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A182440
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Table, read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array).
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4
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0, 14, 4, 0, 14, 7, 16, 1, 14, 8, 126, 40, 2, 14, 10, 770, 287, 60, 3, 14, 11, 4524, 1730, 420, 72, 4, 14, 12, 26404, 10141, 2522, 497, 88, 5, 14, 13, 153930, 59164, 14774, 2978, 602, 100, 6, 14, 14, 897206
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OFFSET
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0,2
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COMMENTS
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This is a table related to A001477 interpreted as a square array of the onnegative integers (A001477). Each row k contains A003056(14*k) in column 0 and a corresponding 2nd order recursive sequence G(k) beginning at position a(k,1) such that G(i) = a(k,i+1). If the product 14*k appears in row "r" of the square array A001477, then the product of adjacent terms G(i)*G(i+1) if greater than (r^2 + 3*r - 2)/2, is always in row "r" of square array A001477.
A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For instance, a(k+1,i+1)-a(k,i+1 = A210695(i) if a(k + 1,0) - a(k,0) = 1; while a(k+1,i+1)-a(k,i+1 = A001108(i) if a(k+1,0) - a(k,0) = 0.
A related property is that a(k+1,1+n) - a(k,1+n) is divisible by A143608(n) for all k.
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LINKS
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FORMULA
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a(k,0) equals the positive argument of the largest triangular number equal to or less than 14*k (= A214206(k) which = A003056(14*k)).
a(k,1) equals 14; a(k,2) = k.
For i > 2, a(k,i) = 6*a(k,i-1) -a (k,i-2) + G_k where G_k is a constant equal to 28 + 2*k + 2 + 4*A214206(k).
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EXAMPLE
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For i = 1,2,3,4 ..., a(1,i)*a(1,i+1) = 14*1,1*40,40*287,287*1730, ...; and, each product is 4 more than a triangular number and thus lies in row 4 of square array A001477.
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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)]];
K1 = 0;
m = 14; table=Reap[While[K1<16, 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; K5 = 6 K4 -K3 + X; K6 = 6*K5 - K4 + X; K7 = 6*K6-K5+X; K8 = 6*K7-K6+X; Sow[J1, c]; Sow[m, d];
Sow[K1, e]; Sow[K2, f]; Sow[K3, g]; Sow[K4, h];
Sow[K5, i]; Sow[K6, j]; Sow[K7, k]; Sow[K8, l];
K1++]][[2]];
a=1;
list5 = Reap[While[a<11, b=a;
While[b>0, Sow[table[[b, a+1-b]]]; b--]; a++]][[2, 1]];
list5
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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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