A051846 Digits 1..n in strict descending order n..1 interpreted in base n+1.

1, 7, 57, 586, 7465, 114381, 2054353, 42374116, 987654321, 25678050355, 736867805641, 23136292864686, 789018236134297, 29043982525261081, 1147797409030816545, 48471109094902544776, 2178347851919531492065, 103805969587115219182431

Offset: 1

Author

Antti Karttunen, Dec 13 1999

Keywords

• easy
• base
• nonn

Comments

All odd-indexed (2n+1) terms are divisible by (2n+1). See A051847.

All even-indexed (2n) terms are divisible by n. - Alexander R. Povolotsky, Oct 20 2022

Examples

			a(1) = 1,
a(2) = 2*3 + 1 = 7,
a(3) = 3*(4^2) + 2*4 + 1 = 57,
a(4) = 4*(5^3) + 3*(5^2) + 2*5 + 1 = 586.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Crossrefs

The right edge of A051845.

Cf. A023811, A051847, A058128, A062806, A199969.

Programs

Maple

a(n) := proc(n) local i; add(i*((n+1)^(i-1)),i=1..n); end;

Mathematica

Array[Sum[i*(# + 1)^(i - 1), {i, #}] &, 18] (* Michael De Vlieger, Apr 04 2024 *)

Maxima

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

PARI

a(n)=((n+1)^(n+1)*(n-1)+1)/n^2

Python

def a(n): return sum((i+1)*(n+1)**i for i in range(n))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 10 2022

Formula

a(n) = Sum_{i=1..n} i*(n+1)^(i-1).

a(n) = ((n+1)^(n+1)*(n-1) + 1)/n^2 = A062806(n+1)/(n+1) - (n+1)^(n+1). - Benoit Cloitre, Sep 28 2002

a(n) = A028310(n-1) * A023811(n+1) + A199969(n+1). - M. F. Hasler, Jan 22 2013

a(n) = (n-1) * A058128(n+1) + 1. - Seiichi Manyama, Apr 10 2022

Extensions

Minor edits in formulas by M. F. Hasler, Oct 11 2019