A265746 -id:A265746 - cp's OEIS frontend

A265745 a(n) is the number of Jacobsthal numbers (A001045) needed to sum to n using the greedy algorithm.

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

Offset: 0

Antti Karttunen, Dec 17 2015

Sum of digits in "Jacobsthal greedy base", A265747.
It would be nice to know for sure whether this sequence gives also the least number of Jacobsthal numbers that add to n, i.e., that there cannot be even better nongreedy solutions.
The integer 63=21+21+21 has 3 for its 'non-greedy' solution, and a(63) = 5 for its greedy solution 63=43+11+5+3+1. - Yuriko Suwa, Jul 11 2021
Positions where a(n) is different from A372555(n) are n=63, 84, 148, 169, 191, 212, 234, 255, etc. See A372557. - Antti Karttunen, May 07 2024

			a(0) = 0, because no numbers are needed to form an empty sum, which is zero.
For n=1 we need just A001045(2) = 1, thus a(1) = 1.
For n=2 we need A001045(2) + A001045(2) = 1 + 1, thus a(2) = 2.
For n=4 we need A001045(3) + A001045(2) = 3 + 1, thus a(4) = 2.
For n=6 we form the greedy sum as A001045(4) + A001045(2) = 5 + 1, thus a(6) = 2. Alternatively, we could form the sum as A001045(3) + A001045(3) = 3 + 3, but the number of summands in that case is no less.
For n=7 we need A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = 3.
For n=8 we need A001045(4) + A001045(3) = 5 + 3, thus a(8) = 2.
For n=10 we need A001045(4) + A001045(4) = 5 + 5, thus a(10) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..10923

Cf. A001045, A130249, A197911, A003158, A265746, A265747, A364377, A364379, A364380, A372288, A372555, A372557.
Cf. A054111 (apparently the positions of the first occurrence of each n > 0).
Cf. also A007895, A014420, A053610, A265404, A265743, A265744.

jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265745[n_] := A265745[n] = 1 + A265745[n - jacob[maxInd[n]]]; A265745[0] = 0; Array[A265745, 100, 0] (* Amiram Eldar, Jul 21 2023 *)

A130249(n) = floor(log(3*n + 1)/log(2));
A001045(n) = (2^n - (-1)^n) / 3;
A265745(n) = {if(n == 0, 0, my(d = n - A001045(A130249(n))); if(d == 0, 1, 1 + A265745(d)));} \\ Amiram Eldar, Jul 21 2023

def greedyJ(n): n1 = (3*n+1).bit_length() - 1; return (2**n1 - (-1)**n1)//3
def a(n): return 0 if n == 0 else 1 + a(n - greedyJ(n))
print([a(n) for n in range(107)]) # Michael S. Branicky, Jul 11 2021

a(0) = 0; for n >= 1, a(n) = 1 + a(n - A001045(A130249(n))). [This formula uses a simple greedy algorithm.]

A265747 Numbers written in Jacobsthal greedy base.

Original entry on oeis.org

0, 1, 2, 10, 11, 100, 101, 102, 110, 111, 200, 1000, 1001, 1002, 1010, 1011, 1100, 1101, 1102, 1110, 1111, 10000, 10001, 10002, 10010, 10011, 10100, 10101, 10102, 10110, 10111, 10200, 11000, 11001, 11002, 11010, 11011, 11100, 11101, 11102, 11110, 11111, 20000, 100000, 100001, 100002, 100010, 100011, 100100

Offset: 0

Antti Karttunen, Dec 17 2015

These are called "Jacobsthal Representation Numbers" in Horadam's 1996 paper.
Sum_{i=0..} digit(i)*A001045(2+digit(i)) recovers n from such representation a(n), where digit(0) stands for the least significant digit (at the right), and A001045(k) gives the k-th Jacobsthal number.
No larger digits than 2 will occur, which allows representing the same sequence in a more compact form by base-3 coding in A265746.
Sequence A197911 gives the terms with no digit "2" in their representation, while its complement A003158 gives the terms where "2" occurs at least once.
Numbers beginning with digit "2" in this representation are given by A020988(n) [= 2*A002450(n) = 2*A001045(2n)].

			For n=7, when selecting the terms of A001045 with the greedy algorithm, we need terms A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = "102".
For n=10, we need A001045(4) + A001045(4) = 5+5, thus a(10) = "200".

Antti Karttunen, Table of n, a(n) for n = 0..10923
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34 (1996), 40-54. (See especially page 50, which is page 11 in PDF.)

Cf. A001045, A002450, A020988, A007089, A197911, A003158.
Cf. A265745 (sum of digits).
Cf. A265746 (same numbers interpreted in base-3, then shown in decimal).
Cf. A084639 (positions of repunits).
Cf. A007961, A014417, A014418, A244159 for analogous sequences.

jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265747[n_] := A265747[n] = 10^(maxInd[n] - 2) + A265747[n - jacob[maxInd[n]]]; A265747[0] = 0; Array[A265747, 100, 0] (* Amiram Eldar, Jul 21 2023 *)

A130249(n) = floor(log(3*n + 1) / log(2));
A001045(n) = (2^n - (-1)^n) / 3;
A265747(n) = {if(n==0, 0, my(d=n - A001045(A130249(n))); 10^(A130249(n)-2) + if(d == 0, 0, A265747(d)));} \\ Amiram Eldar, Jul 21 2023

def greedyJ(n): m = (3*n+1).bit_length() - 1; return (m, (2**m-(-1)**m)//3)
def a(n):
    if n == 0: return 0
    place, value = greedyJ(n)
    return 10**(place-2) + a(n - value)
print([a(n) for n in range(49)]) # Michael S. Branicky, Jul 11 2021

a(0) = 0; for n >= 1, a(n) = 10^(A130249(n)-2) + a(n-A001045(A130249(n))).

a(n) = A007089(A265746(n)).

A147612 If n is a Jacobsthal number then 1 else 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Nov 08 2008

nonn

a(A001045(n)) = 1; a(A147613(n)) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..43692
Index entries for characteristic functions

Cf. A001045, A105348, A265746, A293431, A293432, A293433, A293434.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[If[FreeQ[s, #], 0, 1] &, 105, 0]] (* Michael De Vlieger, Oct 09 2017 *)

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1)); \\ Antti Karttunen, Oct 09 2017

def A147612(n): return int((m:=3*n+1).bit_length()>(m-3).bit_length()) if n else 1 # Chai Wah Wu, Apr 18 2025

a(n) = 0^(j(n,1)*j(n,-1)) with j(n,i) = if n mod 2 = 0 then n else j((n+i)/2,-i).
a(n) = A105348(n), for n <> 1. - R. J. Mathar, Nov 19 2008
For n > 0, a(n) = A000035(A281228(A265746(n))), where A000035(A281228(n)) is the characteristic function of powers of 3 (A000244). - Antti Karttunen, Oct 09 2017

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A265745 a(n) is the number of Jacobsthal numbers (A001045) needed to sum to n using the greedy algorithm.

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A265747 Numbers written in Jacobsthal greedy base.

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A147612 If n is a Jacobsthal number then 1 else 0.

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