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A182118
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Table of triangular arguments such that if A002262(14*k) = "r" then the product A182440(k,i + 1) *A182440(k,i + 2) equals "r" + a(k,i)*(a(k,i)+1)/2.
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-1, 0, -5, 63, 8, -8, 440, 151, 15, -9, 0, 996, 224, 20, -11, 0, 0, 1455, 267, 26, -12, 0, 0, 0, 1720, 325, 31, -13, 0, 0, 0, 0, 2082, 368, 36, -14, 0, 0, 0, 0, 0, 2347, 411, 41, -15, 0, 0, 0, 0, 0, 0, 2612, 454, 46
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OFFSET
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0,3
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COMMENTS
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It is noted that the difference between adjacent rows of the respective elements, depends on the difference between the elements of column 0 in the respective rows. It is apparent that the series of differences corresponding to a difference of d in column 0, i.e. A(k+1,0) - A(k,0) = d, is defined as follows: D(0) = d, D(1) = 4 - d, D(n) = 6*D(n-1) - D(n-2) -8*d + 4. The sequence of differences corresponding to a difference of -1 or 0 in column 0 form related series A182191 and A182190.
The Mathematica program below first calculates an array containing only the first four nonnegative triangular arguments P of each row then changes at most 2 of the arguments to the corresponding negative value, N = -P -1 in order to obtain the relation a(k,i) -7*a(k,i+1) + 7*a(k,i+2) - a(k,i+3) = 0, then chooses the appropriate argument to continue this relationship with the remainder of the row. In this way, the sequence is finally determined. Thus in this table a few 0's have been changed to -1.
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LINKS
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MATHEMATICA
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highTri = Compile[{{S1, _Integer}}, Module[{xS0=0, xS1=S1},
While[xS1-xS0*(xS0+1)/2>xS0, xS0++];
xS0]];
overTri = Compile[{{S2, _Integer}}, Module[{xS0=0, xS2=S2},
While[xS2-xS0*(xS0+1)/2>xS0, xS0++];
xS2 - (xS0*(1+xS0)/2)]];
tt = SparseArray[{{12, 1} -> 0, {1, 12} -> 0}];
K1 = 0;
m = 14; While[K1<12, J1=highTri[m*K1]; X =2*(m+K1+(J1*2+1));
K2 = 6 K1 - m + X; K3 = 6 K2 - K1 + X; K4 = 6 K3 - K2 + X;
o = overTri[m*K1]; tt[[1, K1+1]] =highTri[m*K1];
tt[[2, K1+1]] = highTri[K1*K2-o]; tt[[3, K1+1]] = highTri[K2*K3-o]; tt[[4, K1+1]] = highTri[K3*K4-o];
K1++]; k = 1;
While[k<13, z = 1; xx = 99; While[z<5 && xx == 99,
If[tt[[1, k]]+ 7 tt[[3, k]] - 7 tt[[2, k]] - tt[[4, k]] == 0, Break[]];
If[z == 1, t = -tt[[z, k]]-1; tt[[z, k]] = t, s = -tt[[z-1, k]]-1; tt[[z-1, k]]=s; t =-tt[[z, k]]-1]; tt[[z, k]] = t;
w = 1; While[w<5 && xx == 99, If[tt[[1, k]]+ 7 tt[[3, k]] - 7 tt[[2, k]] - tt[[4, k]] == 0, xx =0; Break[]]; If[w==z, w++];
t=-tt[[w, k]] - 1; tt[[w, k]]=t; If[tt[[1, k]]+ 7 tt[[3, k]] - 7 tt[[2, k]] - tt[[4, k]] == 0, xx =0; Break[],
t = -tt[[w, k]] - 1]; tt[[w, k]] = t; w++]; z++]; cc = tt[[1, k]] -6 tt[[2, k]] + tt[[3, k]]; p = 5; While[p < 14-k,
tt[[p, k]] = 6 tt[[p-1, k]] - tt[[p-2, k]] + cc; p++]; k++];
a=1; list2 = Reap[While[a<12, b=a; While[b>4, Sow[0]; b--]; While[b>0, Sow[tt[[b, a+1-b]]]; b--]; a++]][[2, 1]]; list2
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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