A100230 -id:A100230 - cp's OEIS frontend

A006497 a(n) = 3*a(n-1) + a(n-2) with a(0) = 2, a(1) = 3.

2, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, 510117, 1684802, 5564523, 18378371, 60699636, 200477279, 662131473, 2186871698, 7222746567, 23855111399, 78788080764, 260219353691, 859446141837, 2838557779202

Offset: 0

N. J. A. Sloane

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, Sequence L_{3,n}.
A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.
Haruo Hosoya, What Can Mathematical Chemistry Contribute to the Development of Mathematics?, HYLE--International Journal for Philosophy of Chemistry, Vol. 19, No.1 (2013), pp. 87-105.
Tanya Khovanova, Recursive Sequences
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
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
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2).
Index entries for sequences related to Chebyshev polynomials..
Index entries for linear recurrences with constant coefficients, signature (3,1).

Cf. A006190, A100230, A001622, A014176, A080039, A098316.

a006497 n = a006497_list !! n
a006497_list = 2 : 3 : zipWith (+) (map (* 3) $ tail a006497_list) a006497_list
-- Reinhard Zumkeller, Feb 19 2011

[ n eq 1 select 2 else n eq 2 select 3 else 3*Self(n-1)+Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 20 2011

a:= n-> (<<0|1>, <1|3>>^n. <<2, 3>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2018

Table[LucasL[n, 3], {n, 0, 30}] (* Zerinvary Lajos, Jul 09 2009 *)
LucasL[Range[0, 30], 3] (* Eric W. Weisstein, Apr 17 2018 *)
LinearRecurrence[{3,1},{2,3},30] (* Harvey P. Dale, Feb 17 2020 *)

my(x='x+O('x^30)); Vec((2-3*x)/(1-3*x-x^2)) \\ G. C. Greubel, Jul 05 2017

apply( {A006497(n)=[2,3]*([0,1;1,3]^n)[,1]}, [0..30]) \\ M. F. Hasler, Mar 06 2020

[lucas_number2(n,3,-1) for n in range(0, 30)] # Zerinvary Lajos, Apr 30 2009

G.f.: (2-3*x)/(1-3*x-x^2). - Simon Plouffe in his 1992 dissertation
From Gary W. Adamson, Jun 15 2003: (Start)
a(n) = ((3 + sqrt(13))/2)^n + ((3 - sqrt(13))/2)^n. See bronze mean (A098316).
A006190(n-2) + A006190(n) = a(n-1).
a(n)^2 - 13*A006190(n)^2 = 4(-1)^n. (End)
From Paul Barry, Nov 15 2003: (Start)
E.g.f.: 2*exp(3*x/2)*cosh(sqrt(13)*x/2).
a(n) = 2^(1-n)*Sum_{k=0..floor(n/2)} C(n, 2*k)* (13)^k * 3^(n-2*k).
a(n) = 2*T(n, 3i/2)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
From Hieronymus Fischer, Jan 02 2009: (Start)
fract(((3+sqrt(13))/2)^n) = (1/2)*(1+(-1)^n) - (-1)^n*((3+sqrt(13))/2)^(-n) = (1/2)*(1+(-1)^n) - ((3-sqrt(13))/2)^n.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).
a(n) = round(((3+sqrt(13))/2)^n) for n > 0. (End)
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 3*A097783(n), a(2n) = A057076(n).
a(3n+1) = A041018(5n), a(3n+2) = A041018(5n+3) and a(3n+3) = 2*A041018(5n+4).
Limit_{k -> infinity} a(n+k)/a(k) = (a(n) + A006190(n)*sqrt(13))/2.
Limit_{n -> infinity} a(n)/A006190(n) = sqrt(13).
(End)
a(n) = sqrt(13*(A006190(n))^2 + 4*(-1)^n). - Vladimir Shevelev, Mar 13 2013
G.f.: G(0), where G(k) = 1 + 1/(1 - (x*(13*k-9))/((x*(13*k+4)) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 13*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = Lucas(n,3), Lucas polynomials, L(n,x), evaluated at x=3. - G. C. Greubel, Jun 06 2019
a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 3 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024
a(n) = 2*A006190(n+1) - 3*A006190(n). - R. J. Mathar, Feb 14 2024
a(2*n+1) = 3 + 3*Sum_{k=1..n} a(2*k). - Greg Dresden and Canran Wang, Jul 11 2024
From Peter Bala, Jul 14 2025: (Start)
The following series telescope (Cf. A000032):
For  k >= 1, Sum_{n >= 1} (-1)^((k+1)*(n+1)) * a(2*n*k)/(a((2*n-1)*k)*a((2*n+1)*k)) = 1/a(k)^2.
For positive even k, Sum_{n >= 1} 1/(a(k*n) - (a(k) + 2)/a(k*n)) = 1/(a(k) - 2) and
Sum_{n >= 1} (-1)^(n+1)/(a(k*n) + (a(k) - 2)/a(k*n)) = 1/(a(k) + 2).
For positive odd k, Sum_{n >= 1} 1/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) + 2)/(2*(a(2*k) - 2)) and
Sum_{n >= 1} (-1)^(n+1)/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) - 2)/(2*(a(2*k) - 2)). (End)

