A329013 -id:A329013 - cp's OEIS frontend

A329011 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

1, 2, 7, 26, 521, 434, 13021, 8138, 36169, 813802, 8138021, 3390842, 203450521, 508626302, 1695421007, 1589457194, 127156575521, 35321270978, 3178914388021, 3973642985026, 26490953233507, 198682149251302, 1986821492513021, 413921144273546, 49670537312825521

Offset: 1

Clark Kimberling, Nov 23 2019

a(n) is a strong divisibility sequence; i.e., gcd(a(h),a(k)) = a(gcd(h,k)).

			See Example in A327322.

Cf. A327320, A327321, A329012, A329013.

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];
r = Sqrt[5]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]];  (* A327322 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329011 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329012 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329013 *)
(* Peter J. C. Moses, Nov 01 2019 *)

A329012 a(n) = p(1,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

Original entry on oeis.org

1, 7, 52, 406, 16496, 27664, 1663936, 2081968, 18513664, 833245952, 16665967616, 13888655872, 1666655481856, 8333310963712, 55555495903232, 104166621927424, 16666663803355136, 9259258622967808, 1666666620853682176, 4166666620853682176, 55555555311219638272

Offset: 1

Clark Kimberling, Nov 23 2019

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).

			See Example in A327322.

Cf. A327320, A327321, A329011, A329013.

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];
r = Sqrt[5]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]];  (* A327322 *)
Table[f[x, n] /. x -> 0, {n, 1, 30}]   (* A329011 *)
Table[f[x, n] /. x -> 1, {n, 1, 30}]   (* A329012 *)
Table[f[x, n] /. x -> 2, {n, 1, 30}]   (* A329013 *)
(* Peter J. C. Moses, Nov 01 2019 *)

cp's OEIS Frontend

A329011 a(n) = p(0,n), where p(x,n) is the strong divisibility sequence of polynomials based on sqrt(5) as in A327322.

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