A332082 a(n) = Sum_{1 <= m <= n} Sum_{1 <= k <= n+1-m} m*R(k,n+1), where R(k,b) = (b^k - 1)/(b - 1) is the base-b repunit of length k.

0, 1, 7, 42, 295, 2675, 31122, 447188, 7661370, 152415765, 3452271185, 87693358654, 2468455488681, 76256200336407, 2564715882332660, 93281313241869480, 3647955866777821668, 152635100350763019705, 6803550294289868214315, 321844061970058547739730, 16103630469426364324556635

Offset

0,3

Comments

This is the sum of all elements of a triangular matrix (of size n given by the index) where the k-th diagonal is filled with k-repdigits (in base n+1, as to have digits up to n) of increasing length, as in

( 1 0 0 . . . . . . . . 0 )

( 2 11 0 : )

( 3 22 111 ˙ · . : )

( : 33 222 ˙ · . ˙ · . : )

( : ˙ · . ˙ · . ˙ · . 0 )

( n . . . . . 3..3 2...2 1....1 )

with numbers written in base n+1.

Links

Table of n, a(n) for n=0..20.

Example

a(2) = 2 + 1 + 11[3] = 3 + 4 = 7.
a(3) = 3 + 2 + 22[4] + 1 + 11[4] + 111[4] = 6 + 15 + 21 = 42.
a(4) = 4 + 3 + 33[5] + 2 + 22[5] + 222[5] + 1 + 11[5] + 111[5] + 1111[5] = 10 + 36 + 93 + 156 = 295.

Mathematica

a[n_] := Sum[(n + 1 - m)*((n + 1)*((n + 1)^m - 1) - m*n)/n^2, {m, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Aug 24 2020 *)

Prog

(PARI) apply( {A332082(n)=sum(m=1, n, (n+1-m)*((n+1)*((n+1)^m-1)-m*n)\n^2)}, [0..20])

Crossrefs

For base 10 repunits and repdigits, cf. A002275 (repunits), A010785 (repdigits) and A014824 (partial sums of repunits).

Sequence in context: A187246 A278152 A366222 * A271427 A073506 A025593

Adjacent sequences: A332079 A332080 A332081 * A332083 A332084 A332085

Keyword

nonn,base

Author

M. F. Hasler, Aug 19 2020

Status

approved