A329012 - OEIS

%I #7 Dec 07 2019 01:43:59

%S 1,7,52,406,16496,27664,1663936,2081968,18513664,833245952,

%T 16665967616,13888655872,1666655481856,8333310963712,55555495903232,

%U 104166621927424,16666663803355136,9259258622967808,1666666620853682176,4166666620853682176,55555555311219638272

%N a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

%C a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)). Conjecture: there is no upper bound for the number of consecutive equal digits among numbers in this sequence, as suggested, for example, by 34 straight 1's in a(96) and 38 straight 6's in a(97).

%e See Example in A327322.

%t c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@Variables /@ #1 &)[List @@ poly], 0], poly];

%t r = Sqrt[5]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];

%t Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]];  (* A327322 *)

%t Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329011 *)

%t Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329012 *)

%t Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329013 *)

%t (* _Peter J. C. Moses_, Nov 01 2019 *)

%Y Cf. A327320, A327321, A329011, A329013.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Nov 23 2019