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 A016779 a(n) = (3n + 1)^3. 10
 1, 64, 343, 1000, 2197, 4096, 6859, 10648, 15625, 21952, 29791, 39304, 50653, 64000, 79507, 97336, 117649, 140608, 166375, 195112, 226981, 262144, 300763, 343000, 389017, 438976, 493039, 551368, 614125, 681472, 753571, 830584, 912673, 1000000, 1092727, 1191016 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The inverse binomial transform is 1, 63, 216, 162, 0, 0, 0 (0 continued). R. J. Mathar, May 07 2008 Perfect cubes with digital root 1 in base 10. Proof: perfect cubes are one of (3*s)^3, (3*s+1)^3 or (3*s+2)^3. Digital roots of (3*s)^3 are 0, digital roots of (3*s+1)^3 are 1, and digital roots of (3*s+2)^3 are 8, using trinomial expansion and the multiplicative property of digits roots. - R. J. Mathar, Jul 31 2010 REFERENCES S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.3. Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published) LINKS Harry J. Smith, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA Sum_{n>=0} 1/a(n) = 2*Pi^2 / (81*sqrt(3)) + 13*zeta(3)/27. O.g.f.: (1+60*x+93*x^2+8*x^3)/(1-x)^4. - R. J. Mathar, May 07 2008 E.g.f.: (1 + 63*x + 108*x^2 + 27*x^3)*exp(x). - Ilya Gutkovskiy, Jun 16 2016 EXAMPLE 343 = (3*2+1)^3, 6859 = (3*6+1)^3. MATHEMATICA Table[(3n+1)^3, {n, 0, 100}] (* Mohammad K. Azarian, Jun 15 2016 *) LinearRecurrence[{4, -6, 4, -1}, {1, 64, 343, 1000}, 40] (* Harvey P. Dale, Oct 31 2016 *) PROG (PARI) { b=0; for (n=0, 1000, until (s==1, b++; s=b^3; s-=9*(s\9)); write("b016779.txt", n, " ", b^3) ) } \\ Harry J. Smith, Jul 18 2009 (MAGMA) [(3*n+1)^3: n in [0..30]]; // Vincenzo Librandi, May 09 2011 (PARI) a(n)=(3*n+1)^3 \\ Charles R Greathouse IV, Jan 02 2012 CROSSREFS Cf. A016791, A054966. Sequence in context: A186441 A297642 A061102 * A298220 A299349 A299096 Adjacent sequences:  A016776 A016777 A016778 * A016780 A016781 A016782 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified March 19 19:32 EDT 2019. Contains 321330 sequences. (Running on oeis4.)