A281859 Curious identities based on the Armstrong number 407 = A005188(13).

407, 340067, 334000667, 333400006667, 333340000066667, 333334000000666667, 333333400000006666667, 333333340000000066666667, 333333334000000000666666667, 333333333400000000006666666667, 333333333340000000000066666666667, 333333333334000000000000666666666667

Offset: 1

Author

Wolfdieter Lang, Feb 08 2017

Keywords

• nonn
• easy

Comments

See a comment in A093137.

Examples

			Curious cubic identities: 407 = 4^3 + 0^3 + 7^3, 340067 = 34^3 + (00)^3 + 67^3, 334000677 = 334^3 + (000)^3 + 677^3, ...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..333
• Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000).

Programs

Mathematica

Table[FromDigits@ Join[ReplacePart[ConstantArray[3, n], -1 -> 4], ConstantArray[0, n], ReplacePart[ConstantArray[6, n], -1 -> 7]], {n, 12}] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{1111,-112110,1111000,-1000000},{407,340067,334000667,333400006667},20] (* Harvey P. Dale, May 10 2018 *)

PARI

Vec(x*(407 - 112110*x + 1815000*x^2 - 2000000*x^3) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^30)) \\ Colin Barker, Feb 08 2017

Formula

From Colin Barker, Feb 08 2017: (Start)

G.f.: x*(407 - 112110*x + 1815000*x^2 - 2000000*x^3) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).

a(n) = (1 + 2^(1+n)*5^n + 2^(1+2*n)*25^n + 1000^n) / 3.

a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4) for n>4. (End)