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 A011900 a(n) = 6*a(n-1) - a(n-2) - 2 with a(0) = 1, a(1) = 3. 20
 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) = 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 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. A001541, A001653, A011906, A046090, A053141, A156035. Sequence in context: A202336 A093593 A212201 * A346374 A118342 A084209 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 December 11 00:22 EST 2023. Contains 367717 sequences. (Running on oeis4.)