Definition completed by M. F. Hasler, Mar 06 2020

A098316 Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.

Original entry on oeis.org

3, 3, 0, 2, 7, 7, 5, 6, 3, 7, 7, 3, 1, 9, 9, 4, 6, 4, 6, 5, 5, 9, 6, 1, 0, 6, 3, 3, 7, 3, 5, 2, 4, 7, 9, 7, 3, 1, 2, 5, 6, 4, 8, 2, 8, 6, 9, 2, 2, 6, 2, 3, 1, 0, 6, 3, 5, 5, 2, 2, 6, 5, 2, 8, 1, 1, 3, 5, 8, 3, 4, 7, 4, 1, 4, 6, 5, 0, 5, 2, 2, 2, 6, 0, 2, 3, 0, 9, 5, 4, 1, 0, 0, 9, 2, 4, 5, 3, 5, 8, 8, 3

Offset: 1

Eric W. Weisstein, Sep 02 2004

For reasons following from the formula section, this constant could be called "the bronze ratio". For this, compare with A001622 and A014176.
If c is this constant and n > 0, then for n even, c^n = [A100230(n), 1, A100230(n)-1, 1, A100230(n)-1, 1, A100230(n)-1, 1, ...], for n odd, c^n = [A100230(n)+1, A100230(n)+1, A100230(n)+1, ...]. - Gerald McGarvey, Dec 15 2007
This is the shape of a 3-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011
From Vladimir Shevelev, Mar 02 2013: (Start)
An analog of Fermat theorem: for prime p, round(c^p) == 3 (mod p).
A generalization for "metallic" constants c_N = (N+sqrt(N^2+4))/2, N>=1: for prime p, round((c_N)^p) == N (mod p). (End)
This is the positive real algebraic number c of degree 2 with minimal polynomial x^3 - x - 1. The other negative root is 3 - c. - Wolfdieter Lang, Aug 29 2022
c^n = c*A006190(n) + A006190(n-1). - Gary W. Adamson, Apr 02 2024

			3.30277563...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A001622, A014176, A098317, A098318, A000032, A006497, A080039, A049310.

