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a(1)=1, a(n) = smallest odd number such that all sums of pairs of (not necessarily distinct) terms in the sequence are distinct.
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%I #25 Sep 05 2023 18:26:05

%S 1,3,7,15,25,41,61,89,131,161,193,245,295,363,407,503,579,721,801,949,

%T 1129,1185,1323,1549,1643,1831,1939,2031,2317,2623,2789,3045,3143,

%U 3641,3791,4057,4507,4757,5019,5559,5849,6309,6707,7181,7593

%N a(1)=1, a(n) = smallest odd number such that all sums of pairs of (not necessarily distinct) terms in the sequence are distinct.

%C a(1) = 1, a(n) = least number such that every difference a(i)-a(j) is a distinct even number. - _Amarnath Murthy_, Apr 07 2004

%H Reinhard Zumkeller, <a href="/A034757/b034757.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = 2*A005282(n)-1. (David Wasserman)

%e 5 is not in the sequence since 5+1 is already obtainable from 3+3, 9 is excluded since 1, 3 and 7 are in the sequence and would collide with 1+9

%t seq2={1, 3}; Do[le=Length[seq2]; t=Last[seq2]+2; While[Length[Expand[(Plus @@ (x^seq2) + x^t)^2]] < Pochhammer[3, le]/le!, t=t+2]; AppendTo[seq2, t], {20}]; Print@seq2

%o (Haskell)

%o a034757 = (subtract 1) . (* 2) . a005282 -- _Reinhard Zumkeller_, Dec 18 2012

%o (Python)

%o from itertools import count, islice

%o def A034757_gen(): # generator of terms

%o aset1, aset2, alist = set(), set(), []

%o for k in count(1,2):

%o bset2 = {k<<1}

%o if (k<<1) not in aset2:

%o for d in aset1:

%o if (m:=d+k) in aset2:

%o break

%o bset2.add(m)

%o else:

%o yield k

%o alist.append(k)

%o aset1.add(k)

%o aset2.update(bset2)

%o A034757_list = list(islice(A034757_gen(),30)) # _Chai Wah Wu_, Sep 05 2023

%Y Cf. A025582, A051912, A055598.

%Y Partial sums of A287178.

%K nonn,nice,easy

%O 1,2

%A _Wouter Meeussen_, Jun 01 2000

%E An incorrect comment from _Amarnath Murthy_, also dated Apr 07 2004, has been deleted.

%E Offset fixed by _Reinhard Zumkeller_, Dec 18 2012