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Listed by antidiagonals, array A[k,n] = digital root of n-th nonzero k-gonal number.
1

%I #22 Mar 30 2012 18:40:59

%S 1,1,3,1,4,6,1,5,9,1,1,6,3,7,6,1,7,6,4,7,3,1,8,9,1,8,9,1,1,9,3,7,9,6,

%T 4,9,1,1,6,4,1,3,7,1,9,1,2,9,1,2,9,1,2,9,1,1,3,3,7,3,6,4,3,9,1,3,1,4,

%U 6,4,4,3,7,4,9,1,4,6,1,5,9,1,5,9,1,5,9,1,5,9,1

%N Listed by antidiagonals, array A[k,n] = digital root of n-th nonzero k-gonal number.

%C The main diagonal begins 1, 4, 3, 1, 1, 6, 1, 7.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>

%e The array begins:

%e ===================================================

%e ....|n=1|n=2|n=3|n=4|n=5|n=6|n=7|n=8|comment

%e ====|===|===|===|===|===|===|===|===|==============

%e k=3.|.1.|.3.|.6.|.1.|.6.|.3.|.1.|.9.|

%e k=4.|.1.|.4.|.9.|.7.|.7.|.9.|.4.|.1.|

%e k=5.|.1.|.5.|.3.|.4.|.8.|.6.|.7.|.2.|A193090

%e k=6.|.1.|.6.|.6.|.1.|.9.|.3.|.1.|.3.|

%e k=7.|.1.|.7.|.9.|.7.|.1.|.9.|.4.|.4.|

%e k=8.|.1.|.8.|.3.|.4.|.2.|.6.|.7.|.5.|

%e k=9.|.1.|.9.|.6.|.1.|.3.|.3.|.1.|.6.|

%e k=10|.1.|.1.|.9.|.7.|.4.|.9.|.4.|.7.|

%e ===================================================

%t dr[n_]:=1+Mod[n-1,9]; A[k_,n_]:=dr[n*(n*(k-2)-(k-4))/2]; Flatten[Table[A[d-i+3,i],{d,13},{i,d}]] (* Ray Chandler, Aug 16 2011 *)

%Y Cf. A010888, A193090.

%K nonn,easy,tabl,base

%O 1,3

%A _Jonathan Vos Post_, Aug 15 2011

%E Corrected and extended by _Ray Chandler_, Aug 16 2011