RealDigits[(3 + Sqrt[13])/2, 10, 100][[1]] (* G. C. Greubel, Apr 16 2017 *)

(3 + sqrt(13))/2 \\ Charles R Greathouse IV, Jul 24 2013

3 plus the constant in A085550. - R. J. Mathar, Sep 02 2008
From Hieronymus Fischer, Jan 02 2009: (Start)
Set c:=(3+sqrt(13))/2. Then the fractional part of c^n equals 1/c^n, if n odd. For even n, the fractional part of c^n is equal to 1-(1/c^n).
c:=(3+sqrt(13))/2 satisfies c-c^(-1)=floor(c)=3, hence c^n + (-c)^(-n) = round(c^n) for n>0, which follows from the general formula of A001622.
1/c=(sqrt(13)-3)/2.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).
Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A014176 (the silver ratio: where floor(x)=2). (End)
c=3+sum{k>=1}(-1)^(k-1)/(A006190(k)*A006190(k+1)). - Vladimir Shevelev, Feb 23 2013
A generalization for "metallic" constants c_N = (N+sqrt(N^2+4))/2, N>=1. Let {A_N(n), n>=0} be the sequence 0, 1, N, N^2+1, N^3+2*N, N^4+3*N^2+1,..., a(N) = N*a(N-1) + a(N-2). Then c_N = N + sum_{n>=1} (-1)^(n-1)/(A_N(n)*A_N(n+1)) (cf. A001622, A014176, A098316, A098317, A098318). - Vladimir Shevelev, Feb 23 2013
Equals lim_{n->oo} S(n, sqrt(13))/S(n-1, sqrt(13)), with the S-Chebyshev polynomial (see A049310). - Wolfdieter Lang, Nov 15 2023

A100228 G.f. A(x) satisfies: 4^n - 1 = Sum_{k=0..n} [x^k] A(x)^n and also satisfies: (4+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k] A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 2, 3, -3, -6, 24, 3, -183, 273, 1131, -4407, -3081, 48360, -54750, -396195, 1282551, 1860186, -17122944, 11240049, 166745823, -432682314, -1054472016, 6822994737, -835915197, -76044224139, 152526011235, 587055710271, -2871405804783, -1378878506592, 36081844133766

Offset: 0

Paul D. Hanna, Nov 29 2004

sign

More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.

			From the table of powers of A(x) (A100229), we see that
4^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,2],3,-3,-6,24,3,-183,273,...
A^2=[1,4,10],6,-15,6,75,-174,-276,...
A^3=[1,6,21,35],9,-36,63,72,-612,...
A^4=[1,8,36,92,118],12,-66,192,-147,...
A^5=[1,10,55,185,380,392],15,-105,420,...
A^6=[1,12,78,322,879,1506,1297],18,-153,...

Cf. A100229, A100230.

CoefficientList[Series[(1+3x+Sqrt[1+2x+13x^2])/2,{x,0,30}],x] (* or *) Join[{1},RecurrenceTable[{a[1]==2,a[2]==3,a[n]==-((2n-3)a[n-1]+ 13(n-3)a[n-2])/n},a,{n,30}]] (* Harvey P. Dale, Feb 29 2012 *)

{a(n) = if(n==0,1,(4^n-1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)}
for(n=0,20,print1(a(n),", "))

{a(n) = if(n==0,1,if(n==1,2,if(n==2,3,-((2*n-3)*a(n-1)+13*(n-3)*a(n-2))/n)))}
for(n=0,30,print1(a(n),", "))

{a(n) = polcoeff((1+3*x+sqrt(1+2*x+13*x^2+x^2*O(x^n)))/2,n)}
for(n=0,30,print1(a(n),", "))

a(n) = -((2*n-3)*a(n-1) + 13*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(3)=3.

G.f.: (1+3*x + sqrt(1 + 2*x + 13*x^2))/2.

A100233 a(n) = Lucas(3*n) - 1.

Original entry on oeis.org

1, 3, 17, 75, 321, 1363, 5777, 24475, 103681, 439203, 1860497, 7881195, 33385281, 141422323, 599074577, 2537720635, 10749957121, 45537549123, 192900153617, 817138163595, 3461452808001, 14662949395603, 62113250390417, 263115950957275, 1114577054219521

Offset: 0

Paul D. Hanna, Nov 29 2004

Main diagonal of triangle A100232.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Cf. A000032, A100230, A100232, A016064.

