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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. 35
 0, 1, 5, 20, 76, 285, 1065, 3976, 14840, 55385, 206701, 771420, 2878980, 10744501, 40099025, 149651600, 558507376, 2084377905, 7779004245, 29031639076, 108347552060, 404358569165, 1509086724601, 5631988329240, 21018866592360, 78443478040201, 292755045568445 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 On the previous comment: for m=0 this is actually one third of the same triangular number. - Zak Seidov, Apr 07 2011 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 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) 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 From John P. McSorley, May 26 2020: (Start) 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. Also related to the block sizes of small multi-set designs. (End) REFERENCES R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence. Brian Lawrence and Will Sawin, The Shafarevich conjecture for hypersurfaces in abelian varieties, arXiv:2004.09046 [math.NT], 2020. Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019. L. A. Medina and A. Straub, On multiple and infinite log-concavity, 2013. S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987. - From N. J. A. Sloane, Dec 26 2012 Robert Phillips, A triangular number result, 2009. Eric Weisstein, Centered Polygonal Number. Index entries for linear recurrences with constant coefficients, signature (5,-5,1). FORMULA a(n) = 4*a(n-1) - a(n-2) + 1. a(n) = A001075(n) - a(n-1) - 1. a(n) = (A001835(n+1) - 1)/2 = (A001353(n+1) - A001353(n) - 1)/2. a(n) = a(n-1) + A001353(n), i.e., partial sum of A001353. From Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002: (Start) a(n+2) = 4*a(n+1) - a(n) + 1 for a(0)=0, a(1)=1. G.f.: x/((1 - x)*(1 - 4*x + x^2)). a(n) = (1/12)*((3 - sqrt(3))*(2 - sqrt(3))^n + (3 + sqrt(3))*(2 + sqrt(3))^n-6). (End) a(n) = (1/12)*(A003500(n) + A003500(n+1)-6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 11 2003 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 G.f.: x/(1 - 5*x + 5*x^2 - x^3). a(n) = A079935(n+1) + A001571(n) for n>0, a(0)=0. - Gerry Martens, Jun 05 2015 a(n)*a(n-2) = a(n-1)*(a(n-1) - 1) for n>1. - Bruno Berselli, Nov 29 2016 From John P. McSorley, May 25 2020: (Start) a(n)^2 - a(n) + a(n-1)^2 - a(n-1) - 4*a(n)*a(n-1) = 0. a(n) = A001834(n-1) + a(n-2). (End) (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 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 EXAMPLE a(2)=5 and T(5)=15 which is 1/3 of 45=T(9). MAPLE 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): map(f, [\$0..50]); # Robert Israel, Jun 05 2015 MATHEMATICA CoefficientList[Series[x/(1 - 5*x + 5*x^2 - x^3), {x, 0, nn}], x] (* T. D. Noe, Jun 04 2012 *) LinearRecurrence[{5, -5, 1}, {0, 1, 5}, 30] (* Harvey P. Dale, Dec 23 2012 *) PROG (PARI) M = [1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=1, 30, print1(([1, 0, 0]*M^i), ", ")) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005 (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 CROSSREFS Cf. A001075, A001353, A001571, A001834, A001835, A079935.  Also cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3). Sequence in context: A275908 A290909 A270023 * A000758 A005283 A057552 Adjacent sequences:  A061275 A061276 A061277 * A061279 A061280 A061281 KEYWORD nonn,easy AUTHOR Henry Bottomley, Jun 04 2001 EXTENSIONS More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005 STATUS approved

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Last modified April 11 09:24 EDT 2021. Contains 342886 sequences. (Running on oeis4.)