The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A001922 Numbers k such that 3*k^2 - 3*k + 1 is both a square (A000290) and a centered hexagonal number (A003215). (Formerly M4569 N1946) 7
 1, 8, 105, 1456, 20273, 282360, 3932761, 54776288, 762935265, 10626317416, 148005508553, 2061450802320, 28712305723921, 399910829332568, 5570039304932025, 77580639439715776, 1080558912851088833, 15050244140475527880, 209622859053806301481 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also larger of two consecutive integers whose cubes differ by a square. Defined by a(n)^3 - (a(n)-1)^3 = square. Let m be the n-th ratio 2/1, 7/4, 26/15, 97/56, 362/209, ... Then a(n) = m*(2-m)/(m^2-3). The numerators 2, 7, 26, ... of m are A001075. The denominators 1, 4, 15, ... of m are A001353. From Colin Barker, Jan 06 2015: (Start) Also indices of centered triangular numbers (A005448) which are also centered square numbers (A001844). Also indices of centered hexagonal numbers (A003215) which are also centered octagonal numbers (A016754). Also positive integers x in the solutions to 3*x^2 - 4*y^2 - 3*x + 4*y = 0, the corresponding values of y being A156712. (End) REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..800 J. Brenner and E. P. Starke, Problem E702, Amer. Math. Monthly, 53 (1946), 465. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Sociedad Magic Penny Patagonia, Leonardo en Patagonia Index entries for linear recurrences with constant coefficients, signature (15,-15,1). FORMULA a(n) = 15a(n-1) - 15a(n-2) + a(n-3). a(n) = (s1*t1^n + s2*t2^n + 6)/12 where s1 = 3 + 2*sqrt(3), s2 = 3 - 2*sqrt(3), t1 = 7 + 4*sqrt(3), t2 = 7 - 4*sqrt(3). a(n) = A001075(n)*A001353(n+1). G.f.: (1-7*x)/((1-x)*(1-14*x+x^2)). - Simon Plouffe (in his 1992 dissertation) and Colin Barker, Jan 01 2012 a(n) = A076139(n+1) - 7*A076139(n). - R. J. Mathar, Jul 14 2015 EXAMPLE 8 is in the sequence because 3*8^2 - 3*8 + 1 = 169 is a square and also a centered hexagonal number. - Colin Barker, Jan 07 2015 MATHEMATICA With[{s1=3+2Sqrt, s2=3-2Sqrt, t1=7+4Sqrt, t2=7-4Sqrt}, Simplify[ Table[(s1 t1^n+s2 t2^n+6)/12, {n, 0, 20}]]] (* or *) LinearRecurrence[ {15, -15, 1}, {1, 8, 105}, 21] (* Harvey P. Dale, Aug 14 2011 *) CoefficientList[Series[(1-7*x)/(1-15*x+15*x^2-x^3), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 16 2012 *) PROG (MAGMA) I:=[1, 8, 105]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Apr 16 2012 (PARI) Vec((1-7*x)/(1-15*x+15*x^2-x^3) + O(x^100)) \\ Colin Barker, Jan 06 2015 CROSSREFS Cf. A001921, A001570, A006051. Cf. A001844, A003215, A005448, A156712, A016754, A076139. Sequence in context: A302804 A303465 A239400 * A264014 A222839 A113551 Adjacent sequences:  A001919 A001920 A001921 * A001923 A001924 A001925 KEYWORD nonn,easy AUTHOR EXTENSIONS Additional comments from James R. Buddenhagen, Mar 04 2001 Name improved by Colin Barker, Jan 07 2015 Edited by Robert Israel, Feb 20 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 4 08:01 EDT 2022. Contains 355070 sequences. (Running on oeis4.)