A098158 -id:A098158 - cp's OEIS frontend

A002533 a(n) = 2a(n-1) + 5a(n-2), with a(0) = a(1) = 1.

1, 1, 7, 19, 73, 241, 847, 2899, 10033, 34561, 119287, 411379, 1419193, 4895281, 16886527, 58249459, 200931553, 693110401, 2390878567, 8247309139, 28449011113, 98134567921, 338514191407, 1167701222419, 4027973401873, 13894452915841, 47928772841047, 165329810261299

Offset: 0

N. J. A. Sloane

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - Cino Hilliard, Sep 25 2005
a(n), n>0 = term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 2,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010
Pisano period lengths: 1, 1, 1, 4, 4, 1, 24, 4, 3, 4, 120, 4, 56, 24, 4, 8, 288, 3, 18, 4, ... - R. J. Mathar, Aug 10 2012
a(k*m) is divisible by a(m) if k is odd.  - Robert Israel, May 03 2024

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,5).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

[(1/2)*Floor((1+Sqrt(6))^n+(1-Sqrt(6))^n): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

[n le 2 select 1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

A002533:=(-1+z)/(-1+2*z+5*z**2); # conjectured by Simon Plouffe in his 1992 dissertation

f[n_] := Simplify[((1 + Sqrt[6])^n + (1 - Sqrt[6])^n)/2]; Array[f, 28, 0] (* Or *)
LinearRecurrence[{2, 5}, {1, 1}, 28] (* Or *)
Table[ MatrixPower[{{1, 2}, {3, 1}}, n][[1, 1]], {n, 0, 25}]
(* Robert G. Wilson v, Sep 18 2013 *)

a(n)=([0,1; 5,2]^n*[1;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

x='x+O('x^30); Vec((1-x)/(1-2*x-5*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number2(n,2,-5)/2 for n in range(0, 21)] # Zerinvary Lajos, Apr 30 2009

a(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
From Mario Catalani (mario.catalani(AT)unito.it), May 03 2003: (Start)
G.f.: (1-x)/(1-2*x-5*x^2).
a(n) = (1/2)*((1+sqrt(6))^n + (1-sqrt(6))^n).
a(n)/A083694(n) converges to sqrt(3/2).
a(n)/A083695(n) converges to sqrt(2/3).
a(n) = a(n-1) + 3*A083694(n-1).
a(n) = a(n-1) + 2*A083695(n-1), n>0. (End)
Binomial transform of expansion of cosh(sqrt(6)*x) (A000400, with interpolated zeros). E.g.f.: exp(x)*cosh(sqrt(6)*x) - Paul Barry, May 09 2003
From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)
a(2*n+1) = 2*a(n)*a(n+1) - (-5)^n.
a(n)^2 - 6*A002532(n)^2 = (-5)^n. (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 6^k. - Paul Barry, Jul 25 2004
a(n) = Sum_{k=0..n} A098158(n,k)*6^(n-k). - Philippe Deléham, Dec 26 2007
If p(1)=1, and p(I)=6, for i>1, and if A is the Hessenberg matrix of order n defined by: A(i,j) = p(j-i+1) for i<=j, A(i,j)=-1 for i=j+1, and A(i,j)=0 otherwise. Then, for n>=1, a(n) = det A. - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A083098 a(n) = 2a(n-1) + 6a(n-2).

Original entry on oeis.org

1, 1, 8, 22, 92, 316, 1184, 4264, 15632, 56848, 207488, 756064, 2757056, 10050496, 36643328, 133589632, 487039232, 1775616256, 6473467904, 23600633344, 86042074112, 313687948288, 1143628341248, 4169384372224, 15200538791936

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

a(n+1) = a(n) + 7*A083099(n-1); a(n+1)/A083099(n) converges to sqrt(7).
Binomial transform of expansion of cosh(sqrt(7)x) (A000420 with interpolated zeros: 1, 0, 7, 0, 49, 0, 343, 0, ...).
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard, Sep 25 2005
a(n) is the number of compositions of n when there are 1 type of 1 and 7 types of other natural numbers. - Milan Janjic, Aug 13 2010

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Project Euler, Problem 752, sequence alpha(n).
Index entries for linear recurrences with constant coefficients, signature (2,6).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

