A001571 a(n) = 4*a(n-1) - a(n-2) + 1, with a(0) = 0, a(1) = 2.

0, 2, 9, 35, 132, 494, 1845, 6887, 25704, 95930, 358017, 1336139, 4986540, 18610022, 69453549, 259204175, 967363152, 3610248434, 13473630585, 50284273907, 187663465044, 700369586270, 2613814880037, 9754889933879, 36405744855480, 135868089488042

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Second member of the Diophantine pair (m,k) that satisfies 3(m^2 + m) = k^2 + k: a(n) = k. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
• Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
• Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
• Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
• Vladimir Pletser, Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2103.03019 [math.GM], 2021.
• Vladimir Pletser, Searching for multiple of triangular numbers being triangular numbers, 2021.
• Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2021.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Jamie Radcliffe and Adam Volk, Generalized saturation problems for cliques, paths, and stars, arXiv:2101.04213 [math.CO], 2021.
• V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.
• Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Crossrefs

Cf. A001353, A001834, A061278, A071954, A082840.

Programs

Magma

I:=[0,2]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2)+1: n in [1..30]]; // Vincenzo Librandi, Jun 07 2015

Magma

[(Evaluate(ChebyshevU(n+1), 2) + Evaluate(ChebyshevU(n), 2) - 1)/2 : n in [0..30]]; // G. C. Greubel, Feb 02 2022

Maple

f := gfun:-rectoproc({a(0) = 0, a(1) = 2, a(n) = 4*a(n - 1) - a(n - 2) + 1}, a(n), remember): map(f, [$ (0 .. 40)])[]; # Vladimir Pletser, Jul 25 2020

Mathematica

a[0]=0; a[1]=2; a[n_]:= a[n]= 4a[n-1] -a[n-2] +1; Table[a[n], {n, 0, 24}] (* Robert G. Wilson v, Apr 24 2004 *)
Table[(ChebyshevU[n,2] +ChebyshevU[n-1,2] -1)/2, {n,0,30}] (* G. C. Greubel, Feb 02 2022 *)

Sage

[(chebyshev_U(n,2) + chebyshev_U(n-1,2) - 1)/2 for n in (0..30)] # G. C. Greubel, Feb 02 2022

Formula

a(n) = (A001834(n) - 1)/2.

G.f.: x*(2-x)/( (1-x)*(1-4*x+x^2) ). - Simon Plouffe in his 1992 dissertation.

a(n) = sqrt((-2 + (2 - sqrt(3))^n + (2 + sqrt(3))^n)*(2 + (2 - sqrt(3))^(1 + n) + (2 + sqrt(3))^(1 + n)))/(2*sqrt(2)). - Gerry Martens, Jun 05 2015

E.g.f.: (exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - exp(x))/2. - Franck Maminirina Ramaharo, Nov 12 2018

a(n) = 2*A061278(n) - A061278(n-1). - R. J. Mathar, Feb 06 2020

a(n) = ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n)/4 - (1/2). - Vladimir Pletser, Jan 15 2021

a(n) = (ChebyshevU(n, 2) + ChebyshevU(n-1, 2) - 1)/2. - G. C. Greubel, Feb 02 2022

Extensions

Better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002

More terms and new description from Robert G. Wilson v, Apr 24 2004