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EXAMPLE
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a(1)=1 because there is one solution (a,b,c) as (4,5,6) with 0<c<=10^1.
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MATHEMATICA
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Courtesy of Daniel Lichtblau of Wolfram Research: countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2, c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]], 4], #[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1, aa_}:>aa+1]; fax = Ceiling[Apply[Times, fax]/2]; total += fax; , {c, m}]; total]
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