A281858 Curious cubic identities based on the Armstrong number 370.

370, 336700, 333667000, 333366670000, 333336666700000, 333333666667000000, 333333366666670000000, 333333336666666700000000, 333333333666666667000000000, 333333333366666666670000000000, 333333333336666666666700000000000, 333333333333666666666667000000000000

Offset: 1

Author

Wolfdieter Lang, Feb 08 2017

Keywords

• nonn
• easy

Comments

See a comment in A067275, and the analog to the Armstrong number 153 = A005188(10) treated in A281857, 370 = A005188(11).

Examples

			n=1: 370 =  3^3 + 7^3 + 0^3; n=2: 336700 = 33^3 + 67^3 + (00)^3; n=3: 333667000 = 333^3 + 667^3 + (000)^3.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..333
• Index entries for linear recurrences with constant coefficients, signature (1110,-111000,1000000).

Crossrefs

Cf. A002277, A005188, A067275, A246057, A281857.

Programs

Mathematica

Table[FromDigits@ Join[ConstantArray[3, n], ReplacePart[ConstantArray[6, n], -1 -> 7], ConstantArray[0, n]], {n, 12}] (* Michael De Vlieger, Feb 08 2017 *)

PARI

Vec(10*x*(37 - 7400*x + 100000*x^2) / ((1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^30)) \\ Colin Barker, Feb 08 2017

Formula

a(n) = A002277(n)^3 + A067275(n+1)^3 + 0(n)^3, n >= 1, with 0(n) standing for n 0's.

From Colin Barker, Feb 08 2017: (Start)

G.f.: 10*x*(37 - 7400*x + 100000*x^2) / ((1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).

a(n) = 10^n*(1 + 10^n + 100^n) / 3.

a(n) = 1110*a(n-1) - 111000*a(n-2) + 1000000*a(n-3) for n>3. (End)