Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, 8 to 10 minutes. Drain and place in a large serving bowl.
Place a grill pan over medium-high heat or preheat a gas or charcoal grill.
Brush the corn with the oil, using a pastry brush. Grill, turning frequently, until the corn is tender, about 10 minutes. Allow the corn to cool slightly. Remove the kernels with a large knife and add to the serving bowl.
For the pesto: In a food processor, combine 2 cups artichokes, oil, walnuts, oregano, parsley, lemon juice, garlic, lemon zest, salt and pepper. Blend, scraping down the sides of the bowl, as needed, until the mixture is thick and smooth.
Chop the remaining 1 cup artichoke hearts into 3/4-inch pieces and add to the serving bowl. Add the pesto, tomatoes and Parmesan. Toss until all the ingredients are coated. Add salt and pepper to taste and serve.
Cook's Note: To toast the walnuts, preheat the oven to 350 degrees F. Arrange the walnuts in a single layer on a baking sheet. Bake until lightly toasted, 8 to 10 minutes. Cool completely before using.
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