cp's OEIS Frontend

A271527 a(n) = 1000^n + 1.

%I A271527 #17 Dec 07 2019 12:18:28
%S A271527 2,1001,1000001,1000000001,1000000000001,1000000000000001,
%T A271527 1000000000000000001,1000000000000000000001,1000000000000000000000001,
%U A271527 1000000000000000000000000001,1000000000000000000000000000001,1000000000000000000000000000000001
%N A271527 a(n) = 1000^n + 1.
%C A271527 All terms in this sequence are palindromes (A002113).
%C A271527 Also, A062395 written in base 2 (see example).
%C A271527 a(n) minus one gives the number of nodes at n-th level of a 1000-ary tree.
%C A271527 More generally, the ordinary generating function for sequences of the form k^n + m, is (1 + m - (1 + k*m)*x)/((1 - x)*(1 - k*x)), and the exponential generating function is exp(k*x) + m*exp(x).
%H A271527 Ilya Gutkovskiy, <a href="/A271527/a271527.pdf">Examples of the ordinary generating function for the sequences of the form k^n + m</a>
%H A271527 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1001,-1000)
%F A271527 G.f.: (2 - 1001*x)/((1 - x)*(1 - 1000*x)).
%F A271527 E.g.f.: exp(1000*x) + exp(x).
%F A271527 a(n) = 1001*a(n-1) - 1000*a(n-2).
%F A271527 a(n) = A060365(n) + 1.
%F A271527 a(n) = A000533(3n), n>0.
%F A271527 a(n) = A007088(A062395(n)).
%F A271527 A007953(a(n)) = A007395(n).
%F A271527 A000035(a(n)) = A057427(n).
%F A271527 Sum_{n>=0} 1/a(n) = 0.501000001999002...
%F A271527 Lim_{n->infinity} a(n + 1)/a(n) = 1000.
%e A271527 a(n), n>0, is the binary representation of A062395(n)
%e A271527 n  ------------------------------------------
%e A271527 0  2........................................2
%e A271527 1  1001.....................................9
%e A271527 2  1000001.................................65
%e A271527 3  1000000001.............................513
%e A271527 4  1000000000001.........................4097
%e A271527 5  1000000000000001.....................32769
%e A271527 6  1000000000000000001.................262145
%e A271527 7  1000000000000000000001.............2097153
%e A271527 8  1000000000000000000000001.........16777217
%e A271527 9  1000000000000000000000000001.....134217729
%t A271527 Table[1000^n + 1, {n, 0, 11}]
%t A271527 LinearRecurrence[{1001, -1000}, {2, 1001}, 12]
%o A271527 (PARI) x='x+O('x^99); Vec((2-1001*x)/((1-x)*(1-1000*x))) \\ _Altug Alkan_, Apr 09 2016
%o A271527 (Python)
%o A271527 for n in range(0,10**4):print(1000**n+1)
%o A271527 # _Soumil Mandal_, Apr 10 2016
%Y A271527 Cf. A000035, A000533, A007088, A007395, A007953, A057427, A060365, A062395, A152756.
%K A271527 nonn,base,easy
%O A271527 0,1
%A A271527 _Ilya Gutkovskiy_, Apr 09 2016