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We spoke to a very helpful man at YESCO (Young Electric Sign Company, which owns the sign but leases it to Clark County) and we have great news for you. Just like the sign itself, which is uncopyrighted (hence all the imitations and spinoffs), the shade of blue you're after is anything but proprietary. Its official name, as defined by the international Pantone standard, is Process Blue. It's one of the fundamental standard colors and, as a result, does not have a number in the PMS (Pantone Paint Matching System) because it's 100% cyan.
You'll find this shade on the various ColorCheckers sold by Pantone -- a checkerboard array of 24 scientifically prepared colored squares in a wide range of colors that represent natural objects of special interest, such as human skin, foliage and -- in the case of Process Blue -- blue sky (see photograph below, although we have a sneaking suspicion that this picture has been slightly enhanced...) The ColorChecker squares are not only the same color as their counterparts, but also reflect light the same way in all parts of the visible spectrum. This unique feature means they'll match the colors of natural objects under any illumination and with any color reproduction process.
Process Blue should therefore be available in whatever paint type you need (for metal, brickwork, wood, ceramic etc.) -- here's a link to the Process Blue page on the Pantone website, which should enable you to find all the information you need.
If you're "theming" your office, you might perhaps also want to invest in an authentic lightbulb from the Welcome sign, available for $69 each from Official Las Vegas Light.
(Image appears courtesy of the Las Vegas News Bureau)