CoefficientList[Series[(1+6x)/(1-2x-6x^2), {x, 0, 25}], x]
LinearRecurrence[{2, 6}, {1, 1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)
a[n_] := Simplify[((1 + Sqrt[7])^n + (1 - Sqrt[7])^n)/2]; Array[a, 25, 0] (* Robert G. Wilson v, Sep 18 2013 *)

x='x+O('x^30); Vec((1-x)/(1-2*x-6*x^2)) \\ G. C. Greubel, Jan 08 2018

[lucas_number2(n,2,-6)/2 for n in range(0, 25)] # Zerinvary Lajos, Apr 30 2009

G.f.: (1-x)/(1-2*x-6*x^2).
a(n) = (1+sqrt(7))^n/2 + (1-sqrt(7))^n/2.
E.g.f.: exp(x)*cosh(sqrt(7)x).
a(n) = Sum_{k=0..n} A098158(n,k)*7^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=7, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n) = det A. - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).

Original entry on oeis.org

1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 237, 6469, 11753, -8839, -76443, -108691, 164833, 873121, 922077, -2521451, -9653287, -6699319, 34867797, 103232189, 32125393, -451910159, -1064447283, 130656229, 5583548873

Offset: 0

N. J. A. Sloane

Row sums of the Euler related triangle A117411. Partial sums are A006495. - Paul Barry, Mar 16 2006
Binomial transform of [1, 0, -4, 0, 16, 0, -64, 0, 256, 0, ...], i.e. powers of -4 with interpolated zeros. - Philippe Deléham, Dec 02 2008
The absolute values of these numbers are the odd numbers y such that x^2 + y^2 = 5^n with x and y coprime. See A098122. - T. D. Noe, Apr 14 2011
Pisano period lengths: 1, 1, 8, 1, 4, 8, 48, 4, 24, 4, 60, 8, 12, 48, 8, 8, 16, 24, 90, 4, ... - R. J. Mathar, Aug 10 2012
Multiplied by a signed sequence of 2's we obtain 2, -2, -6, 22, -14, -82, 234, -58, -1054, 2398, 474, -12938, ..., the Lucas V(-2,5) sequence. - R. J. Mathar, Jan 08 2013

			1 + x - 3*x^2 - 11*x^3 - 7*x^4 + 41*x^5 + 117*x^6 + 29*x^7 - 527*x^8 + ...

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..200 (a(88) onwards corrected by Sean A. Irvine, Apr 29 2019)
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
G. Berzsenyi, Gaussian Fibonacci numbers, Fib. Quart., 15 (1977), 233-236.
Wikipedia, Lucas sequence.
Index entries for Lucas sequences.
Index entries for Gaussian integers and primes.
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A006496, A045873 (partial sums).

A006495:=func< n | Integers()!Real((1+2*Sqrt(-1))^n) >; [ A006495(n): n in [0..30] ]; // Klaus Brockhaus, Feb 04 2011

a := n -> hypergeom([1/2 - n/2, -n/2], [1/2], -4):
seq(simplify(a(n)), n=0..28); # Peter Luschny, Jul 26 2020

Table[Re[(1+2I)^n],{n,0,29}] (* Giovanni Resta, Mar 28 2006 *)

{a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (4*k + 1) * A[k-1] - 8 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n) = real( (1 + 2*I)^n ) \\ Charles R Greathouse IV, Nov 21 2014

{a(n) = my(A=1);
A = sum(m=0, n+1, (1 + (-1)^m*I)^m * x^m / (1 - (-1)^m*I*x +x*O(x^n))^(m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Mar 09 2019

[lucas_number2(n,2,5)/2 for n in range(0,30)] # Zerinvary Lajos, Jul 08 2008

a(n) = (1/2)*((1+2*i)^n + (1-2*i)^n). - Benoit Cloitre, Oct 28 2002
From Paul Barry, Mar 16 2006: (Start)
G.f.: (1-x)/(1 - 2*x + 5*x^2);
a(n) = 2*a(n-1) - 5*a(n-2);
a(n) = 5^(n/2)*cos(n*atan(1/3) + Pi*n/4);
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,k-j)*C(j,n-k)*(-4)^(n-k). (End)
A000351(n) = a(n)^2 + A006496(n)^2. - Fabrice Baubet (intih(AT)free.fr), May 28 2007
a(n) = upper left and lower right terms of the 2 X 2 matrix [1,-2; 2,1]^n. - Gary W. Adamson, Mar 28 2008
a(n) = Sum_{k=0..n} A124182(n,k)*(-5)^(n-k). - Philippe Deléham, Nov 01 2008
a(n) = Sum_{k=0..n} A098158(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 14 2008
a(n) = (4*n+5)*a(n-1) - 8*Sum_{k=1..n} a(k-1)*a(n-k) if n > 0. - Michael Somos, Jul 23 2011
E.g.f.: exp(x)*cos(2*x). - Sergei N. Gladkovskii, Jul 22 2012
a(n) = 5^(n/2) * cos(n*arctan(2)). - Sergei N. Gladkovskii, Aug 13 2012
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k+1)/(x*(4*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
From Paul D. Hanna, Mar 09 2019: (Start)
G.f.: Sum_{n>=0} (1 + (-1)^n*i)^n * x^n / (1 - (-1)^n*i*x)^(n+1).
G.f.: Sum_{n>=0} (1 - (-1)^n*i)^n * x^n / (1 + (-1)^n*i*x)^(n+1).
(End)
a(n) = hypergeom([1/2 - n/2, -n/2], [1/2], -4). - Peter Luschny, Jul 26 2020

