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%I #47 May 09 2023 05:28:10
%S 0,0,4,14,1,7,110,14,2,8,672,95,14,3,10,3948,568,84,14,4,11,23042,
%T 3325,492,81,14,5,12,134330,19394,2870,472,74,14,6,13,782964,113051,
%U 16730,2751,424,71,14,7,14,4563480,658924,97512,16034,2464,404,68,14,8,15
%N Table a(k,i), 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 - see below).
%C 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).
%C 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.
%C Also, a(k+1,2+n) - a(k,2+n) is divisible by A143608(n) for n>0 for all k.
%F a(k,0) equals the largest m such that m*(m+1)/2 is equal to or less than 14*k, A003056(14*k).
%F a(k,1) = k; a(k,2) = 14.
%F 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).
%F a(k,i) = 7*a(k,i-1)-7*a(k,i-2)+a(k,i-3). - _R. J. Mathar_, Jul 09 2012
%e 0, 0, 14, 110, 672, 3948, 23042,134330,782964,
%e 4, 1, 14, 95, 568, 3325, 19394,113051,658924,
%e 7, 2, 14, 84, 492, 2870, 16730, 97512,568344,
%e 8, 3, 14, 81, 472, 2751, 16034, 93453,544684,
%e 10, 4, 14, 74, 424, 2464, 14354, 83654,487564,
%e 11, 5, 14, 71, 404, 2345, 13658, 79595,463904,
%e 12, 6, 14, 68, 384, 2226, 12962, 75536,440244.
%e Note that 0*14, 14*110, 110*672, etc. are all triangular numbers and thus appear in row 0 of square array A001477; while, 1*14, 14*95, 95*568, 568*3325, etc. are all 4 more than a triangular number and appear in row 4 of square array A001477.
%p A182439 := proc(n,k)
%p if k = 0 then
%p A003056(14*n) ;
%p elif k = 1 then
%p n;
%p elif k = 2 then
%p 14;
%p else
%p 6*procname(n,k-1)-procname(n,k-2)+ 28+2*n-2-4*procname(n,0) ;
%p end if;
%p end proc: # _R. J. Mathar_, Jul 09 2012
%t highTri = Compile[{{S1,_Integer}}, Module[{xS0=0, xS1=S1}, While[xS1-xS0*(xS0+1)/2 > xS0, xS0++]; xS0]];
%t overTri = Compile[{{S2,_Integer}}, Module[{xS0=0, xS2=S2}, While[xS2-xS0*(xS0+1)/2 > xS0, xS0++]; xS2 - (xS0*(1+xS0)/2)]];
%t 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;
%t 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];
%t Sow[K2,f]; Sow[K3,g]; Sow[K4,h];
%t 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,
%t Sow[tab[[b,a+1-b]]]; b--]; a++]][[2,1]]; list5
%t (* Second program: *)
%t A003056[n_] := Floor[(Sqrt[1 + 8n] - 1)/2];
%t 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]];
%t Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}] (* _Jean-François Alcover_, May 09 2023, after _R. J. Mathar_ *)
%Y Cf. A001108, A001652, A182431, A182440, A182441, A143608.
%K nonn,tabl
%O 0,3
%A _Kenneth J Ramsey_, Apr 28 2012
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