Negalucas Numbers Generator

The Lucas sequence 2, 1, 3, 4, 7, 11, 18, 29... ends the story too early — mathematics insists on extending recurrences backwards. The negalucas numbers do exactly that: L(-1) = -1, L(-2) = 3, L(-3) = -4, L(-4) = 7, L(-5) = -11, L(-6) = 18... The mirror rule L(-n) = (-1)^n L(n) alternates signs, always returning the positive Lucas values with a twist of sign, and the recurrence L(k) = L(k-1) + L(k-2) keeps holding on both sides of zero.

Negalucas Numbers Generator builds the extended sequence to any range you specify — from negative indexes to positive — with BigInt-exact values and a formula note on every negative row. Choose a start index from -500 to 500, take up to 200 terms, and copy or download the whole indexed list. All computation happens in the browser.

• Extended sequence: Lucas values on both sides of zero via L(-n) = (-1)^n L(n), mirrored exactly.
• Any range: Start from -500 to 500 with up to 200 consecutive terms in one run.
• BigInt exactness: No floating point — values stay exact at any magnitude.
• Formula notes: Every negative row shows the reflection that produced its sign.
• Indexed output: L(i) = value lines, ready to paste into spreadsheets or reports.
• Copy or download: One click takes the complete run to the clipboard or a text file.
• Reference tab: The mirror rule, the closed form, and the Fibonacci connection.
• Free and private: All computation runs locally — nothing is uploaded, stored or logged.

1. Set the start index — any integer from -500 to 500; negatives give the negalucas half.
2. Set the count — how many consecutive terms to generate (1 to 200).
3. Generate — the indexed run appears with sign notes on the negative rows.
4. Trace a sign — read the (-1)^n x L(|n|) note on any negative value.
5. Copy or download — the complete run for analysis, teaching or datasets.

Example 1 — The classic mirror run. Start at -6, count 13. The first six rows L(-6) = 18, L(-5) = -11, L(-4) = 7, L(-3) = -4, L(-2) = 3, L(-1) = -1 alternate signs exactly, then L(0) = 2 and L(1) = 1 return to the standard sequence.

Example 2 — Only the negative half. Start at -8, count 8. You get rows L(-8) down to L(-1): 47, -29, 18, -11, 7, -4, 3, -1 — the Lucas numbers in reverse order, sign-twisted.

Example 3 — Round trip symmetry. Start -4, count 9. The two ends L(-4) = 7 and L(4) = 7 are equal: the 4th power of -1 is +1. The middle rows at odd |i| are negatives — a mirror you can eyeball.

Example 4 — Identity check. Run L(-5) = -11 next to L(5) = 11: the pair sums to zero, exactly the (-1)^5 flip the formula predicts.

Example 5 — Big stretch. Start 90, count 12 to watch values reach 78-digit scale — exact, because the generator builds each term by BigInt addition of the two before it.

• Complete symmetry: The full extended sequence, not just the positive half.
• Instant mirroring: The (-1)^n rule applied automatically with a note on every row.
• Exact at every size: BigInt addition means zero rounding, even past 70 digits.
• Flexible ranges: Negative starts, positive starts, or a run that crosses zero.
• Text-ready output: Indexed lines paste directly into any spreadsheet.
• Private by design: Nothing leaves the browser; no uploads, no logs.