Signs from Christian G. Bower, Nov 15 1998

Corrected by Giovanni Resta, Mar 28 2006

A119900 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 0, 6, 20, 6, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 1, 55, 330, 462, 165, 11, 0, 0, 0, 0, 0, 0, 12, 220, 792, 792, 220, 12, 0, 0, 0, 0, 0, 0, 1, 78

Offset: 0

Emeric Deutsch, May 27 2006

Sum of terms in row n is 2^n (A000079). Sum of terms in column k is A001906(k+1) (the even-indexed Fibonacci numbers). Row n contains 1+floor(n/2) nonzero terms. Sum_{k=0..n} k*T(n,k) = (3n+1)*2^(n-2) = A066373(n+1) for n>=1.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0, 0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 02 2008
From R. Bagula's comment in A053122 (cf. Damianou link), the columns of this array give the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
Odd rows contain the Pascal triangle numbers A091042. See A034867 and A034839 for some relations to tan(x). - Tom Copeland, Oct 15 2014

			The binary word 1/0/01/01/1/1/01 has 7 strictly increasing runs.
T(5,3)=6 because we have 0/01/01, 01/0/01, 01/01/0, 01/1/01, 01/01/1 and 1/01/01 (the runs are separated by /).
Triangle starts:
  1;
  0,2;
  0,1,3;
  0,0,4,4;
  0,0,1,10,5;
  0,0,0,6,20,6;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
A. Collins et al., Binary words, n-color compositions and bisection of the Fibonacci numbers, Fib. Quarterly, 51 (2013), 130-136.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.
R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A000079, A001906, A066373, A034867.
Cf. A098158. - Philippe Deléham, Dec 02 2008
Cf. A000292, A000389, A000580, A001263, A049310, A053122, A134264, A207543.

/* triangle */ [[Binomial(n+1, 2*k-n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Oct 22 2017

T:=(n,k)->binomial(n+1,2*k-n): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Table[Binomial[n + 1, 2 k - n], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 21 2016 *)

for(n=0,10, for(k=0,n, print1(binomial(n+1, 2*k-n), ", "))) \\ G. C. Greubel, Oct 22 2017

T(n,k) = binomial(n+1,2k-n).
G.f.: 1/(1 - 2*t*z - t*(1-t)*z^2).
T(n,k) = A034867(n,n-k)
From Tom Copeland, Sep 30 2011: (Start)
With K(x,t) = 1/{d/dx{x/[t-1+1/(1-x)]}} = [t-1+1/(1-x)]^2/{t-[x/(1-x)]^2}, the g.f. of A119900 = K(x*t,t)-t+1.
From formulas in A134264: K(x,t)d/dx is a generator for A001263. A refinement of A119900 to partition polynomials is given by umbralizing
  K(x,t) roughly as K(h.x,h_0) and precisely as in A134264 as
  W(x)= 1/{d/dx[f(x)]}=1/{d/dx[x/h(x)]}. (End)
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2). - Philippe Deléham, Oct 02 2011
From Tom Copeland, Dec 07 2015: (Start)
An alternate o.g.f. is (1/(x*t)) {-1 + 1 / [1 - (1/t)[x*t/(1-x*t)]^2]} = Sum_{n>0} x^(2(n-1)+1) t^(n-1) / (1-t*x)^(2n) = x + 2t x^2 + (t+3t^2) x^3 + ... .
The n-th diagonal has elements binomial(2n+1+k,k), starting with k=0 for the first non-vanishing element, with o.g.f. (1-x)^(-2(n+1)). The first few subdiagonals are shifted versions of A000292, A000389, and A000580. Cf. A049310.
See A034867 for the matrix representation for the infinitesimal generator K(x,t) d/dx for the Narayana polynomials. (End)
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k + 1,n) = (n*(n + 1)/2)^(p+1).
For example, for row 6 we find S(7,n) + 21*S(9,n) + 35*S(11,n) + 7*S(13,n) = (n*(n + 1)/2)^7.
There appears to be a similar result for the even power sums S(2*k,n) involving A207543. (End)

