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%I #29 Feb 24 2021 09:10:11
%S 1,3,5,343,729,161051,371293,170859375,410338673,322687697779,
%T 794280046581,952809757913927,2384185791015625,4052555153018976267,
%U 10260628712958602189,23465261991844685929951,59938945498865420543457,177482997121587371826171875,456487940826035155404146917
%N a(n) = (2*n + 1)^(2*floor((n-1)/2) + 1).
%C All members are odd, therefore:
%C ........................
%C | k | a(n) mod k |
%C |.......|..............|
%C | n+1 | A001477(n) |
%C | 2*n+2 | A005408(n) |
%C | 2 | A000012(n) |
%C | 3 | A080425(n+2)|
%C | 4 | A010684(n) |
%C | 6 | A130793(n) |
%C ........................
%C 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 Ilya Gutkovskiy, <a href="/A271390/b271390.txt">Table of n, a(n) for n = 0..75</a>
%F a(n) = (2*n + 1)^(n - 1 + (1 + (-1)^(n-1))/2).
%F a(n) = A005408(n)^A109613(n-1).
%F a(n) = (2*n + 1)^(n - 1/2 - (-1)^n/2). - _Wesley Ivan Hurt_, Apr 10 2016
%e a(0) = 1;
%e a(1) = 3^1 = 3;
%e a(2) = 5^1 = 5;
%e a(3) = 7^3 = 343;
%e a(4) = 9^3 = 729;
%e a(5) = 11^5 = 161051;
%e a(6) = 13^5 = 371293;
%e a(7) = 15^7 = 170859375;
%e a(8) = 17^7 = 410338673;
%e ...
%e a(10000) = 1.644...*10^43006;
%e ...
%e a(100000) = 8.235...*10^530097, etc.
%e This sequence can be represented as a binary tree:
%e 1
%e ................../ \..................
%e 3^1 5^1
%e 7^3......../ \......9^3 11^5....../ \.......13^5
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e 15^7 17^7 19^9 21^9 23^11 25^11 27^13 29^13
%p A271390:=n->(2*n + 1)^(n - 1/2 - (-1)^n/2): seq(A271390(n), n=0..30); # _Wesley Ivan Hurt_, Apr 10 2016
%t Table[(2 n + 1)^(2 Floor[(n - 1)/2] + 1), {n, 0, 18}]
%t Table[(2 n + 1)^(n - 1 + (1 + (-1)^(n - 1))/2), {n, 0, 18}]
%o (PARI) a(n) = (2*n + 1)^(2*((n-1)\2) + 1); \\ _Altug Alkan_, Apr 06 2016
%o (Python)
%o for n in range(0,10**3):print((int)((2*n+1)**(2*floor((n-1)/2)+1)))
%o # _Soumil Mandal_, Apr 10 2016
%Y Cf. A005408, A092503, A109613.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Apr 06 2016
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