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A271390
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a(n) = (2*n + 1)^(2*floor((n-1)/2) + 1).
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1
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1, 3, 5, 343, 729, 161051, 371293, 170859375, 410338673, 322687697779, 794280046581, 952809757913927, 2384185791015625, 4052555153018976267, 10260628712958602189, 23465261991844685929951, 59938945498865420543457, 177482997121587371826171875, 456487940826035155404146917
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OFFSET
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0,2
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COMMENTS
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All members are odd, therefore:
........................
| k | a(n) mod k |
|.......|..............|
........................
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).
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LINKS
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FORMULA
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a(n) = (2*n + 1)^(n - 1 + (1 + (-1)^(n-1))/2).
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EXAMPLE
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a(0) = 1;
a(1) = 3^1 = 3;
a(2) = 5^1 = 5;
a(3) = 7^3 = 343;
a(4) = 9^3 = 729;
a(5) = 11^5 = 161051;
a(6) = 13^5 = 371293;
a(7) = 15^7 = 170859375;
a(8) = 17^7 = 410338673;
...
a(10000) = 1.644...*10^43006;
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a(100000) = 8.235...*10^530097, etc.
This sequence can be represented as a binary tree:
1
................../ \..................
3^1 5^1
7^3......../ \......9^3 11^5....../ \.......13^5
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
15^7 17^7 19^9 21^9 23^11 25^11 27^13 29^13
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MAPLE
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MATHEMATICA
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Table[(2 n + 1)^(2 Floor[(n - 1)/2] + 1), {n, 0, 18}]
Table[(2 n + 1)^(n - 1 + (1 + (-1)^(n - 1))/2), {n, 0, 18}]
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PROG
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(PARI) a(n) = (2*n + 1)^(2*((n-1)\2) + 1); \\ Altug Alkan, Apr 06 2016
(Python)
for n in range(0, 10**3):print((int)((2*n+1)**(2*floor((n-1)/2)+1)))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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