|
| |
|
|
A001571
|
|
a(n) = 4*a(n-1) - a(n-2) + 1, with a(0) = 0, a(1) = 2.
(Formerly M1928 N0762)
|
|
24
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
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).
|
|
|
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) = -(1/2) - (1/4)*sqrt(3)*(2 - sqrt(3))^n + (1/4)*sqrt(3)*(2 + sqrt(3))^n + (1/4)*(2 - sqrt(3))^n + (1/4)*(2 + sqrt(3))^n, with n >= 0. - Paolo P. Lava, Jul 31 2008
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
|
|
|
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 *)
|
|
|
PROG
|
(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
(Sage) [(chebyshev_U(n, 2) + chebyshev_U(n-1, 2) - 1)/2 for n in (0..30)] # G. C. Greubel, Feb 02 2022
|
|
|
CROSSREFS
|
Cf. A001353, A001834, A061278, A071954, A082840.
Sequence in context: A032601 A255410 A083141 * A092431 A147762 A077837
Adjacent sequences: A001568 A001569 A001570 * A001572 A001573 A001574
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
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
|
|
|
STATUS
|
approved
|
| |
|
|
