16, 16, -17, -15, 15, 17, -18, -14, 14, 18, -20, -12, 12, 20, -21, -19, -13, -11, 11, 13, 19, 21, -32, 32, -33, -31, 31, 33, -34, -30, 30, 34, -36, -28, 28, 36

Offset: 0

Author

Rémy Sigrist, Dec 16 2024

Keywords

• sign
• tabf
• base

Comments

A permutation of the integers (Z).

For any n >= 0:

- in the Zeckendorf expansion of n,

- replace each Fibonacci number, say A000045(2+i) with i >= 0, by 2^i or -2^i,

- the various values obtained make up the n-th row.

Examples

			Triangle T(n, k) begins:
  n   n-th row
  --  ----------------------------------
   0  0
   1  -1, 1
   2  -2, 2
   3  -4, 4
   4  -5, -3, 3, 5
   5  -8, 8
   6  -9, -7, 7, 9
   7  -10, -6, 6, 10
   8  -16, 16
   9  -17, -15, 15, 17
  10  -18, -14, 14, 18
  11  -20, -12, 12, 20
  12  -21, -19, -13, -11, 11, 13, 19, 21
  13  -32, 32
  14  -33, -31, 31, 33
  15  -34, -30, 30, 34
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10922 (rows for n = 0..609 flattened)
• Index entries for sequences related to Zeckendorf expansion of n

Crossrefs

Cf. A000045, A003714, A007895, A184617, A379175.

Programs

PARI

tozeck(n) = { for (i=0, oo, if (n<=fibonacci(2+i), my (v=0, f); forstep (j=i, 0, -1, if (n>=f=fibonacci(2+j), n-=f; v+=2^j;); if (n==0, return (v););););); }
row(n) = { my (z = tozeck(n), r = [0], b); while (z, z -= b = 2^valuation(z, 2); r = concat([v - b | v <- r], [v + b | v <- r]);); return (r); }

Formula

T(n, 1) = -A003714(n).

T(n, 2^A007895(n)) = A003714(n).

T(n, k) = -T(n, 2^A007895(n)+1-k) for k = 1..2^A007895(n).