I:=[1, 3, 17]; [n le 3 select I[n] else 5*Self(n-1) -3*Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[3*n] - 1, {n,0,50}] (* or *) LinearRecurrence[{5,-3,-1}, {1,3,17}, 30] (* G. C. Greubel, Dec 21 2017 *)

a(n)=if(n==0,1,n*polcoeff(log((1-x)/(1-4*x-x^2)+x*O(x^n)),n))

Vec((1-2*x+5*x^2)/((1-x)*(1-4*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 02 2016

a(n) = A014448(n) - 1.
a(n) = 4*a(n-1) + a(n-2) + 4 for n>1, with a(0)=1, a(1)=3.
G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-4*x-x^2)).
a(n) = [x^n] ( 1 + 2*x + sqrt(1 + 2*x + 5*x^2) )^n. Cf. A016064. - Peter Bala, Jun 23 2015
From Colin Barker, Jun 02 2016: (Start)
a(n) = -1+(2-sqrt(5))^n+(2+sqrt(5))^n.
a(n) = 5*a(n-1)-3*a(n-2)-a(n-3) for n>2.
G.f.: (1-2*x+5*x^2) / ((1-x)*(1-4*x-x^2)).
(End)

New definition from Ralf Stephan, Dec 01 2004

A100229 Triangle, read by rows, of the coefficients of [x^k] in G100228(x)^n such that the row sums are 4^n-1 for n>0, where G100228(x) is the g.f. of A100228.

Original entry on oeis.org

1, 1, 2, 1, 4, 10, 1, 6, 21, 35, 1, 8, 36, 92, 118, 1, 10, 55, 185, 380, 392, 1, 12, 78, 322, 879, 1506, 1297, 1, 14, 105, 511, 1715, 3948, 5803, 4286, 1, 16, 136, 760, 3004, 8536, 17020, 21904, 14158, 1, 18, 171, 1077, 4878, 16344, 40395, 71109, 81387, 46763

Offset: 0

Paul D. Hanna, Nov 29 2004

The main diagonal forms A100230. Secondary diagonal is T(n+1,n) = (n+1)*A052924(n). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

			Rows begin:
[1],
[1,2],
[1,4,10],
[1,6,21,35],
[1,8,36,92,118],
[1,10,55,185,380,392],
[1,12,78,322,879,1506,1297],
[1,14,105,511,1715,3948,5803,4286],
[1,16,136,760,3004,8536,17020,21904,14158],...
where row sums form 4^n-1 for n>0:
4^1-1 = 1+2 = 3
4^2-1 = 1+4+10 = 15
4^3-1 = 1+6+21+35 = 63
4^4-1 = 1+8+36+92+118 = 255
4^5-1 = 1+10+55+185+380+392 = 1023.
The main diagonal forms A100230 = [1,2,10,35,118,392,1297,...],
where Sum_{n>=1} A100230(n)/n*x^n = log((1-x)/(1-3*x-x^2)).

Cf. A100228, A100230, A052924.

PARI
```
T(n,k,m=4)=if(n
				
```

G.f.: A(x, y)=(1-2*x*y+4*x^2*y^2)/((1-x*y)*(1-3*x*y-x^2*y^2-x*(1-x*y))).

cp's OEIS Frontend

A006497 a(n) = 3*a(n-1) + a(n-2) with a(0) = 2, a(1) = 3.

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A098316 Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.

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A100228 G.f. A(x) satisfies: 4^n - 1 = Sum_{k=0..n} [x^k] A(x)^n and also satisfies: (4+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k] A(x)^n denotes the coefficient of x^k in A(x)^n.

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A100233 a(n) = Lucas(3*n) - 1.

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A100229 Triangle, read by rows, of the coefficients of [x^k] in G100228(x)^n such that the row sums are 4^n-1 for n>0, where G100228(x) is the g.f. of A100228.

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