A182102 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)).

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

Offset: 0

Author

Kenneth J Ramsey, Apr 11 2012

Keywords

• tabl
• sign

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A182189, A182190, A182431.

Programs

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)]];
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

Formula

a(k,0) = Floor[(Sqrt[1 + 112*k] - 1)/2]

a(k,i) = A003056(A182431(k,i+1)*A182431(k,i+2) - A002262(14*k)) or

-1 - A003056(A182431(k,i+1)*A182431(k,i+2) - A002262(14*k)) .

For i>2, a(k,i) = 7*a(k,i-1)-7*a(k,i-2)+a(k,i-3).