%I #13 Oct 26 2019 16:24:33

%S 560,736,1541,3201,5461,6816,7400,9976,11041,11408,13333,14981,15408,

%T 15841,19521,21000,21505,25761,28616,30401,41536,45141,50440,51221,

%U 52008,54405,56856,61920,63656,65416,69008,75525,76480,81345,82336,85345,87381,89441

%N Octagonal numbers which are the sums of exactly two positive octagonal numbers.

%C For sums of two positive octagonal numbers, see A136345. This is to octagonal numbers A000567 as A089982 is to triangular numbers A000217, as A009000 is to squares A000290, as A136117 is to pentagonal numbers A000326, as A133215 is to hexagonal numbers A000384, and as A117104 is to heptagonal numbers A000566. If Oc(a) + Oc(b) = Oc(c) then a(3a-2) + b(3b+2) = c(3c+2), so solving the quadratic equations for c we have (when an integer): c = (2 + sqrt(4 + 36a^2 + 36b^2 - 24a - 24b))/6.

%H B. D. Swan, <a href="/A136346/b136346.txt">Table of n, a(n) for n = 1..1800</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>

%F A000567 INTERSECTION {A000567(i) + A000567(j), i, j > 0}. {i*(3*i-2)} INTERSECTION {i*(3*i-2) + j(3*j-2), i > 0}.

%e Where Oc(n) = A000567(n) = n-th octagonal number:

%e a(1) = 560 = Oc(14) = 280 + 280 = Oc(10) + Oc(10).

%e a(2) = 736 = Oc(16) = 560 + 176 = Oc(14) + Oc(8).

%e a(3) = 1541 = Oc(23) = 1408 + 133 = Oc(22) + Oc(7).

%e a(4) = 3201 = Oc(33) = 2465 + 736 = Oc(29) + Oc(16).

%e a(5) = 5461 = Oc(43) = 2821 + 2640 = Oc(31) + Oc(30).

%t Module[{nn=300,ono},ono=PolygonalNumber[8,Range[nn]];Union[Select[ Total/@ Tuples[ono,2],MemberQ[ono,#]&]]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 26 2019 *)

%Y Cf. A000567, A009000, A089982, A136117, A136345.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Dec 25 2007

%E Corrected and edited by B. D. Swan (bdswan(AT)gmail.com), Dec 20 2008