A152112 -id:A152112 - cp's OEIS frontend

A167810 Number of admissible basis in the postage stamp problem for n denominations and h = 3 stamps.

1, 3, 13, 86, 760, 8518, 116278, 1911198, 37063964, 835779524, 21626042510, 635611172160, 21033034941826, 777710150809009

Offset: 1

Yogy Namara (yogy.namara(AT)gmail.com), Nov 12 2009

A basis 1 = b_1 < b_2 ... < b_n is admissible if all the values 1 <= x <= b_n is obtainable as a sum of at most h (not necessarily distinct) numbers in the basis.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis, Two new techniques for computing extremal h-bases A_k, Comp J 36(2) (1993) 117-126
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
S. Mossige, Algorithms for Computing the h-Range of the Postage Stamp Problem, Math. Comp. 36 (1981) 575-582

Other enumerations with different parameters: A167809 (h = 2), A167810 (h = 3), A167811 (h = 4), A167812 (h = 5), A167813 (h = 6), A167814 (h = 7).
For h = 2, cf. A008932.
A152112 is essentially the same sequence by definition. [From Herbert Kociemba, Jul 14 2010]

Terms a(1) to a(12) verified and new terms a(13) and a(14) added by Herbert Kociemba, Jul 14 2010

A152111 An increasing basis of order 3. See Comments for full definition.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 65, 72, 73, 128, 130, 144, 146, 256, 260, 288, 292, 512, 513, 520, 521, 576, 577, 584, 585, 1024, 1026, 1040, 1042, 1152, 1154, 1168, 1170, 2048, 2052, 2080, 2084, 2304, 2308, 2336, 2340, 4096, 4097, 4104, 4105, 4160

Offset: 1

David S. Newman, Mar 22 2009

Using the terminology of A008932, call a set A a basis of order h if every number can be written as the sum of h (not necessarily distinct) elements of A. Call a basis an increasing basis of order h if its elements are arranged in increasing order, a0 < a1 < a2 < ...
This sequence is constructed as follows: Take the union of the following three sets: (1) the set of all nonnegative numbers which can be written in base two as sums of powers, k, of 2, where k is congruent to 0 mod 3; (2) the set of all nonnegative numbers which can be written in base two as sums of powers, k, of 2, where k is congruent to 1 mod 3; (3) the set of all nonnegative numbers which can be written in base two as sums of powers, k, of 2, where k is congruent to 2 mod 3.
Numbers of the form A033045(k), or 2*A033045(k), or 4*A033045(k). - R. J. Mathar, Sep 21 2009
There are 3*2^i - 1 terms up to 8^i. - David A. Corneth, Aug 02 2017

David A. Corneth, Table of n, a(n) for n = 1..12287 (terms up to 8^12).
Index entries for sequences that are an additive basis, order 3.

Cf. A008932, A033045, A152112, A284116.

ismod3 := proc(n,m) b := convert(n,base,2) ; for i from 1+((m+1) mod 3) to nops(b) by 3 do if op(i,b) <> 0 then RETURN(false) ; fi; od: for i from 1 + ((m+2) mod 3) to nops(b) by 3 do if op(i,b) <> 0 then RETURN(false) ; fi; od: true ; end: for n from 0 to 20700 do if ismod3(n,0) or ismod3(n,1) or ismod3(n,2) then printf("%d,",n); fi; od: # R. J. Mathar, Sep 21 2009

upto(n) = {my(i = 1, r, res = List()); while(1, b = binary(i); r = sum(i=1, #b, 8^i*b[#b+1-i])>>3; if(r > n, break); listput(res, r); i+=2); q = #res; for(i=1,  q, e = res[i] << 1; while(e <= n, listput(res, e); e=e<<1)); listput(res, 0); listsort(res); res} \\ David A. Corneth, Aug 02 2017

More terms from R. J. Mathar, Sep 21 2009

cp's OEIS Frontend

A167810 Number of admissible basis in the postage stamp problem for n denominations and h = 3 stamps.

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