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A113170
Ascending descending base exponent transform of odd numbers A005408.
2
1, 4, 33, 376, 5665, 115356, 3014209, 95722288, 3619661121, 161338248820, 8349617508961, 493959321484584, 33041900704133473, 2479933070973253516, 207343189445230918785, 19175058576632809926496, 1949302342535131018462849, 216707770770991401785821668
OFFSET
1,2
COMMENTS
A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The parity of this sequence cycles odd, even, odd, even, ... There is no nontrivial integer fixed point of the transform.
LINKS
FORMULA
a(1) = 1. For n>1: a(n) = Sum_{i=1..n} (2n+1)^(2n-i).
EXAMPLE
a(2) = 4 because 1^3 + 3^1 = 1 + 3 = 4.
a(3) = 33 because 1^5 + 3^3 + 5^1 = 1 + 27 + 5 = 33.
a(4) = 406 because 1^7 + 3^5 + 5^3 + 7^1 = 1 + 243 + 125 + 7 = 376.
a(5) = 5665 because 1^9 + 3^7 + 5^5 + 7^3 + 9^1 = 5665.
a(6) = 115356 = 1^11 + 3^9 + 5^7 + 7^5 + 9^3 + 11^1.
a(7) = 3014209 = 1^13 + 3^11 + 5^9 + 7^7 + 9^5 + 11^3 + 13^1.
a(8) = 95722288 = 1^15 + 3^13 + 5^11 + 7^9 + 9^7 + 11^5 + 13^3 + 15^1.
a(9) = 3619661121 = 1^17 + 3^15 + 5^13 + 7^11 + 9^9 + 11^7 + 13^5 + 15^3 + 17^1.
a(10) = 161338248820 = 1^19 + 3^17 + 5^15 + 7^13 + 9^11 + 11^9 + 13^7 + 15^5 + 17^3 + 19^1.
MATHEMATICA
Table[Sum[(2 k + 1)^(2 n - 2 k + 1), {k, 1, n}], {n, 0, 10}] + 1 (* G. C. Greubel, May 18 2017 *)
PROG
(PARI) for(n=0, 25, print1(1 + sum(k=1, n, (2*k+1)^(2*n-2*k+1)), ", ")) \\ G. C. Greubel, May 18 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jan 06 2006
STATUS
approved