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A011900
a(n) = 6*a(n-1) - a(n-2) - 2 with a(0) = 1, a(1) = 3.
21
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
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 61.
Mario Velucchi "The Pell's equation ... an amusing application" in Mathematics and Informatics Quarterly, to appear 1997.
LINKS
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]
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
a(n) = A053141(n) + 1 = A000194(A029549(n))+1 = A002024(A075528(n))+1. - Pontus von Brömssen, Sep 11 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, 40}] (* Michael De Vlieger, Dec 05 2015 *)
Table[(Fibonacci[2n + 1, 2] + 1)/2, {n, 0, 40}] (* Vladimir Reshetnikov, Sep 16 2016 *)
LinearRecurrence[{7, -7, 1}, {1, 3, 15}, 40] (* 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^50)) \\ 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..40]]; // Vincenzo Librandi, Dec 05 2015
(SageMath) [(1+lucas_number1(2*n+1, 2, -1))//2 for n in range(41)] # G. C. Greubel, Oct 17 2024
KEYWORD
nonn,easy
AUTHOR
Mario Velucchi (mathchess(AT)velucchi.it)
EXTENSIONS
More terms and comments from Wolfdieter Lang
STATUS
approved