This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271390 #29 Feb 24 2021 09:10:11
%S A271390 1,3,5,343,729,161051,371293,170859375,410338673,322687697779,
%T A271390 794280046581,952809757913927,2384185791015625,4052555153018976267,
%U A271390 10260628712958602189,23465261991844685929951,59938945498865420543457,177482997121587371826171875,456487940826035155404146917
%N A271390 a(n) = (2*n + 1)^(2*floor((n-1)/2) + 1).
%C A271390 All members are odd, therefore:
%C A271390 ........................
%C A271390 | k | a(n) mod k |
%C A271390 |.......|..............|
%C A271390 | n+1 | A001477(n) |
%C A271390 | 2*n+2 | A005408(n) |
%C A271390 | 2 | A000012(n) |
%C A271390 | 3 | A080425(n+2)|
%C A271390 | 4 | A010684(n) |
%C A271390 | 6 | A130793(n) |
%C A271390 ........................
%C A271390 Final digit of (2*n + 1)^(2*floor((n-1)/2) + 1) gives periodic sequence -> period 20: repeat [1,3,5,3,9,1,3,5,3,9,1,7,5,7,9,1,7,5,7,9], defined by the recurrence relation b(n) = b(n-2) - b(n-4) + b(n+5) + b(n+6) - b(n-7) - b(n-8) + b(n-9) - b(n-11) + b(n-13).
%H A271390 Ilya Gutkovskiy, <a href="/A271390/b271390.txt">Table of n, a(n) for n = 0..75</a>
%F A271390 a(n) = (2*n + 1)^(n - 1 + (1 + (-1)^(n-1))/2).
%F A271390 a(n) = A005408(n)^A109613(n-1).
%F A271390 a(n) = (2*n + 1)^(n - 1/2 - (-1)^n/2). - _Wesley Ivan Hurt_, Apr 10 2016
%e A271390 a(0) = 1;
%e A271390 a(1) = 3^1 = 3;
%e A271390 a(2) = 5^1 = 5;
%e A271390 a(3) = 7^3 = 343;
%e A271390 a(4) = 9^3 = 729;
%e A271390 a(5) = 11^5 = 161051;
%e A271390 a(6) = 13^5 = 371293;
%e A271390 a(7) = 15^7 = 170859375;
%e A271390 a(8) = 17^7 = 410338673;
%e A271390 ...
%e A271390 a(10000) = 1.644...*10^43006;
%e A271390 ...
%e A271390 a(100000) = 8.235...*10^530097, etc.
%e A271390 This sequence can be represented as a binary tree:
%e A271390 1
%e A271390 ................../ \..................
%e A271390 3^1 5^1
%e A271390 7^3......../ \......9^3 11^5....../ \.......13^5
%e A271390 / \ / \ / \ / \
%e A271390 / \ / \ / \ / \
%e A271390 / \ / \ / \ / \
%e A271390 15^7 17^7 19^9 21^9 23^11 25^11 27^13 29^13
%p A271390 A271390:=n->(2*n + 1)^(n - 1/2 - (-1)^n/2): seq(A271390(n), n=0..30); # _Wesley Ivan Hurt_, Apr 10 2016
%t A271390 Table[(2 n + 1)^(2 Floor[(n - 1)/2] + 1), {n, 0, 18}]
%t A271390 Table[(2 n + 1)^(n - 1 + (1 + (-1)^(n - 1))/2), {n, 0, 18}]
%o A271390 (PARI) a(n) = (2*n + 1)^(2*((n-1)\2) + 1); \\ _Altug Alkan_, Apr 06 2016
%o A271390 (Python)
%o A271390 for n in range(0,10**3):print((int)((2*n+1)**(2*floor((n-1)/2)+1)))
%o A271390 # _Soumil Mandal_, Apr 10 2016
%Y A271390 Cf. A005408, A092503, A109613.
%K A271390 nonn,easy
%O A271390 0,2
%A A271390 _Ilya Gutkovskiy_, Apr 06 2016