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Reference.org
Centered triangular number
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<p>A centered (or centred) triangular number is a <a href="/facts/Centered_number/FAB00ppP">centered</a> <a href="/facts/Figurate_number/f1gdGcja">figurate number</a> that represents an <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a> with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. </p><p>This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to n {\displaystyle n} . </p><p>The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). </p>