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A061278 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) with a(1) = 1 and a(k) = 0 if k <= 0. 37

%I #152 Jul 07 2022 02:24:15

%S 0,1,5,20,76,285,1065,3976,14840,55385,206701,771420,2878980,10744501,

%T 40099025,149651600,558507376,2084377905,7779004245,29031639076,

%U 108347552060,404358569165,1509086724601,5631988329240,21018866592360,78443478040201,292755045568445

%N a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) with a(1) = 1 and a(k) = 0 if k <= 0.

%C Indices m of triangular numbers T(m) which are one-third of another triangular number: 3*T(m) = T(k); the k's are given by A001571. - Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002

%C On the previous comment: for m=0 this is actually one third of the same triangular number. - _Zak Seidov_, Apr 07 2011

%C Also numbers n such that the n-th centered 24-gonal number 12*n*(n+1)+1 is a perfect square A001834(n)^2, where A001834(n) is defined by the recursion: a(0) = 1, a(1) = 5, a(n) = 4*a(n-1) - a(n-2) + 1. - _Alexander Adamchuk_, Apr 21 2007

%C Also numbers n such that RootMeanSquare(5,...,6*n-1) is an integer. - _Ctibor O. Zizka_, Dec 17 2008 (Corrected by _Robert K. Moniot_, Jul 22 2020)

%C Also numbers n such that n*(n+1) = Sum_{i=1..x} n+i for some x. (This does not apply to the first term.). - _Gil Broussard_, Dec 23 2008

%C From _John P. McSorley_, May 26 2020: (Start)

%C Consecutive terms (a(n-1), a(n)) = (u,v) give all points on the hyperbola u^2 - u + v^2 - v - 4*u*v = 0 in quadrant I with both coordinates an integer.

%C Also related to the block sizes of small multi-set designs. (End)

%C If a(n) white balls and a(n+1) black balls are mixed in a bag, and a pair of balls is drawn without replacement, the probability that one ball of each color is drawn is exactly 1/3. These are the only integers for which the probability is 1/3. For example, if there are 20 white balls and 76 black balls, the probability of drawing one of each is (20/96)*(76/95) + (76/96)*(20/95) = 1/3. - _Elliott Line_, May 13 2022

%D R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.

%H Vincenzo Librandi, <a href="/A061278/b061278.txt">Table of n, a(n) for n = 0..1000</a>

%H Niccolò Castronuovo, <a href="https://arxiv.org/abs/2102.02739">On the number of fixed points of the map gamma</a>, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.

%H Brian Lawrence and Will Sawin, <a href="https://arxiv.org/abs/2004.09046">The Shafarevich conjecture for hypersurfaces in abelian varieties</a>, arXiv:2004.09046 [math.NT], 2020.

%H Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.

%H L. A. Medina and A. Straub, <a href="http://emmy.uprrp.edu/lmedina/papers/logconcave/logconcavity.pdf">On multiple and infinite log-concavity</a>, 2013.

%H S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, <a href="http://aif.cedram.org/item?id=AIF_2012__62_3_937_0">2-frieze patterns and the cluster structure of the space of polygons</a>, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987. - From _N. J. A. Sloane_, Dec 26 2012

%H Robert Phillips, <a href="https://web.archive.org/web/20100713033404/http://www.usca.edu/math/~mathdept/bobp/pdf/result.pdf">A triangular number result</a>, 2009.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2101.00998">Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers</a>, arXiv:2101.00998 [math.NT], 2021.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.12392">Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers</a>, arXiv:2102.12392 [math.GM], 2021.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.13494">Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations</a>, arXiv:2102.13494 [math.NT], 2021.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2103.03019">Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers</a>, arXiv:2103.03019 [math.GM], 2021.

%H Vladimir Pletser, <a href="https://doi.org/10.13140/RG.2.2.35428.91527">Searching for multiple of triangular numbers being triangular numbers</a>, 2021.

%H Vladimir Pletser, <a href="https://www.researchgate.net/profile/Vladimir-Pletser/publication/359808848_USING_PELL_EQUATION_SOLUTIONS_TO_FIND_ALL_TRIANGULAR_NUMBERS_MULTIPLE_OF_OTHER_TRIANGULAR_NUMBERS/">Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers</a>, 2022.

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).

%F a(n) = 4*a(n-1) - a(n-2) + 1.

%F a(n) = A001075(n) - a(n-1) - 1.

%F a(n) = (A001835(n+1) - 1)/2 = (A001353(n+1) - A001353(n) - 1)/2.

%F a(n) = a(n-1) + A001353(n), i.e., partial sum of A001353.

%F From Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002: (Start)

%F a(n+2) = 4*a(n+1) - a(n) + 1 for a(0)=0, a(1)=1.

%F G.f.: x/((1 - x)*(1 - 4*x + x^2)).

%F a(n) = (1/12)*((3 - sqrt(3))*(2 - sqrt(3))^n + (3 + sqrt(3))*(2 + sqrt(3))^n-6). (End)

%F a(n) = (1/12)*(A003500(n) + A003500(n+1)-6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 11 2003

%F a(n+1) = Sum_{k=0..n} U(k, 2)} = Sum_{k=0..n} S(k, 4), where U(n,x) and S(n,x) are Chebyshev polynomials. - _Paul Barry_, Nov 14 2003

%F G.f.: x/(1 - 5*x + 5*x^2 - x^3).

%F a(n) = A079935(n+1) + A001571(n) for n>0, a(0)=0. - _Gerry Martens_, Jun 05 2015

%F a(n)*a(n-2) = a(n-1)*(a(n-1) - 1) for n>1. - _Bruno Berselli_, Nov 29 2016

%F From _John P. McSorley_, May 25 2020: (Start)

%F a(n)^2 - a(n) + a(n-1)^2 - a(n-1) - 4*a(n)*a(n-1) = 0.

%F a(n) = A001834(n-1) + a(n-2). (End)

%F (T(a(n)-1) + T(a(n+1)-1))/T(a(n) + a(n+1) - 1) = 2/3 where T(i) is the i-th triangular number. - _Robert K. Moniot_, Oct 11 2020

%F E.g.f.: exp(x)*(exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 3)/6. - _Stefano Spezia_, Feb 05 2021

%F a(n) = A101265(n) - 1. - _Jon E. Schoenfield_, Jan 01 2022

%e a(2)=5 and T(5)=15 which is 1/3 of 45=T(9).

%p f:= gfun:-rectoproc({a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3),a(1)=1,a(0)=0,a(-1)=0},a(n),remember):

%p map(f, [$0..50]); # _Robert Israel_, Jun 05 2015

%t CoefficientList[Series[x/(1 - 5*x + 5*x^2 - x^3), {x, 0, nn}], x] (* _T. D. Noe_, Jun 04 2012 *)

%t LinearRecurrence[{5,-5,1},{0,1,5},30] (* _Harvey P. Dale_, Dec 23 2012 *)

%o (PARI) M = [1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=1, 30, print1(([1, 0, 0]*M^i)[3], ",")) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005

%o (Magma) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2) + 1: n in [1..30]]; // _Vincenzo Librandi_, Dec 23 2012

%Y Cf. A001075, A001353, A001571, A001834, A001835, A079935, A101265. Also cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

%K nonn,easy

%O 0,3

%A _Henry Bottomley_, Jun 04 2001

%E More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005

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Last modified April 23 22:36 EDT 2024. Contains 371917 sequences. (Running on oeis4.)