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 A193922 Mirror of the triangle A193921. 2
 1, 1, 1, 2, 2, 1, 4, 4, 3, 2, 7, 7, 6, 5, 3, 12, 12, 11, 10, 8, 5, 20, 20, 19, 18, 16, 13, 8, 33, 33, 32, 31, 29, 26, 21, 13, 54, 54, 53, 52, 50, 47, 42, 34, 21, 88, 88, 87, 86, 84, 81, 76, 68, 55, 34, 143, 143, 142, 141, 139, 136, 131, 123, 110, 89, 55, 232, 232, 231 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A193922 is obtained by reversing the rows of the triangle A193921. Also, triangle read by rows: T(n,k) = Fibonacci(n+2) - Fibonacci(k+1) with T(0,0) = 1, 0 <= k <= n. - Arkadiusz Wesolowski, Aug 05 2012 LINKS Arkadiusz Wesolowski, Rows n = 0..140 of triangle, flattened FORMULA Write w(n,k) for the triangle at A193921. The triangle at A193922 is then given by w(n,n-k). G.f.: 1-(x*y-y-x)/((x^2+x-1)*(y^2+y-1)). - Vladimir Kruchinin, Jan 12 2024 EXAMPLE First six rows: 1 1 1 2 2 1 4 4 3 2 7 7 6 5 3 12 12 11 10 8 5 MATHEMATICA z = 12; p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}]; q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1; t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0; w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1 g[n_] := CoefficientList[w[n, x], {x}] TableForm[Table[Reverse[g[n]], {n, -1, z}]] Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193921 *) TableForm[Table[g[n], {n, -1, z}]] Flatten[Table[g[n], {n, -1, z}]] (* this sequence *) Factor[w[7, x]] Factor[w[8, x]] Table[Expand[p[n, x]], {n, 0, 4}] Table[Expand[q[n, x]], {n, 0, 4}] Prepend[Flatten@Table[Fibonacci[n + 2] - Fibonacci[k + 1], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Aug 05 2012 *) CROSSREFS Cf. A193921. Sequence in context: A182222 A225639 A110664 * A319534 A061436 A214095 Adjacent sequences: A193919 A193920 A193921 * A193923 A193924 A193925 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Aug 09 2011 STATUS approved

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Last modified April 17 12:12 EDT 2024. Contains 371763 sequences. (Running on oeis4.)