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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

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

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 *)

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 *)

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 July 25 02:49 EDT 2017. Contains 289779 sequences.