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)

1, 8, 105, 1456, 20273, 282360, 3932761, 54776288, 762935265, 10626317416, 148005508553, 2061450802320, 28712305723921, 399910829332568, 5570039304932025, 77580639439715776, 1080558912851088833, 15050244140475527880, 209622859053806301481

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) = 15*a(n-1) - 15*a(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

a(n) = (1/2)*(1 + ChebyshevU(n, 7) + ChebyshevU(n-1, 7)). G. C. Greubel, Oct 07 2022

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

Maple

seq(simplify((1 +ChebyshevU(n, 7) +ChebyshevU(n-1, 7))/2), n=0..30); # G. C. Greubel, Oct 07 2022

Mathematica

With[{s1=3+2Sqrt[3], s2=3-2Sqrt[3], t1=7+4Sqrt[3], t2=7-4Sqrt[3]}, 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
(SageMath) [(1+chebyshev_U(n, 7) +chebyshev_U(n-1, 7))/2 for n in range(30)] # G. C. Greubel, Oct 07 2022

Crossrefs

Cf. A001075, A001353, A001570, A001844, A001921, A003215.

Cf. A005448, A006051, A016754, A076139, A156712.

Sequence in context: A302804 A303465 A239400 * A264014 A222839 A113551

Adjacent sequences: A001919 A001920 A001921 * A001923 A001924 A001925

Keyword

nonn,easy

Author

N. J. A. Sloane

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