A221741 a(n) = -4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n).

1, 5, 9, 97, 373, 7625, 48913, 1513361, 13717421, 570623341, 6698798233, 350549891889, 5057809205989, 319164643134737, 5465701947765793, 403925909124187873, 8008631808527689309, 678470389458269406421, 15287592943577781017641

Offset: 1

Author

Alexander R. Povolotsky, Jan 23 2013

Keywords

• nonn
• easy

Comments

Per exhaustive program, written for bases from 2 to 10, the number of permutations pairs, which have the same ratio, equal to A221740(n)/a(n) = (n^2 (n+1)^n-(n+1)^n+1) / (-n^2+n (n+1)^n+(n+1)^n-n-1), is: {2,2,3,3,5,3,7,5,7,...} for n>=1 where n=r-1 and r is the base radix. Judging by above sequence it appears that the number of such permutations pairs is related to phi, which is the Euler totient function - according to A039649, A039650, A214288 (see bullet 1 of the analysis in the answer section of the StackExchange link). Alexander R. Povolotsky, Jan 26 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 1..385
• NMBRTHRY, Number of specific permutations pairs relates to Euler Phi totient function ?
• StackExchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.

Crossrefs

Cf. A221740, A023811.

Programs

Mathematica

Table[-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), {n,1,50}] (* G. C. Greubel, Feb 19 2017 *)

Maxima

makelist(-4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n), n, 1, 20); /* Martin Ettl, Jan 25 2013 */

PARI

for(n=1,25, print1(-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), ", ")) \\ G. C. Greubel, Feb 19 2017

Formula

a(n) = -4*A023811(n+1)/((-3 + (-1)^n)*n).