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 A035507 Inverse Stolarsky array read by antidiagonals. 7
 1, 2, 4, 3, 7, 12, 5, 9, 20, 33, 6, 14, 25, 54, 88, 8, 17, 38, 67, 143, 232, 10, 22, 46, 101, 177, 376, 609, 11, 27, 59, 122, 266, 465, 986, 1596, 13, 30, 72, 156, 321, 698, 1219, 2583, 4180, 15, 35, 80, 190, 410, 842, 1829, 3193, 6764, 10945, 16, 41, 93, 211, 499 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The inverse Stolarky array is the dispersion of the sequence u given by u(n) = floor(n*x + x + n + 1 - x/2), where x=(golden ratio).  For a discussion of dispersions, see A191426. LINKS C. Kimberling, Interspersions C. Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society 117 (1993) 313-321. N. J. A. Sloane, Classic Sequences FORMULA The term in row n and column k of the inverse Stolarsky array has the following expression: a(n, k) = F(2k-3) - 1 - c1(n)*F(2k-4) + c2(n)*F(2k-2), where F is the Fibonacci sequence; c1(n)=1 if n=1, [(n-1)*tau] if n>1 (first column of the Inverse Stolarsky array) and c2(n) = c1(n) + 1 + floor((2*c1(n)+1)*tau/2) (second column of the Inverse Stolarsky array). tau = (1+sqrt(5))/2 and [] denotes the nearest integer function. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004 Also, the following recurrence holds: a(n, k) = 3*a(n, k-1) - a(n, k-2) + 1 with a(n, 1)=c1(n) and a(n, 2)=c2(n). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004 EXAMPLE Top left hand corner of array:   1,    4,   12,   33,   88,  232, ...   2,    7,   20,   54,  143,  376, ...   3,    9,   25,   67,  177,  465, ...   5,   14,   38,  101,  266,  698, ...   6,   17,   46,  122,  321,  842, ... MAPLE with(combinat, fibonacci): gold:=(1+sqrt(5))/2: c1:=n->piecewise(n<>1, round((n-1)*gold), 1): c2:=n->c1(n)+floor((2*c1(n)+1)*gold/2)+1: inv_stol:=(n, k)->fibonacci(2*k-3)-1-c1(n)*fibonacci(2*k-4)+c2(n)*fibonacci(2*k-2): seq(seq(inv_stol(n+1-k, k), k=1..n), n=1..11); inv_stol2:=(n, k)->(1+c0(n))*fibonacci(2*k-3)+(1+floor((2*c0(n)+1)*gold/2))*fibonacci(2*k-2)-1:seq(seq(inv_stol2(n+1-k, k), k=1..n), n=1..11);  # C. Ronaldo, Dec 31 2004 MATHEMATICA (* program generates the dispersion array T of the complement of increasing sequence f[n] *) r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *) c = 40; c1 = 12;   (* c=# cols of T, c1=# cols to show *) x = GoldenRatio; f[n_] :=  Floor[n*x + x + n + 1 - x/2] (* f(n) is complement of column 1 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,   Length[Union[list]]] rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]];  (* the array T *) TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* Inverse Stolarsky array, A035507 *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *) (* Program by Peter J. C. Moses, Jun 01 2011 *) CROSSREFS Cf. A035506 (Stolarsky array), A191426. Sequence in context: A207625 A238953 A238964 * A138612 A246680 A294244 Adjacent sequences:  A035504 A035505 A035506 * A035508 A035509 A035510 KEYWORD nonn,tabl,easy,nice AUTHOR EXTENSIONS More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004 Mathematica program, extended example, and comments from Clark Kimberling, Jun 03 2011 STATUS approved

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Last modified October 22 17:39 EDT 2019. Contains 328319 sequences. (Running on oeis4.)