OFFSET
0,2
COMMENTS
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.
FORMULA
EXAMPLE
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.
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Kenneth J Ramsey, Apr 28 2012
STATUS
approved