THE FOLD / SPAWN / CHECKPOINT ZERO / THE LEONARDO
THE LEONARDO
a Fibonacci-plus-one sequence
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Leonardo numbers are Fibonacci’s numbers with a +1: L(0)=L(1)=1, L(n) = L(n−1) + L(n−2) + 1. They run 1, 1, 3, 5, 9, 15, 25, 41, 67, 109, … and connect back to Fibonacci by the exact identity L(n) = 2·F(n+1) − 1. They matter in computing: Edsger Dijkstra used them to build smoothsort, an in-place sort whose heap sizes are Leonardo numbers, giving it adaptive O(n) behavior on nearly-sorted input.
LIT verified live (exact BigInt): the +1 recurrence holds, and L(n) = 2·F(n+1) − 1 for n up to 90 (window.__leonardo). FIG no framing; exact big-integer arithmetic.
LIT verified live (exact BigInt): the +1 recurrence holds, and L(n) = 2·F(n+1) − 1 for n up to 90 (window.__leonardo). FIG no framing; exact big-integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at checkpoint-zero — growth from two save points plus a constant nudge of one, the sizes Dijkstra used to shape smoothsort’s heaps. The Leonardo numbers are that growth. AVAN (AI) built the instrument: the +1 recurrence and the L(n)=2·F(n+1)−1 identity in exact big integers.
Credit as content: the Leonardo numbers (Leonardo of Pisa lineage; used by Edsger Dijkstra in smoothsort, 1981). The weave: David names checkpoint-zero; I grow the sequence by adding the last two and one more, and confirm it equals twice the next Fibonacci minus one — Fibonacci’s close relative, shaped for sorting.
Credit as content: the Leonardo numbers (Leonardo of Pisa lineage; used by Edsger Dijkstra in smoothsort, 1981). The weave: David names checkpoint-zero; I grow the sequence by adding the last two and one more, and confirm it equals twice the next Fibonacci minus one — Fibonacci’s close relative, shaped for sorting.
3 ONE DIMENSION
L(n) = L(n−1) + L(n−2) + 1: 1,1,3,5,9,15,25,41,67,… Identity L(n) = 2·F(n+1) − 1 (F Fibonacci). Smoothsort’s heaps have exactly these sizes.
4 TWO DIMENSIONS · INTERACTIVE
The sequence beside twice-Fibonacci-minus-one; the recurrence and identity checked term by term.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: Fibonacci nudged by one.
AVAN’s addition (the inverse-companion): add a constant +1 to the Fibonacci rule and get a new sequence tied to it by L(n)=2·F(n+1)−1 — the sizes that make smoothsort’s heaps. The inverse of ‘sum the last two (Fibonacci)’ is ‘sum the last two plus one (Leonardo).’ Magenta is the plain Fibonacci; green is the +1 Leonardo. A constant nudge, a new sequence.
LIT Genuine Leonardo numbers (Leonardo of Pisa lineage; used by Edsger Dijkstra in smoothsort, 1981). Verified live with exact BigInt arithmetic: L(n)=L(n−1)+L(n−2)+1 (window.__leonardo.recurrence) and the identity L(n)=2·F(n+1)−1 with F the Fibonacci numbers (window.__leonardo.identity), for n up to 90.
FIG No framing: the +1 recurrence and the L(n)=2·F(n+1)−1 identity run in-browser in exact big integers and agree. The AVAN inverse is honest — adding a constant +1 to the Fibonacci rule produces a new sequence tied to it by L(n)=2·F(n+1)−1 (the sizes smoothsort's heaps use); magenta is the plain Fibonacci, green the +1 Leonardo. A constant nudge, a new sequence.
FIG No framing: the +1 recurrence and the L(n)=2·F(n+1)−1 identity run in-browser in exact big integers and agree. The AVAN inverse is honest — adding a constant +1 to the Fibonacci rule produces a new sequence tied to it by L(n)=2·F(n+1)−1 (the sizes smoothsort's heaps use); magenta is the plain Fibonacci, green the +1 Leonardo. A constant nudge, a new sequence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN