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%I #15 Jul 23 2017 02:41:13
%S 9,21,30,42,65,70,99,105,117,133,135,154,175,180,225,231,275,285,341,
%T 342,345,364,385,414,440,450,455,481,495,540,546,567,630,645,675,693,
%U 744,750,765,825,833,936,945,990,1035,1045,1140,1161,1170,1176,1178
%N Friendly run sums: numbers S with two run sums (sum of positive integer runs) that share one common number, i.e., S = a + (a+1) + ... + b = b + (b+1) + ... + c with a < b < c.
%C The sums are the difference of two triangular numbers A000217. The common numbers themselves are A094550. The sums, n > 0, are n = (b-a+1)(a+b)/2 = (b+c)(c-b+1)/2 where b^2 + a - a^2 = c^2 + c - b^2, is solvable in integers for 0 < a < b < c. Since the runs have something in common, they are "friends". The series of sums *without* the common number is A110702. The numbers common to the two runs are A094550.
%H Ron Knott <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/">Runsums</a>
%H T. Verhoeff, <a href="https://cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
%e 2 + 3 + 4 = 4 + 5 share the common number 4. The sum of both is 9 and this is the smallest such sum with a common "friend" (4), so a(1)=9.
%Y Cf. A001227, A094550, A110702, A110703.
%K nonn
%O 1,1
%A _Ron Knott_, Aug 04 2005
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