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 A084231 Numbers n such that root-mean-square value of 1, 2, ..., n, sqrt(Sum(k^2, k, 1, n)/n), is an integer. 6
 1, 337, 65521, 12710881, 2465845537, 478361323441, 92799630902161, 18002650033695937, 3492421306906109761, 677511730889751597841, 131433783371304903871537, 25497476462302261599480481, 4946378999903267445395341921, 959572028504771582145096852337 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equivalently, sqrt((n+1)*(2*n+1)/6) is an integer. LINKS Indranil Ghosh, Table of n, a(n) for n = 1..437 Peter Khoury and Gerard D. Koffi, Continued fractions and their application to solving Pell’s equations (2009) Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (195,-195,1). FORMULA a(n) = ((7/2 + 2sqrt(3))(97 + 56sqrt(3))^n + (7/2 - 2sqrt(3))(97 - 56sqrt(3))^n - 3)/4 a(n)=([(7/2 + 2sqrt(3))(97 + 56rq(3))^n] - 2)/4, [x] = integer part of x. a(n+3)=195(a(n+2) - a(n+1)) + a(n). G.f.: x(1+142x+x^2)/[(1-x)(1-194x+x^2)]. a(n) = ((7-4sqrt(3))^(1+2n)+(7+4sqrt(3))^(1+2n)-6)/8. - Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005 a(n) = 195*a(n-1)- 195*a(n-2)+ a(n-3), with a(0)=0, a(1)=1, a(2)=337, a(3)=65521. - Harvey P. Dale, Jul 14 2011 EXAMPLE a(1)=337 because sqrt(Sum(k^2, k, 1, 337)/337) is integer (195=A084232(1)). MATHEMATICA a[n_]:=Expand[((7-4Sqrt[3])^(1+2n)+(7+4Sqrt[3])^(1+2n)-6)/8] (Pein) CoefficientList[Series[x (1+142x+x^2)/((1-x)(1-194x+x^2)), {x, 0, 30}], x] (* or *) Join[{0}, LinearRecurrence[{195, -195, 1}, {1, 337, 65521}, 30]] (* Harvey P. Dale, Jul 14 2011 *) CROSSREFS Cf. A084232. Sequence in context: A263865 A184202 A194478 * A243483 A234625 A226539 Adjacent sequences:  A084228 A084229 A084230 * A084232 A084233 A084234 KEYWORD nonn,easy AUTHOR Ignacio Larrosa Cañestro, May 20 2003 EXTENSIONS One more term from Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005 STATUS approved

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Last modified August 17 03:29 EDT 2018. Contains 313810 sequences. (Running on oeis4.)