A centered octagonal number is a centeredfigurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered octagonal number for n is given by the formula
8Tn _ 1 + 1
where T is a regular triangular number, or much more simply, by squaring the odd numbers:
All centered octagonal numbers are odd, and in base 10 one can notice the one's digits follow the pattern 1-9-5-9-1.
Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number.
A centeredoctagonalnumber is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.
All centeredoctagonalnumbers are odd, and in base 10 one can notice the one's digits follow the pattern 1-9-5-9-1.
Calculating Ramanujan's tau function on a centeredoctagonalnumber yields an odd number, whereas for any other number the function yields an even number.
st triangular number is represented by the white triangles, the nth triangular number is represented by the fl triangles, and the total number of triangles is the square number
The following numbers cannot be represented using fewer than five distinct squares: 55, 88, 103, 132, 172, 176, 192, 240, 268, 288, 304, 368, 384, 432, 448, 496, 512, and 752, together with all numbers obtained by multiplying these numbers by a power of 4.
The numbers that are not the difference of two squares are 2, 6, 10, 14, 18,...
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