OFFSET
0,3
COMMENTS
This is a square array related to the square array of nonnegative integers, A001477. Each row k contains the positive argument of the largest triangular number equal to or less than 14*k in column 0 and a corresponding 2nd-order recursive sequence G(k) in the rest of the row. Each second-order recursive series term G(i) corresponds to 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. If the product is less than (r^2 + 3*r -2)/2 then assuming the row can take negative indices, the product can still be said to lie in the same row r. For instance, 0, 1, 3, and 6 are each a triangular number and appear as the first 4 terms of row 0 of square array A001477. Note that in the next row and to the left of the 1, 3, and 6 are 2, 4 and 7 so going down a row and to the left in the square array increases the value by 1. Going down to the next row and to the left again would be 3, 5, and 8 so 3 which is 2 more than 1 would be in row 2 if that row were made to take the indices (2,-1).
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 example, a(k,2+n) - a(k,2+n) = A001652(n) for n=0,1,2,3,... whereever a(k+1,0) - a(k,0) = 1.
Also, a(k+1,2+n) - a(k,2+n) is divisible by A143608(n) for n>0 for all k.
FORMULA
a(k,0) equals the largest m such that m*(m+1)/2 is equal to or less than 14*k, A003056(14*k).
a(k,1) = k; a(k,2) = 14.
For i > 2, a(k,i) = 6*a(k,i-1) -a (k,i-2) + G_k where G_k = 28 + 2*k - 2 - 4*a(k,0).
a(k,i) = 7*a(k,i-1)-7*a(k,i-2)+a(k,i-3). - R. J. Mathar, Jul 09 2012
EXAMPLE
0, 0, 14, 110, 672, 3948, 23042,134330,782964,
4, 1, 14, 95, 568, 3325, 19394,113051,658924,
7, 2, 14, 84, 492, 2870, 16730, 97512,568344,
8, 3, 14, 81, 472, 2751, 16034, 93453,544684,
10, 4, 14, 74, 424, 2464, 14354, 83654,487564,
11, 5, 14, 71, 404, 2345, 13658, 79595,463904,
12, 6, 14, 68, 384, 2226, 12962, 75536,440244.
MAPLE
A182439 := proc(n, k)
if k = 0 then
A003056(14*n) ;
elif k = 1 then
n;
elif k = 2 then
14;
else
6*procname(n, k-1)-procname(n, k-2)+ 28+2*n-2-4*procname(n, 0) ;
end if;
end proc: # R. J. Mathar, Jul 09 2012
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; tab=Reap[While[K1<16, J1=highTri[m*K1]; X = 2*(m+K1-(J1*2+1)); K2 = (6 m - K1 + X); K3 = 6 K2 - m + 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[K1, d]; Sow[m, 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[tab[[b, a+1-b]]]; b--]; a++]][[2, 1]]; list5
(* Second program: *)
A003056[n_] := Floor[(Sqrt[1 + 8n] - 1)/2];
T[n_, k_] := Switch[k, 0, A003056[14n], 1, n, 2, 14, _, 6T[n, k-1] - T[n, k-2] + 28 + 2n - 2 - 4T[n, 0]];
Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}] (* Jean-François Alcover, May 09 2023, after R. J. Mathar *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Kenneth J Ramsey, Apr 28 2012
STATUS
approved