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A093137
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Expansion of (1-7*x)/((1-x)(1-10*x)).
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16
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1, 4, 34, 334, 3334, 33334, 333334, 3333334, 33333334, 333333334, 3333333334, 33333333334, 333333333334, 3333333333334, 33333333333334, 333333333333334, 3333333333333334, 33333333333333334, 333333333333333334
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OFFSET
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0,2
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COMMENTS
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Second binomial transform of 3*A001045(3n)/3+(-1)^n. Partial sums of A093138. A convex combination of 10^n and 1. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1,1+k,1+11k,1+111k,... This is the case for k=3.
a(n) is the number of n-length sequences of decimal digits whose sum is divisible by 3. - Geoffrey Critzer, Jan 18 2014
This sequence appears in a family of curious cubic identities based on the Armstrong number 407 = A005188(13). See the formula section. For the analog identities based on 153 = A005188(10) see a comment on A246057 with the van der Poorten et al. reference and A281857. For those based on 370 = A005188(11) see A067275, A002277 and A281858. - Wolfdieter Lang, Feb 08 2017
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LINKS
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FORMULA
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a(n) = 3*10^n/9 + 6/9.
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EXAMPLE
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a(1)^2 = 16
a(2)^2 = 1156
a(3)^2 = 111556
a(4)^2 = 11115556
a(5)^2 = 1111155556
a(6)^2 = 111111555556
a(7)^2 = 11111115555556
a(8)^2 = 1111111155555556
Curious cubic identities: 407 = 4^3 + 0^3 + 7^3, 340067 = 34^3 + (00)^3 + 67^3, 334000677 = 334^3 + (000)^3 + 677^3, ... - Wolfdieter Lang, Feb 08 2017
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MATHEMATICA
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nn=20; r=Solve[{s==4x s+3 x a+3x b+1, a==4x a+3x s+3x b, b==4x b+3x s+3x a}, {s, a, b}]; CoefficientList[Series[s/.r, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 18 2014 *)
Table[3*10^n/9 + 6/9, {n, 0, 20}] (* or *) NestList[10 # - 6 &, 1, 20] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{11, -10}, {1, 4}, 20] (* Harvey P. Dale, Oct 07 2017 *)
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PROG
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(PARI) Vec((1-7*x)/((1-x)*(1-10*x)) + O (x^30)) \\ Michel Marcus, Feb 09 2017
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CROSSREFS
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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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