A122571 a(n) = 14*a(n-1) - a(n-2), with a(1) = a(2) = 1.

1, 1, 13, 181, 2521, 35113, 489061, 6811741, 94875313, 1321442641, 18405321661, 256353060613, 3570537526921, 49731172316281, 692665874901013, 9647591076297901, 134373609193269601, 1871582937629476513, 26067787517619401581, 363077442309042145621, 5057016404808970637113

Offset: 1

Author

Roger L. Bagula, Sep 17 2006

Keywords

• nonn
• easy

Comments

Essentially the same as A001570: 1 followed by A001570.

Each term is a sum of two consecutive squares, or a(n) = k^2 + (k+1)^2 for some k. Squares of each term are the hex numbers, or centered hexagonal numbers: a(n) = A001570(n-1) for n > 1. - Alexander Adamchuk, Apr 14 2008

References

• Henry MacKean and Victor Moll, Elliptic Curves, Cambridge University Press, New York, 1997, page 22.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..870
• Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (14,-1).

Crossrefs

Cf. A001570 (essentially the same).

Programs

Magma

[n le 2 select 1 else 14*Self(n-1) -Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 29 2024

Mathematica

LinearRecurrence[{14, -1}, {1, 1}, 25] (* Paolo Xausa, Jan 29 2024 *)

SageMath

A122571=BinaryRecurrenceSequence(14,-1,1,1)
[A122571(n-1) for n in range(1,41)] # G. C. Greubel, Oct 29 2024

Formula

G.f.: x*(1-13*x)/(1-14*x+x^2). - Philippe Deléham, Nov 17 2008

a(n+1) = A001570(n). - Ctibor O. Zizka, Feb 26 2010

a(n) = (1/4)*sqrt( 2 + (2-sqrt(3))^(4*n-6) + (2+sqrt(3))^(4*n-6) ). - Gerry Martens, Jun 03 2015

From G. C. Greubel, Oct 29 2024: (Start)

a(n) = (1/4)*( (2 + sqrt(3))^(2*n-3) + (2 - sqrt(3))^(2*n-3) ).

E.g.f.: -13 + exp(7*x)*( 13*cosh(4*sqrt(3)*x) - (15*sqrt(3)/2)*sinh(4*sqrt(3)*x) ). (End)

Extensions

Edited by N. J. A. Sloane, Sep 21 2006 and Dec 04 2006

a(19)-a(21) from Paolo Xausa, Jan 29 2024