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 A182439 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). 5
 0, 0, 4, 14, 1, 7, 110, 14, 2, 8, 672, 95, 14, 3, 10, 3948, 568, 84, 14, 4, 11, 23042, 3325, 492, 81, 14, 5, 12, 134330, 19394, 2870, 472, 74, 14, 6, 13, 782964, 113051, 16730, 2751, 424, 71, 14, 7, 14, 4563480, 658924 (list; table; graph; refs; listen; history; text; internal format)
 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. LINKS 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. 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. 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 CROSSREFS Cf. A001108, A001652, A182431, A182440, A182441, A143608. Sequence in context: A317267 A219366 A270638 * A182441 A107775 A003117 Adjacent sequences:  A182436 A182437 A182438 * A182440 A182441 A182442 KEYWORD nonn,tabl AUTHOR Kenneth J Ramsey, Apr 28 2012 STATUS approved

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Last modified August 16 05:55 EDT 2022. Contains 356160 sequences. (Running on oeis4.)