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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. 27
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) = 4a(n-1) - a(n-2) + 1. - Alexander Adamchuk, Apr 21 2007

Also numbers n such that RootMeanSquare(1,5,...,6*n-1) is an integer. [Ctibor O. Zizka, Dec 17 2008]

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]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

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). (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). - 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

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

Join[{a=0, b=1}, Table[c=4*b-a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

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)[3], ", ")) \\ 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. A001571, A001834.

Sequence in context: A275909 A275908 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 March 28 15:52 EDT 2017. Contains 284243 sequences.