OFFSET

0,3

EXAMPLE

a(1)= 1 since it is not possible to write 1 using only a(0). a(2)=2, since it is not possible to obtain 2 using only a(0) and a(1). The following numbers up to 6 can be represented using these first 3 elements of the sequence: 3 = 1*1 + 1*2, 4 = 0*1 + 2*2, 5 = 1*1 + 2*2, 6 = 2*1 + 2*2. Again we reach a number that cannot be represented as defined above, so that number is appended to the sequence. It happens here when we try to represent 7 using only a(0)=0, a(1)=1, and a(2)=2. So 7 becomes a(3).

A larger example: 216752 = 1*1 + 1*2 + 85*7 + 285*24 + 85*85 + 85*285 + 24*1143 + 24*6268

MATHEMATICA

Nest[Function[a, Append[a, 1 + LengthWhile[Differences@ #, # == 1 &] &@ Union[Total /@ Map[a # &, Tuples[a, Length@ a]]]]], {0}, 8] (* Michael De Vlieger, Jan 09 2018 *)

PROG

(Python)

# Generate all the elements in the sequence, S, necessary to represent all

# numbers until the integer 'last'. It also shows how each integer is

# represented by showing the sequence elements and the respective

# multiplicative factors.

import numpy as np

import itertools

last=100

def generate(i, S):

n=len(S)

s=np.asarray(S, dtype=np.int)

perms = [p for p in itertools.product(S, repeat=n)]

for iks in perms:

t=np.asarray(iks)

if np.dot(t, s) == i:

print('%d=' %i, end=', ')

print(t, 'x', s)

return 0

return -1

S=[0]

for i in range(1, last+1):

if generate(i, S) == -1:

S.append(i)

generate(i, S)

CROSSREFS

KEYWORD

nonn,more,hard

AUTHOR

Luis F.B.A. Alexandre, Dec 28 2017

EXTENSIONS

a(9) from Robert G. Wilson v, Jan 09 2018

a(10)-a(11) from Jon E. Schoenfield, Jan 16 2018

a(12) from Giovanni Resta, Jan 22 2018

STATUS

approved