A113170 Ascending descending base exponent transform of odd numbers A005408.

1, 4, 33, 376, 5665, 115356, 3014209, 95722288, 3619661121, 161338248820, 8349617508961, 493959321484584, 33041900704133473, 2479933070973253516, 207343189445230918785, 19175058576632809926496, 1949302342535131018462849, 216707770770991401785821668

Offset: 1

Author

Jonathan Vos Post, Jan 06 2006

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..295

Crossrefs

Cf. A005408, A113122, A113153, A113154.

Programs

Mathematica

Table[Sum[(2 k + 1)^(2 n - 2 k + 1), {k, 1, n}], {n, 0, 10}] + 1 (* G. C. Greubel, May 18 2017 *)

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

Formula

a(1) = 1. For n>1: a(n) = Sum_{i=1..n} (2n+1)^(2n-i).