A281859 - OEIS

%I #20 May 10 2018 17:03:18

%S 407,340067,334000667,333400006667,333340000066667,333334000000666667,

%T 333333400000006666667,333333340000000066666667,

%U 333333334000000000666666667,333333333400000000006666666667,333333333340000000000066666666667,333333333334000000000000666666666667

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

%C See a comment in A093137.

%H Colin Barker, <a href="/A281859/b281859.txt">Table of n, a(n) for n = 1..333</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1111,-112110,1111000,-1000000).

%F From _Colin Barker_, Feb 08 2017: (Start)

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

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

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

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

%t 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 *)

%t LinearRecurrence[{1111,-112110,1111000,-1000000},{407,340067,334000667,333400006667},20] (* _Harvey P. Dale_, May 10 2018 *)

%o (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

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Feb 08 2017