A011900 a(n) = 6*a(n-1) - a(n-2) - 2 with a(0) = 1, a(1) = 3.

1, 3, 15, 85, 493, 2871, 16731, 97513, 568345, 3312555, 19306983, 112529341, 655869061, 3822685023, 22280241075, 129858761425, 756872327473, 4411375203411, 25711378892991, 149856898154533, 873430010034205, 5090723162050695, 29670908962269963, 172934730611569081

Offset

0,2

Comments

Members of Diophantine pairs.

Solution to b*(b-1) = 2*a*(a-1) in natural numbers; a = a(n), b = b(n) = A046090(n).

Also the indices of centered octagonal numbers which are also centered square numbers. - Colin Barker, Jan 01 2015

Also positive integers y in the solutions to 4*x^2 - 8*y^2 - 4*x + 8*y = 0. - Colin Barker, Jan 01 2015

Also the number of perfect matchings on a triangular lattice of width 3 and length n. - Sergey Perepechko, Jul 11 2019

References

Mario Velucchi "The Pell's equation ... an amusing application" in Mathematics and Informatics Quarterly, to appear 1997.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

H. J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.

S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.

S. N. Perepechko, Number of perfect matchings on triangular lattices of fixed width, DIMA'2015 slides. [see: page 12]

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Formula

a(n) = (A001653(n+1) + 1)/2.

a(n) = (((1+sqrt(2))^(2*n-1) - (1-sqrt(2))^(2*n-1))/sqrt(8)+1)/2.

a(n) = 7*(a(n-1) - a(n-2)) + a(n-3); a(1) = 1, a(2) = 3, a(3) = 15. Also a(n) = 1/2 + ( (1-sqrt(2))/(-4*sqrt(2)) )*(3-2*sqrt(2))^n + ( (1+sqrt(2))/(4*sqrt(2)) )*(3+2*sqrt(2))^n. - Antonio Alberto Olivares, Dec 23 2003

Sqrt(2) = Sum_{n>=0} 1/a(n); a(n) = a(n-1) + floor(1/(sqrt(2) - Sum_{k=0..n-1} 1/a(k))) (n>0) with a(0)=1. - Paul D. Hanna, Jan 25 2004

For n>k, a(n+k) = A001541(n)*A001653(k) - A053141(n-k-1); e.g., 493 = 99*5 - 2. For n<=k, a(n+k)=A001541(n)*A001653(k) - A053141(k-n); e.g., 85 = 3*29 - 2. - Charlie Marion, Oct 18 2004

a(n+1) = 3*a(n) - 1 + sqrt(8*a(n)^2 - 8*a(n) + 1), a(1)=1. - Richard Choulet, Sep 18 2007

a(n+1) = a(n) * (a(n) + 2) / a(n-1) for n>=1 with a(0)=1 and a(1)=3. - Paul D. Hanna, Apr 08 2012

G.f.: (1 - 4*x + x^2)/((1-x)*(1 - 6*x + x^2)). - R. J. Mathar, Oct 26 2009

Sum_{k=a(n)..A001109(n+1)} k = a(n)*A001109(n+1) = A011906(n+1). Example n=2, 3+4+5+6=18, 3*6=18. - Paul Cleary, Dec 05 2015

a(n) = (sqrt(1+8*A001109(n+1)^2)+1)/2 - A001109(n+1). - Robert Israel, Dec 16 2015

a(n) = a(-1-n) for all n in Z. - Michael Somos, Feb 23 2019

E.g.f.: (2*exp(x) + exp(3*x)*(2*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)))/4. - Stefano Spezia, Mar 16 2024

Example

G.f. = 1 + 3*x + 15x^2 + 85*x^3 + 493*x^4 + 2871*x^5 + 16731*x^6 + ... - Michael Somos, Feb 23 2019

Maple

f:= gfun:-rectoproc({a(n)=6*a(n-1)-a(n-2)-2, a(0)=1, a(1)=3}, a(n), remember):
seq(f(n), n=0..40); # Robert Israel, Dec 16 2015

Mathematica

a[0] = 1; a[1] = 3; a[n_] := a[n] = 6 a[n - 1] - a[n - 2] - 2; Table[a@ n, {n, 0, 22}] (* Michael De Vlieger, Dec 05 2015 *)
Table[(Fibonacci[2n + 1, 2] + 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
LinearRecurrence[{7, -7, 1}, {1, 3, 15}, 30] (* Harvey P. Dale, Feb 16 2017 *)
a[ n_] := (4 + ChebyshevT[n, 3] + ChebyshevT[n + 1, 3])/8; (* Michael Somos, Feb 23 2019 *)

Prog

(PARI) Vec((1-4*x+x^2)/((1-x)*(1-6*x+x^2)) + O(x^100)) \\ Altug Alkan, Dec 06 2015
(Magma) I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2) - 2: n in [1..30]]; // Vincenzo Librandi, Dec 05 2015

Crossrefs

Cf. A001109, A001541, A001653, A011906, A046090, A053141, A156035.

Sequence in context: A202336 A093593 A212201 * A371754 A346374 A118342

Adjacent sequences: A011897 A011898 A011899 * A011901 A011902 A011903

Keyword

nonn,easy

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Extensions

More terms and comments from Wolfdieter Lang

Status

approved