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%I #40 Oct 30 2023 14:36:58
%S 0,1,0,0,1,0,1,1,1,0,0,3,2,1,0,1,4,5,3,1,0,0,7,8,7,4,1,0,1,9,13,12,9,
%T 5,1,0,0,13,18,19,16,11,6,1,0,1,16,25,27,25,20,13,7,1,0,0,21,32,37,36,
%U 31,24,15,8,1,0,1,25,41,48,49,45,37,28,17,9,1,0
%N Square array read by antidiagonals with T(n,k) = k*n^2/4+(k-4)*((-1)^n-1)/8, n>=0, k>=0.
%C Also, if k >= 2 and m = 2*k, then column k lists the numbers of the form k*n^2 and the centered m-gonal numbers interleaved.
%C For k >= 3, this is also a table of concentric polygonal numbers. Column k lists the concentric k-gonal numbers.
%C It appears that the first differences of column k are the numbers that are congruent to {1, k-1} mod k, if k >= 3.
%H Muniru A Asiru, <a href="/A195040/b195040.txt">Rows n=0..100, flattened</a>
%e Array begins:
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
%e 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
%e 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
%e 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, ...
%e 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, ...
%e 1, 13, 25, 37, 49, 61, 73, 85, 97, 109, ...
%e 0, 16, 32, 48, 64, 80, 96, 112, 128, 144, ...
%e 1, 21, 41, 61, 81, 101, 121, 141, 161, 181, ...
%e 0, 25, 50, 75, 100, 125, 150, 175, 200, 225, ...
%e ...
%p A195040 := proc(n,k)
%p k*n^2/4+((-1)^n-1)*(k-4)/8 ;
%p end proc:
%p for d from 0 to 12 do
%p for k from 0 to d do
%p printf("%d,",A195040(d-k,k)) ;
%p end do:
%p end do; # _R. J. Mathar_, Sep 28 2011
%t t[n_, k_] := k*n^2/4+(k-4)*((-1)^n-1)/8; Flatten[ Table[ t[n-k, k], {n, 0, 11}, {k, 0, n}]] (* _Jean-François Alcover_, Dec 14 2011 *)
%o (GAP) nmax:=13;; T:=List([0..nmax],n->List([0..nmax],k->k*n^2/4+(k-4)*((-1)^n-1)/8));; b:=List([2..nmax],n->OrderedPartitions(n,2));;
%o a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][2]][b[i][j][1]]))); # _Muniru A Asiru_, Jul 19 2018
%Y Rows n: A000004 (n=0), A000012 (n=1), A001477 (n=2), A005408 (n=3), A008586 (n=4), A016921 (n=5), A008591 (n=6), A017533 (n=7), A008598 (n=8), A215145 (n=9), A008607 (n=10).
%Y Columns k: A000035 (k=0), A004652 (k=1), A000982 (k=2), A077043 (k=3), A000290 (k=4), A032527 (k=5), A032528 (k=6), A195041 (k=7), A077221 (k=8), A195042 (k=9), A195142 (k=10), A195043 (k=11), A195143 (k=12), A195045 (k=13), A195145 (k=14), A195046 (k=15), A195146 (k=16), A195047 (k=17), A195147 (k=18), A195048 (k=19), A195148 (k=20), A195049 (k=21), A195149 (k=22), A195058 (k=23), A195158 (k=24).
%K nonn,tabl,easy
%O 0,12
%A _Omar E. Pol_, Sep 27 2011
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