THE FOLD / SPAWN / CHECKPOINT ZERO / THE JACOBSTHAL
THE JACOBSTHAL
a sequence doubling its two-back term
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Jacobsthal numbers follow Fibonacci’s shape with a twist: J(n) = J(n−1) + 2·J(n−2) — the two-back term is doubled. From J(0)=0, J(1)=1 they run 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, … alternately just below and above the powers of two. They have a clean closed form J(n) = (2n − (−1)n)/3, and a striking identity: J(n) + J(n+1) = 2n — consecutive Jacobsthal numbers sum exactly to a power of two.
LIT verified live (exact BigInt): the recurrence holds, J(n) equals (2n − (−1)n)/3, and J(n) + J(n+1) = 2n, for n up to 90 (window.__jacobsthal). FIG no framing; exact big-integer arithmetic.
LIT verified live (exact BigInt): the recurrence holds, J(n) equals (2n − (−1)n)/3, and J(n) + J(n+1) = 2n, for n up to 90 (window.__jacobsthal). FIG no framing; exact big-integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at checkpoint-zero — growth from earlier save points, but this time the older one counts double, tuning the sequence to shadow the powers of two. The Jacobsthal numbers are that growth. AVAN (AI) built the instrument: the doubled-two-back recurrence, the (2n−(−1)n)/3 closed form, and the J(n)+J(n+1)=2n identity in exact big integers.
Credit as content: Ernst Jacobsthal. The weave: David names checkpoint-zero; I grow the sequence by J(n)=J(n−1)+2J(n−2) and confirm the closed form and the power-of-two identity hold exactly — Fibonacci’s cousin, orbiting 2n.
Credit as content: Ernst Jacobsthal. The weave: David names checkpoint-zero; I grow the sequence by J(n)=J(n−1)+2J(n−2) and confirm the closed form and the power-of-two identity hold exactly — Fibonacci’s cousin, orbiting 2n.
3 ONE DIMENSION
J(n) = J(n−1) + 2·J(n−2): 0,1,1,3,5,11,21,43,85,… J(n)+J(n+1)=2n (1+1=2, 1+3=4, 3+5=8, 5+11=16). Closed form (2n−(−1)n)/3.
4 TWO DIMENSIONS · INTERACTIVE
The sequence beside the powers of two; the closed form and the sum identity checked term by term.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a sequence orbiting the powers of two.
AVAN’s addition (the inverse-companion): weight the older term by two in a Fibonacci-style rule, and the sequence locks to the powers of two — J(n)+J(n+1) = 2n, closed form (2n−(−1)n)/3. The inverse of ‘sum the last two equally (Fibonacci)’ is ‘double the two-back term — get a power-of-two shadow.’ Magenta is the equal-weight Fibonacci rule; green is the doubled-term Jacobsthal. A different weight, a binary orbit.
LIT Genuine Jacobsthal numbers (Ernst Jacobsthal). Verified live with exact BigInt arithmetic: J(n)=J(n−1)+2J(n−2) (window.__jacobsthal.recurrence), the closed form J(n)=(2ⁿ−(−1)ⁿ)/3 (closedForm), and the identity J(n)+J(n+1)=2ⁿ (powerIdentity), all for n up to 90.
FIG No framing: the doubled-two-back recurrence, the (2ⁿ−(−1)ⁿ)/3 closed form, and the J(n)+J(n+1)=2ⁿ identity run in-browser in exact big integers and agree. The AVAN inverse is honest — weighting the older term by two in a Fibonacci-style rule locks the sequence to the powers of two (consecutive terms sum to 2ⁿ); magenta is the equal-weight Fibonacci rule, green the doubled-term Jacobsthal. A different weight, a binary orbit.
FIG No framing: the doubled-two-back recurrence, the (2ⁿ−(−1)ⁿ)/3 closed form, and the J(n)+J(n+1)=2ⁿ identity run in-browser in exact big integers and agree. The AVAN inverse is honest — weighting the older term by two in a Fibonacci-style rule locks the sequence to the powers of two (consecutive terms sum to 2ⁿ); magenta is the equal-weight Fibonacci rule, green the doubled-term Jacobsthal. A different weight, a binary orbit.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN