A067870 - OEIS

%I #26 Feb 27 2023 04:00:28

%S 7,13,27,33,47,53,67,73,87,93,107,113,127,133,147,153,167,173,187,193,

%T 207,213,227,233,247,253,267,273,287,293,307,313,327,333,347,353,367,

%U 373,387,393,407,413,427,433,447,453,467,473,487,493,507,513,527,533

%N Numbers k such that k and 3^k end with the same digit.

%C Also numbers k such that k^k ends with 3. - _Bruno Berselli_, Dec 11 2018

%C Numbers congruent to {7, 13} mod 20. - _Amiram Eldar_, Feb 27 2023

%H Colin Barker, <a href="/A067870/b067870.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(2*n+1) = 20*n-13, a(2*n) = 20*n-7.

%F a(n) = 20*(n-1)-a(n-1) for n>1, a(1)=7. - _Vincenzo Librandi_, Aug 08 2010

%F From _Colin Barker_, Apr 06 2020: (Start)

%F G.f.: x*(7 + 6*x + 7*x^2) / ((1 - x)^2*(1 + x)).

%F a(n) = -5 - 2*(-1)^n + 10*n for n>0.

%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. (End)

%F Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/20)*Pi/20. - _Amiram Eldar_, Feb 27 2023

%e 3^13 = 1594323 hence 13 is in the sequence.

%t LinearRecurrence[{1, 1, -1}, {7, 13, 27}, 50] (* _Amiram Eldar_, Feb 27 2023 *)

%o (PARI) a(n) = (5*(2*n-1)*(-1)^n - 2)*(-1)^n; \\ _Jinyuan Wang_, Apr 06 2020

%o (PARI) Vec(x*(7 + 6*x + 7*x^2) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ _Colin Barker_, Apr 06 2020

%K nonn,base,easy

%O 1,1

%A _Benoit Cloitre_, Mar 07 2002