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 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. 1
 0, 1, 7, 42, 295, 2675, 31122, 447188, 7661370, 152415765, 3452271185, 87693358654, 2468455488681, 76256200336407, 2564715882332660, 93281313241869480, 3647955866777821668, 152635100350763019705, 6803550294289868214315, 321844061970058547739730, 16103630469426364324556635 (list; graph; refs; listen; history; text; internal format)
 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 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: A221794 A187246 A278152 * A271427 A073506 A025593 Adjacent sequences:  A332079 A332080 A332081 * A332083 A332084 A332085 KEYWORD nonn,base AUTHOR M. F. Hasler, Aug 19 2020 STATUS approved

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Last modified January 28 01:34 EST 2022. Contains 350654 sequences. (Running on oeis4.)