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A011900 a(n) = 6*a(n-1) - a(n-2) - 2 with a(0) = 1, a(1) = 3. 16
1, 3, 15, 85, 493, 2871, 16731, 97513, 568345, 3312555, 19306983, 112529341, 655869061, 3822685023, 22280241075, 129858761425, 756872327473, 4411375203411, 25711378892991, 149856898154533, 873430010034205, 5090723162050695, 29670908962269963 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Members of Diophantine pairs.

Solution to b(b-1) = 2a(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

Sum_{x=a(n)..A001109(n+1)} x = a(n)*A001109(n+1) =A011906(n+1). Example n=2, 3+4+5+6=18, 3x6=18. - Paul Cleary, Dec 05 2015

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

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

FORMULA

a(n) = (A001653(n) + 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. - 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

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

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 *)

PROG

(PARI) {a(n)=if(n<0, 0, if(n==0, 1, if(n==1, 3, a(n-1)*(a(n-1)+2)/a(n-2))))} /* Paul D. Hanna, Apr 08 2012 */

(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

(PARI) Vec((1-4*x+x^2)/((1-x)*(1-6*x+x^2)) + O(x^100)) \\ Altug Alkan, Dec 06 2015

CROSSREFS

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

Sequence in context: A202336 A093593 A212201 * A118342 A084209 A182016

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

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Last modified August 26 02:44 EDT 2016. Contains 275847 sequences.