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A055017
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Difference between sums of alternate digits of n starting with the last, i.e. (Sum of ultimate digit of n, antepenultimate digit of n,...)-(sum of penultimate digit of n, preantepenultimate digit of n,...).
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11
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, -8, -7, -6, -5, -4, -3
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| n is divisible by 11 iff a(n) is a multiple of 11
Digital sum with alternating signs starting with a positive sign for the rightmost digit. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
For n<100 equal to (n mod 10 - floor(n/10)) = -A076313(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
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FORMULA
| a(n)=n+11*sum{k>0,(-1)^k*floor(n/10^k)}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
a(10n+k)=k-a(n), 0<=k<10. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
G.f. g(x)=sum{k>0, (x^k-x^(k+10^k)+(-1)^k*11*x^(10^k))/(1-x^(10^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
a(n)=n+11*sum{10<=k<=n, sum{j|k,j>=10, (-1)^floor(log_10(j))*(floor(log_10(j))-floor(log_10(j-1)))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
G.f. expressed in terms of Lambert series: g(x)=(x/(1-x)+11*L[b(k)](x))/(1-x) where L[b(k)](x)=sum{k>=0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k)=(-1)^floor(log_10(k)), if k>1 is a power of 10, else b(k)=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25, 2007
G.f.: g(x)=sum{k>0, (1+11*c(k))*x^k}/(1-x), where c(k)=sum{j>1,j|k, (-1)^floor(log_10(j))*(floor(log_10(j))-floor(log_10(j-1)))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25, 2007
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EXAMPLE
| a(123)=3-2+1=2, a(9875)=5-7+8-9=-3.
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MAPLE
| sumodigs := proc(n) local dg; dg := convert(n, base, 10) ; add(op(1+2*i, dg), i=0..floor(nops(dg)-1)/2) ; end proc:
sumedigs := proc(n) local dg; dg := convert(n, base, 10) ; add(op(2+2*i, dg), i=0..floor(nops(dg)-2)/2) ; end proc:
A055017 := proc(n) sumodigs(n)-sumedigs(n) ; end proc: # R. J. Mathar, Aug 26 2011
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CROSSREFS
| Unsigned version differs from A040114 and A040115 when n=100 and from A040997 when n=101.
Cf. A076313, A076314, A007953, A003132.
Sequence in context: A189823 A001073 A076313 * A040997 A177894 A175398
Adjacent sequences: A055014 A055015 A055016 * A055018 A055019 A055020
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KEYWORD
| base,easy,sign
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), May 31 2000
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