Keyword tabl added by Philippe Deléham, Jan 26 2010

A087455 Expansion of (1 - x)/(1 - 2x + 3x^2) in powers of x.

Original entry on oeis.org

1, 1, -1, -5, -7, 1, 23, 43, 17, -95, -241, -197, 329, 1249, 1511, -725, -5983, -9791, -1633, 26107, 57113, 35905, -99529, -306773, -314959, 290401, 1525679, 2180155, -216727, -6973919, -13297657, -5673557, 28545857, 74112385, 62587199, -97162757, -382087111, -472685951

Offset: 0

Simone Severini, Oct 23 2003

Type 2 generalized Gaussian Fibonacci integers.
Binomial transform of A077966. - Philippe Deléham, Dec 02 2008
The real component of Q^n, where Q is the quaternion 1 + 0*i + 1*j + 1*k. - Stanislav Sykora, Jun 11 2012
If entries are multiplied by 2*(-1)^n, which gives 2, -2, -2, 10, -14, -2, 46, -86, 34, 190, -482, 394, ..., we obtain the Lucas V(-2,3) sequence. - R. J. Mathar, Jan 08 2013
The real component of (1 + sqrt(-2))^n. - Giovanni Resta, Apr 01 2014
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017
Given an alternated cubic honeycomb with a planar dissection along a plane from edge to opposite edge of the containing cube. The sequence (1 + sqrt(-2))^n contains a real component representing distance along the edge of the tetrahedron/octahedron and an imaginary component representing the orthogonal distance along the sqrt(2) axis in a tetrahedron/octahedron, this generates a unique cevian (line from the apical vertex to a vertex on the triangular tiling composing the opposite face) in this plane with length (sqrt(3))^n. - Jason Pruski, Sep 04 2017, Jan 08 2018
From Peter Bala, Apr 01 2018: (Start)
This sequence is the Lucas sequence V(n,2,3). The companion Lucas sequence U(n,2,3) is A088137.
Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 1 o 1 o ... o 1 (n terms) = A088137(n)/a(n). Cf. A025172 and A127357. (End)

			G.f. = 1 + x - x^2 - 5*x^3 - 7*x^4 + x^5 + 23*x6 + 43*x^7 + 17*x^8 - 95*x^9 + ...

Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).

Robert Israel, Table of n, a(n) for n = 0..3500
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-3).
Index entries for sequences related to Benford's law

Cf. A025172, A048473, A077966, A084102, A088137, A088138, A098158, A124182, A127357.

[n le 2 select 1 else 2*Self(n-1) -3*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 03 2024

Digits:=100; a:=n->round(abs(evalf((3^(n/2))*cos(n*arctan(sqrt(2))))));
# alternative:
a:= gfun:-rectoproc({a(n) = 2*a(n-1) - 3*a(n-2),a(0)=1,a(1)=1},a(n),remember):
map(a, [$0..100]); # Robert Israel, Jun 23 2015

CoefficientList[Series[(1-x)/(1-2*x+3*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, Apr 01 2014 *)
a[ n_] := ChebyshevT[ n, 1/Sqrt[3]] Sqrt[3]^n // Simplify; (* Michael Somos, May 15 2015 *)
LinearRecurrence[{2,-3},{1,1},50] (* Harvey P. Dale, Jul 30 2019 *)

{a(n) = real( (1 + quadgen(-8))^n )}; /* Michael Somos, Jul 26 2006 */

{a(n) = real( subst( poltchebi(n), 'x, quadgen(12) / 3) * quadgen(12)^n)}; /* Michael Somos, Jul 26 2006 */

a(n)=simplify(polchebyshev(n,,quadgen(12)/3)*quadgen(12)^n) \\ Charles R Greathouse IV, Jun 26 2013

