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Does the following number triangle occur in the literature? Consider the sequences $(1,{c_1},{c_2}, \cdots ) = (1,2,4,5,7,8, \cdots )$ of all integers not divisible by 3 and $\left( {1,{d_1},{d_2}, \cdots } \right) = (1,1,2,2,3,3, \cdots )$ and form the table

$\begin{array}{*{20}{c}} {}&{}&{}&{}&{}&1&{}&{}&{}&{}&{}\\ {}&{}&{}&{}&1&{{c_1}}&1&{}&{}&{}&{}\\ {}&{}&{}&1&{{c_1}}&{{c_2}}&{{d_1}}&1&{}&{}&{}\\ {}&{}&1&{{c_1}}&{{c_2}}&{{c_3}}&{{d_2}}&{{d_1}}&1&{}&{}\\ {}&1&{{c_1}}&{{c_2}}&{{c_3}}&{{c_4}}&{{d_3}}&{{d_2}}&{{d_1}}&1&{}\\ 1&{{c_1}}&{{c_2}}&{{c_3}}&{{c_4}}&{{c_5}}&{{d_4}}&{{d_3}}&{{d_2}}&{{d_1}}&1 \end{array}$

It has the property that the sum of the $k$’th row is $(k+1)^2$ and the alternating sums of the rows are $1,0,3,0,5,0,7,....$

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  • $\begingroup$ I haven't seen it before (although that is no guarantee that it isn't studied), but I think the numbers are to specific and in a non-trivial ordering for this to be some general triangle (like Pascals one) $\endgroup$ – Ragnar Jun 19 '14 at 16:59
  • $\begingroup$ A good reference for this kind of question is the famous Encyclopedia of Integer Sequences: oeis.org $\endgroup$ – guaraqe Jun 20 '14 at 0:15

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