A281860 Curious identities based on the Armstrong number 371 = A005188(12).

371, 336701, 333667001, 333366670001, 333336666700001, 333333666667000001, 333333366666670000001, 333333336666666700000001, 333333333666666667000000001, 333333333366666666670000000001, 333333333336666666666700000000001, 333333333333666666666667000000000001

Offset: 1

Author

Wolfdieter Lang, Feb 08 2017

Keywords

• nonn
• easy

Comments

See a comment in A067275.

Examples

			n=1: 371 = 3^3 + 7^3 + 1^3;
n=2: 336701 = 33^3 + 67^3 + (01)^3;
n=3: 333667001 = 333^3 + 667^3 + (001)^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).

Crossrefs

Cf. A002277, A067275, A281857, A281858, A281859.

Programs

Mathematica

LinearRecurrence[{1111,-112110,1111000,-1000000},{371,336701,333667001,333366670001},20] (* Harvey P. Dale, May 28 2024 *)

PARI

Vec(x*(371 - 75480*x + 1185000*x^2 - 2000000*x^3) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^30)) \\ Colin Barker, Feb 09 2017

Formula

a(n) = A002277(n) * 10^(2*n) + A067275(n+1) * 10^n + 0(n-1)1, where 0(n-1)1 stands for n-1 0's followed by a 1, for n >= 1.

a(n) = A002277(n)^3 + A067275(n+1)^3 + (0(n-1)1)^3.

From Colin Barker, Feb 09 2017: (Start)

G.f.: x*(371 - 75480*x + 1185000*x^2 - 2000000*x^3)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).

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

a(n) = (3 + 10^n + 100^n + 1000^n)/3. (End)