[sqrt(3)^n*chebyshev_T(n, 1/sqrt(3)) for n in range(41)] # G. C. Greubel, Jan 03 2024

a(n) = (3^(n/2))*cos(n*arctan(sqrt(2))). - Paul Barry, Oct 23 2003
From Paul Barry, Sep 03 2004: (Start)
a(n) = 2*a(n-1) - 3*a(n-2).
a(n) = (-1)^n*Sum_{m=0..n} binomial(n, m)*Sum_{k=0..n} binomial(m, 2k)2^(m-k).
Binomial transform of 1/(1 + 2*x^2), or (1, 0, -2, 0, 4, 0, -8, 0, 16, ...). (End)
a(n+1) = a(n+2) - 2*A088137(n+1), a(n+1) = A088137(n+2) - A088137(n+1). - Creighton Dement, Oct 28 2004
a(n) = upper left and lower right terms of [1,-2, 1,1]^n. - Gary W. Adamson, Mar 28 2008
a(n) = Sum_{k=0..n} A098158(n,k)*(-2)^(n-k). - Philippe Deléham, Nov 14 2008
a(n) = Sum_{k=0..n} A124182(n,k)*(-3)^(n-k). - Philippe Deléham, Nov 15 2008
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = a(-n) * 3^n for all n in Z. - Michael Somos, Aug 25 2014
E.g.f.: (1/2)*(exp((1 - i*sqrt(2))*x) + exp((1 + i*sqrt(2))*x)), where i is the imaginary unit. - Stefano Spezia, Jul 17 2019

The explicit formula was given by Paul Barry.
Corrected and extended by N. J. A. Sloane, Aug 01 2004
More terms from Creighton Dement, Jul 31 2004

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

Original entry on oeis.org

1, 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset: 0

Paul Barry, Mar 06 2008

In general, the expansion of (1-x)/(1 - 2*x + (m+1)*x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.

Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...] = powers of -3 with interpolated zeros. - Philippe Deléham, Dec 02 2008

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A087204, A088138, A098158, A104537, A124182, A128018.

[2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // G. C. Greubel, Feb 11 2023

CoefficientList[Series[(1-x)/(1-2x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,-4},{1,1},30] (* Harvey P. Dale, Nov 11 2014 *)

[2^n*chebyshev_T(n,1/2) for n in range(31)] # G. C. Greubel, Feb 11 2023

From Philippe Deléham, Nov 14 2008: (Start)
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)
a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 15 2008
a(n) = 2^n*cos(Pi*n/3). - Richard Choulet, Nov 19 2008
a(n) = -8*a(n-3). - Paul Curtz, Apr 22 2011
From Sergei N. Gladkovskii, Jul 27 2012: (Start)
G.f.: G(0) where G(k) = 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction).
E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k) = 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction). (End)
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = A088138(n+1) - A088138(n). - R. J. Mathar, Mar 04 2018
a(n) = (-1)^n*A104537(n). - R. J. Mathar, May 21 2019
a(n) = 2^(n-1)*A087204(n). - G. C. Greubel, Feb 11 2023
Sum_{n>=0} 1/a(n) = 4/3. - Amiram Eldar, Feb 14 2023

A083881 a(n) = 6a(n-1) - 6a(n-2), with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 12, 54, 252, 1188, 5616, 26568, 125712, 594864, 2814912, 13320288, 63032256, 298271808, 1411437312, 6678993024, 31605334272, 149558047488, 707716279296, 3348949390848, 15847398669312, 74990695670784, 354859782008832

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A001075.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6).

Cf. A083882.

Cf. A030192.

a:=[1,3];; for n in [3..30] do a[n]:=6*a[n-1]-6*a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

I:=[1,3]; [n le 2 select I[n] else 6*Self(n-1) -6*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019

f[n_]:= Simplify[(3 + Sqrt@3)^n + (3 - Sqrt@3)^n]/2; Array[f, 30, 0] (* Robert G. Wilson v, Oct 31 2010 *)
LinearRecurrence[{6,-6}, {1,3}, 30] (* G. C. Greubel, Aug 01 2019 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-6*x+6*x^2)) \\ G. C. Greubel, Aug 01 2019

((1-3*x)/(1-6*x+6*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = ((3-sqrt(3))^n + (3+sqrt(3))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*3^(n-k).
G.f.: (1-3*x)/(1-6*x+6*x^2).
E.g.f.: exp(3*x) * cosh(x*sqrt(3)).
a(n) = right and left terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 4 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A030192(n) a(n)]. E.g. a(3) = 54 since M^3 * [1 1 1] = [54 144 54] = [a(3) A030192(3) a(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = Sum_{k, 0<=k<=n}3^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
G.f.: A(x) = G(0) where G(k) =  1 + 3*x/((1-3*x) - x*(1-3*x)/(x + (1-3*x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 29 2012.

