A372555 Least number of Jacobsthal numbers that add up to n.

0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 4, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 2

Offset

0,3

Comments

Differs from A265745 for the first time at n=63, where a(63) = 3, while A265745(63) = 5. The next differences occur at n=84, 148, 169, 191, 212, 234, 255, etc. See A372557.

See conjecture in A372556, and also in A372561.

Links

Antti Karttunen, Table of n, a(n) for n = 0..87381

Formula

a(0) = 0, a(1) = 1; for n > 1, a(n) = 1 + a(n-A001045(A372556(n))).

Example

a(5) = 1, because 5 is itself in A001045.
a(7) = 3, because 7 can be expressed as a sum of three Jacobsthal numbers, either as 5+1+1 or 3+3+1, but not as a sum of two Jacobsthal numbers, and neither 7 is itself in A001045.
a(63) = 3, because the least number of Jacobsthal numbers that add up to 63 is obtained when we use A001045(6) = 21 three times, as 21+21+21 = 63. This is the first time this sequence differs from A265745.

Prog

(PARI)
up_to = 87381; \\ = A001045(18).
A001045(n) = (2^n - (-1)^n) / 3;
A130249(n) = (#binary(3*n+1)-1);
A372555_or_556list(up_to_n, return_556_instead) = { my(v372555 = vector(up_to_n), v372556 = vector(up_to_n)); v372555[1] = 1; v372556[1] = 2; for(n=2, #v372556, my(m=-1, mk=-1, s=A130249(n)); if(A001045(s)==n, v372555[n] = 1; v372556[n] = s, forstep(k=s, 1, -1, my(c=v372555[n-A001045(k)]); if(m<0 || c<m, m=c; mk=k)); v372556[n] = mk; v372555[n] = 1+v372555[n-A001045(mk)])); if(return_556_instead, v372556, v372555); };
v372555 = A372555_or_556list(up_to, 0);
A372555(n) = if(!n, n, v372555[n]);
(Scheme)
;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A001045 n) (if (<= n 1) n (+ (A001045 (- n 1)) (* 2 (A001045 (- n 2))))))
(define (A130249 n) (floor->exact (/ (log (+ 1 (* 3 n))) (log 2))))
(define (A147612 n) (if (<= n 1) 1 (if (= (A001045 (A130249 n)) n) 1 0)))
(definec (A372555 n) (if (<= n 1) n (+ 1 (A372555 (- n (A001045 (A372556 n)))))))
(definec (A372556 n) (let ((k (A130249 n))) (if (= 1 (A147612 n)) k (let loop ((k k) (m #f) (mk #f)) (cond ((zero? k) mk) (else (let* ((c (A372555 (- n (A001045 k))))) (if (or (not m) (< c m)) (loop (- k 1) c k) (loop (- k 1) m mk)))))))))

Crossrefs

Cf. A001045, A130249, A147612, A372556, A372557, A372561.

Cf. also A265745.

Sequence in context: A174713 A129985 A085243 * A265745 A092243 A257564

Adjacent sequences: A372552 A372553 A372554 * A372556 A372557 A372558

Keyword

nonn

Author

Antti Karttunen, May 07 2024

Status

approved