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A031973
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a(n) = Sum_{k=0..n} n^k.
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16
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1, 2, 7, 40, 341, 3906, 55987, 960800, 19173961, 435848050, 11111111111, 313842837672, 9726655034461, 328114698808274, 11966776581370171, 469172025408063616, 19676527011956855057, 878942778254232811938, 41660902667961039785743, 2088331858752553232964200
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OFFSET
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0,2
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COMMENTS
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These are the generalized repunits of length n+1 in base n for all n >= 1: a(n) expressed in base n is 111...111 (n+1 1's): a(1) = 1^0 + 1^1 = 2 = A000042(2), a(2) = 2^0 + 2^1 + 2^2 = 7 = A000225(3), a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40 = A003462(4), etc., a(10) = 10^0 + 10^1 + 10^2 + ... + 10^9 + 10^10 = 11111111111 = A002275(11), etc. - Rick L. Shepherd, Aug 26 2004
a(n)=the total number of ordered selections of up to n objects from n types with repetitions allowed. Thus for 2 objects a,b there are 7 possible selections: aa,bb,ab,ba,a,b, and the null set. - J. M. Bergot, Mar 26 2014
a(n)=the total number of ordered arrangements of 0,1,2..n objects, with repetitions allowed, selected from n types of objects. - J. M. Bergot, Apr 11 2014
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LINKS
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FORMULA
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EXAMPLE
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a(3) = 3^0 + 3^1 + 3^2 + 3^3 = 40.
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MAPLE
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a:= proc(n) local c, i; c:=1; for i to n do c:= c*n+1 od; c end:
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MATHEMATICA
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Join[{1}, Table[Total[n^Range[0, n]], {n, 20}]] (* Harvey P. Dale, Nov 13 2011 *)
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PROG
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(Sage) [lucas_number1(n, n, n-1) for n in range(1, 19)] # Zerinvary Lajos, May 16 2009
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CROSSREFS
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Cf. A000042 (unary representations), A000225 (2^n-1: binary repunits shown in decimal), A003462 ((3^n-1)/2: ternary repunits shown in decimal), A002275 ((10^n-1)/9: decimal repunits).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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