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A101265 a(1) = 1, a(2) = 2, a(3) = 6; a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n>3. 10
1, 2, 6, 21, 77, 286, 1066, 3977, 14841, 55386, 206702, 771421, 2878981, 10744502, 40099026, 149651601, 558507377, 2084377906, 7779004246, 29031639077, 108347552061, 404358569166, 1509086724602, 5631988329241, 21018866592361, 78443478040202, 292755045568446 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; then [1,0,0]*M^n = [a(n), A001353(n), A061278(n-1)] for n>1; Further a(n) = A061278(n) + 1; A001353(n) is first difference of a(n) and first difference of A061278(n). Let v(n) = [1,0,0]*M^n, then sum(v_i(n)) = A001075(n) for n>=0; and v_1(n) + v_3(n) = A001835(n) for n>=0; Characteristic polynomial of M = x^3 - 5x^2 + 5x - 1; a(n)/a(n-1) tends to 2 + sqrt(3) = 3.732.... (see A019973) (a root of the polynomial and an eigenvalue of the matrix).

a(n) = 1 + A061278(n). Also numbers n such that the RootMeanSquare(1,...,6*n-5) is an integer. - Ctibor O. Zizka, Dec 17 2008

a(n) blue and b(n) red balls in an urn. Draw 3 balls without replacement. Then Probability(3 red balls)=Probability(1 red and 2 blue balls); binomial(b(n),3)=binomial(b(n),1)*binomial(a(n),2); b(n)=A179167(n). - Paul Weisenhorn, Jul 01 2010

Conjecture: consecutive terms of this sequence and consecutive terms of A032908 provide all the positive integer solutions of (a+b)*(a+b+1) == 0 mod (a*b). - Robert Israel, Aug 26 2015

Conjecture is true: see StackExchange link. - Robert Israel, Sep 06 2015

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

R. Israel, W. Jagy et al, Diophantine equation (x+y)(x+y+1)-kxy=0, Math StackExchange, Sep 1 2015.

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

FORMULA

a(n) = A005246(n)*A005246(n+1). a(n+1) = a(n)*(a(n)+1)/a(n-1). - Franklin T. Adams-Watters, Apr 24 2006

a(n) = (A001835(n) + 1) / 2. - Ralf Stephan, May 16 2007

O.g.f.: x*(1-3*x+x^2)/((1-x)*(1-4*x+x^2)). - R. J. Mathar, Aug 22 2008

a(n) = 4*a(n-1) - a(n-2) - 1. - N. Sato, Jan 21 2010

a(n) = (6+(3+r)*(2+r)^(n-1) + (3-r)*(2-r)^(n-1))/12; r=sqrt(3). - Paul Weisenhorn, Jul 01 2010

a(n+1) = a(n) * (a(n) + 1) / a(n-1) for n>1 with a(0)=1, a(1)=1. - Paul D. Hanna, Apr 08 2012

From Peter Bala, May 01 2012: (Start)

a(n+1) = 1 + Sum {k = 1..n} 2^(k-1)*binomial(n+k,2*k).

Row sums of A211955.

a(n) = T(n,u)*T(n+1,u)/u with u = sqrt(3) and T(n,x) denotes the Chebyshev polynomial of the first kind.

Sum_{n >= 0} 1/a(n) = sqrt(3). In fact, 3 - (Sum_{n = 0..2*N} 1/a(n))^2 = 2/(A001835(N+1))^2 and 3 - (Sum_{n = 0..2*N+1} 1/a(n))^2 = 3/(A001075(N+1))^2. (End)

From Robert Israel, Aug 26 2015: (Start)

(a(n) + a(n+1))*(a(n) + a(n+1) + 1) = 6 * a(n) * a(n+1).

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

a(n) = (ChebyshevU(n, 2) - ChebyshevU(n-1, 2) + 1)/2 = (ChebyshevT(n, 2) + ChebyshevU(n, 2) + 2)/4. - G. C. Greubel, Dec 23 2019

EXAMPLE

For n=5 a(5)=77; b(5)=134; binomial(134,3)=392084; binomial(134,1)*binomial(77,2)=134*2926=392084. - Paul Weisenhorn, Jul 01 2010

MAPLE

r:=sqrt(3): for n from 1 to 100 do a[n]:=(6+(3+r)*(2+r)^(n-1)+(3-r)*(2-r)^(n-1))/12: end do: # Paul Weisenhorn, Jul 01 2010

r:=sqrt(3): a[n]:=round((6+(3+r)*(2+r)^(n-1))/12): # Paul Weisenhorn, Jul 01 2010

f:= proc(n)

  option remember; local x;

  x:= procname(n-1);

  2*x + (sqrt(12*x^2 - 12*x + 1) - 1)/2

end proc:

f(1):= 1:

map(f, [$1..30]); # Robert Israel, Aug 26 2015

seq( simplify((ChebyshevU(n, 2) - Chebyshev(n-1, 2) + 1)/2), n=0..20); # G. C. Greubel, Dec 23 2019

MATHEMATICA

LinearRecurrence[{5, -5, 1}, {1, 2, 6}, 25] (* Ray Chandler, Jan 27 2014 *)

CoefficientList[Series[(1-3x+x^2)/((1-x)(1-4x+x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 07 2015 *)

Table[(ChebyshevU[n, 2] - ChebyshevU[n-1, 2] + 1)/2, {n, 0, 20}] (* G. C. Greubel, Dec 23 2019 *)

PROG

(PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=1, 30, print1(([1, 0, 0]*M^i)[1], ", "))

(PARI) {a(n)=polcoeff(x*(1-3*x+x^2)/((1-x)*(1-4*x+x^2)+x*O(x^n)), n)}

(PARI) {a(n)=if(n==0, 1, if(n==1, 1, a(n-1)*(a(n-1)+1)/a(n-2)))} /* Paul D. Hanna, Apr 08 2012 */

(PARI) vector(21, n, (polchebyshev(n, 2, 2) - polchebyshev(n-1, 2, 2) + 1)/2 ) \\ G. C. Greubel, Dec 23 2019

(Haskell)

a101265 n = a101265_list !! (n-1)

a101265_list = 1 : 2 : 6 : zipWith (+) a101265_list

    (map (* 5) $ tail $ zipWith (-) (tail a101265_list) a101265_list)

-- Reinhard Zumkeller, May 18 2014

(MAGMA) I:=[1, 2, 6]; [n le 3 select I[n] else 5*Self(n-1) - 5*Self(n-2) + Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 07 2015

(Sage) [(chebyshev_U(n, 2) - chebyshev_U(n-1, 2) + 1)/2 for n in (0..20)] # G. C. Greubel, Dec 23 2019

(GAP) a:=[1, 2, 6];; for n in [4..20] do a[n]:=5a[n-1]-5*a[n-2]+a[n-3]; od; a; #

G. C. Greubel, Dec 23 2019

CROSSREFS

Cf. A001075, A001353, A001835, A005246, A019973, A032908, A061278, A211955.

Sequence in context: A294823 A294824 A294825 * A101879 A242622 A279561

Adjacent sequences:  A101262 A101263 A101264 * A101266 A101267 A101268

KEYWORD

nonn,easy

AUTHOR

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Jan 25 2005

EXTENSIONS

a(26)-a(27) from Vincenzo Librandi, Sep 07 2015

STATUS

approved

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Last modified February 21 05:23 EST 2020. Contains 332086 sequences. (Running on oeis4.)