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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

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

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 */

CROSSREFS

Cf. A001653, A046090.

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 September 4 14:51 EDT 2015. Contains 261335 sequences.