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A182102
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Table of triangular arguments such that if A002262(14*k) = "r" then the product A182431(k,i + 1) * A182431(k,i + 2) equals "r" + A000217(a(k,i)).
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3
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0, -1, 4, 48, -1, 7, 343, 16, 0, 8, 2064, 123, -1, 3, 10, 12095, 748, 0, 12, 5, 11, 70560, 4391, 7, 71, 10, 8, 12, 411319, 25624, 48, 416, 45, 23, 11, 13, 2397408, 149379, 287, 2427, 250, 116, 36, 14, 14, 13973183, 870676
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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 A182189 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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FORMULA
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a(k,0) = Floor[(Sqrt[1 + 112*k] - 1)/2]
For i>2, a(k,i) = 7*a(k,i-1)-7*a(k,i-2)+a(k,i-3).
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EXAMPLE
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The table begins as follows:
0 -1 48 343 2064 12095 70560 ...
4 -1 16 123 748 4391 25624 ...
7 0 -1 0 7 48 287 ...
8 3 12 71 416 2427 14148 ...
10 5 10 45 250 1445 8410 ...
11 8 23 116 659 3824 22271 ...
12 11 36 187 1068 6203 36132 ...
13 14 49 258 1477 8582 49993 ...
14 17 62 329 1886 10961 63854 ...
15 20 75 400 2295 13340 77715 ...
16 23 88 471 2704 15719 91576 ...
17 26 101 542 3113 18098 105437 ...
17 30 129 710 4097 23838 138897 ...
...
For n > 1, a(k,n) = 6*a(k,n-1) - a(k,n-2) + G_k where G_k is dependent on k.
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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} -> 1, {1, 12} -> 1}];
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<11, b=a; 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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AUTHOR
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STATUS
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approved
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