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A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

%I A136346 #14 Feb 16 2025 08:33:07
%S A136346 560,736,1541,3201,5461,6816,7400,9976,11041,11408,13333,14981,15408,
%T A136346 15841,19521,21000,21505,25761,28616,30401,41536,45141,50440,51221,
%U A136346 52008,54405,56856,61920,63656,65416,69008,75525,76480,81345,82336,85345,87381,89441
%N A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.
%C A136346 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 A136346 B. D. Swan, <a href="/A136346/b136346.txt">Table of n, a(n) for n = 1..1800</a>
%H A136346 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>
%F A136346 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 A136346 Where Oc(n) = A000567(n) = n-th octagonal number:
%e A136346 a(1) = 560 = Oc(14) = 280 + 280 = Oc(10) + Oc(10).
%e A136346 a(2) = 736 = Oc(16) = 560 + 176 = Oc(14) + Oc(8).
%e A136346 a(3) = 1541 = Oc(23) = 1408 + 133 = Oc(22) + Oc(7).
%e A136346 a(4) = 3201 = Oc(33) = 2465 + 736 = Oc(29) + Oc(16).
%e A136346 a(5) = 5461 = Oc(43) = 2821 + 2640 = Oc(31) + Oc(30).
%t A136346 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 A136346 Cf. A000567, A009000, A089982, A136117, A136345.
%K A136346 easy,nonn
%O A136346 1,1
%A A136346 _Jonathan Vos Post_, Dec 25 2007
%E A136346 Corrected and edited by B. D. Swan (bdswan(AT)gmail.com), Dec 20 2008