A332082 - OEIS

%I #16 Sep 19 2020 23:57:44

%S 0,1,7,42,295,2675,31122,447188,7661370,152415765,3452271185,

%T 87693358654,2468455488681,76256200336407,2564715882332660,

%U 93281313241869480,3647955866777821668,152635100350763019705,6803550294289868214315,321844061970058547739730,16103630469426364324556635

%N 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.

%C 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

%C   ( 1   0    0    . . . . . . . .  0   )

%C   ( 2  11    0                     :   )

%C   ( 3  22   111   ˙ · .            :   )

%C   ( :  33   222   ˙ · .  ˙ · .     :   )

%C   ( :      ˙ · .  ˙ · .  ˙ · .     0   )

%C   ( n  . . . . .   3..3  2...2  1....1 )

%C with numbers written in base n+1.

%e a(2) = 2 + 1 + 11[3] = 3 + 4 = 7.

%e a(3) = 3 + 2 + 22[4] + 1 + 11[4] + 111[4] = 6 + 15 + 21 = 42.

%e a(4) = 4 + 3 + 33[5] + 2 + 22[5] + 222[5] + 1 + 11[5] + 111[5] + 1111[5] = 10 + 36 + 93 + 156 = 295.

%t 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 *)

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

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

%K nonn,base

%O 0,3

%A _M. F. Hasler_, Aug 19 2020