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A265104
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a(n) = A265100(n+1) - 6, n >= 1.
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5
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8, 26, 35, 80, 89, 107, 116, 242, 251, 269, 278, 323, 332, 350, 359, 728, 737, 755, 764, 809, 818, 836, 845, 971, 980, 998, 1007, 1052, 1061, 1079, 1088, 2186, 2195, 2213, 2222, 2267, 2276, 2294, 2303, 2429, 2438, 2456, 2465, 2510, 2519, 2537
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OFFSET
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1,1
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COMMENTS
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In the following, let "gap" and "gap number" be as defined in A265100, and let C(m) denote the m-th Catalan number (A000108).
Conjecture 1: The sequence contains all possible gap numbers.
Conjecture 2: For any gap G, the order |G| of G is the constant |G| = 6.
Conjecture 3: If g is a gap number, then 3*g + 2 is a gap number.
Conjecture 4: If C(m) =!= 0 (mod 3), then C(3*m+1) =!= 0 (mod 3) (=!= means "not congruent") or, what is the same thing, if m lies in a gap, then 3*m + 1 lies in a gap.
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LINKS
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FORMULA
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MATHEMATICA
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a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265100[n_] := 9*a005836[n] + 5; a265104[n_] := a265100[n+1] - 6; Table[a265104[n], {n, 46}]
(* Or: *)
a007814[x_] := IntegerExponent[x, 2]; a003602[x_] := (1 + x/2^a007814[x])/2; a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265100[n_] := 9*a005836[n] + 5; a265104[n_] := (3^(a007814[n] + 2) - 3)/2 + a265100[2^(a007814[n])*(2*a003602[n] - 1)]; Table[a265104[n], {n, 46}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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