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A037123
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a(n) = a(n-1) + sum of digits of n.
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31
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0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 100, 102, 105, 109, 114, 120, 127, 135, 144, 154, 165, 168, 172, 177, 183, 190, 198, 207, 217, 228, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 325, 330, 336, 343, 351, 360, 370, 381
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OFFSET
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0,3
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COMMENTS
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Sum of the digital sum of i for i from 0 to n. - N. J. A. Sloane, Nov 13 2013
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REFERENCES
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N. Agronomof, Sobre una función numérica, Revista Mat. Hispano-Americana 1 (1926), 267-269.
Maurice d'Ocagne, Sur certaines sommations arithmétiques, J. Sciencias Mathematicas e Astronomicas 7 (1886), 117-128.
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LINKS
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P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
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FORMULA
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a(n) = Sum_{k=0..n} s(k) = Sum_{k=0..n} A007953(k), where s(k) denote the sum of the digits of k in decimal representation. Asymptotic expression: a(n-1) = Sum_{k=0..n-1} s(k) = 4.5*n*log_10(n) + O(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
a(n) = n*(n+1)/2 - 9*Sum_{k=1..n} Sum_{i=1..ceiling(log_10(k))} floor(k/10^i). - Benoit Cloitre, Aug 28 2003
G.f.: Sum_{k>=1} ((x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k)))/(1-x)^2.
a(n) = (1/2)*((n+1)*(n - 18*Sum_{k>=1} floor(n/10^k)) + 9*Sum_{k>=1} (1 + floor(n/10^k))*floor(n/10^k)*10^k).
a(n) = (1/2)*((n+1)*(2*A007953(n)-n) + 9*Sum_{k>=1} (1+floor(n/10^k))*floor(n/10^k)*10^k). (End)
a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.
a(n) = a(n mod m) + MSD*a(m - 1) + (MSD*(MSD - 1)/2)*m + MSD*((n mod m) + 1), where m = 10^(A055642(n) - 1), MSD = A000030(n). (End)
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MAPLE
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digsum:=proc(n, B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n, k, B) global digsum; local i;
add( digsum(i, B)^k, i=0..n); end;
lprint([seq(digsum(n, 10), n=0..100)]); # A007953
lprint([seq(f(n, 1, 10), n=0..100)]); #A037123
lprint([seq(f(n, 2, 10), n=0..100)]); #A074784
lprint([seq(f(n, 3, 10), n=0..100)]); #A231688
lprint([seq(f(n, 4, 10), n=0..100)]); #A231689
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MATHEMATICA
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a[0] = 0; a[n_] := a[n - 1] + Plus @@ IntegerDigits@ n; Array[a, 70, 0] (* Robert G. Wilson v, Jul 06 2018 *)
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PROG
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(PARI) a(n)=n*(n+1)/2-9*sum(k=1, n, sum(i=1, ceil(log(k)/log(10)), floor(k/10^i)))
(PARI) a(n)={n++; my(t, i, s); c=n; while(c!=0, i++; c\=10); for(j=1, i, d=(n\10^(i-j))%10; t+=(10^(i-j)*(s*d+binomial(d, 2)+d*9*(i-j)/2)); s+=d); t} \\ David A. Corneth, Aug 16 2013
(Perl) for $i (0..100){ @j = split "", $i; for (@j){ $sum += $_; } print "$sum, "; } __END__ # gamo(AT)telecable.es
(Magma) [ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ]; // Bruno Berselli, May 27 2011
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
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STATUS
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approved
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