A084058 a(n) = 2a(n-1) + 7a(n-2) for n>1, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 9, 25, 113, 401, 1593, 5993, 23137, 88225, 338409, 1294393, 4957649, 18976049, 72655641, 278143625, 1064876737, 4076758849, 15607654857, 59752621657, 228758827313, 875786006225, 3352883803641, 12836269650857, 49142725927201

Offset: 0

Paul Barry, May 10 2003

Binomial transform of expansion of cosh(sqrt(8)x) (A001018 with interpolated zeros : 1, 0, 8, 0, 64, 0, 512, 0, ...); inverse binomial transform of A084128.

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard, Sep 25 2005

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).

Essentially a duplicate of A083100.

a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+7*a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((1+r8)^n+(1-r8)^n)/2: n in [0..30] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 16 2008

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +7*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 01 2019

a[n_]:= Simplify[((1 + Sqrt[8])^n + (1 - Sqrt[8])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-7x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 7}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-7*x^2)) \\ G. C. Greubel, Aug 01 2019

((1-x)/(1-2*x-7*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

a(n) = ((1+sqrt(8))^n + (1-sqrt(8))^n)/2.
G.f.: (1-x)/(1-2*x-7*x^2).
E.g.f.: exp(x) * cosh(sqrt(8)*x).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(n-k). - Philippe Deléham, Dec 26 2007
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
Satisfies recurrence relation system a(n) = 2*b(n-1) - a(n-1), b(n) = 3*b(n-1) + 2*a(n-1), a(0)=1, b(0)=1. - Ilya Gutkovskiy, Apr 11 2017

A002535 a(n) = 2a(n-1) + 9a(n-2), with a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 11, 31, 161, 601, 2651, 10711, 45281, 186961, 781451, 3245551, 13524161, 56258281, 234234011, 974792551, 4057691201, 16888515361, 70296251531, 292589141311, 1217844546401, 5068991364601, 21098583646811, 87818089575031, 365523431971361, 1521409670118001, 6332530227978251

Offset: 0

N. J. A. Sloane

Binomial transform of [1, 0, 10, 0, 100, 0, 1000, 0, 10000, 0, ...]=: powers of 10 (A011557) with interpolated zeros. Inverse binomial transform of A084132. - Philippe Deléham, Dec 02 2008

a(n) is the number of compositions of n when there are 1 type of 1 and 10 types of other natural numbers. - Milan Janjic, Aug 13 2010

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,9).

Cf. A002534 (partial sums), A111015 (primes).

Cf. A011557, A084132, A098158.

a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+9*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019

[Ceiling((1+Sqrt(10))^n/2+(1-Sqrt(10))^n/2): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1)+9*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

A002535:=(-1+z)/(-1+2*z+9*z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Table[ MatrixPower[{{1, 2}, {5, 1}}, n][[1,1]],{n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[n_] := Simplify[((1 + Sqrt[10])^n + (1 - Sqrt[10])^n)/2]; Array[a, 30, 0] (* Or *)
CoefficientList[Series[(1+9x)/(1-2x-9x^2), {x,0,30}], x] (* Or *)
LinearRecurrence[{2, 9}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-9*x^2)) \\ G. C. Greubel, Aug 02 2019

my(p=Mod('x,'x^2-2*'x-9)); a(n) = vecsum(Vec(lift((p^n)))); \\ Kevin Ryde, Jan 28 2023

((1-x)/(1-2*x-9*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

From Paul Barry, May 16 2003: (Start)
a(n) = ((1+sqrt(10))^n + (1-sqrt(10))^n)/2.
G.f.: (1-x)/(1-2*x-9*x^2).
E.g.f.: exp(x)*cosh(sqrt(10)*x). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*10^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=10,(i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A [i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010

A083878 a(0)=1, a(1)=3, a(n) = 6a(n-1) - 7a(n-2), n >= 2.

Original entry on oeis.org

1, 3, 11, 45, 193, 843, 3707, 16341, 72097, 318195, 1404491, 6199581, 27366049, 120799227, 533233019, 2353803525, 10390190017, 45864515427, 202455762443, 893682966669, 3944907462913, 17413664010795, 76867631824379

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A006012. Second binomial transform of A001333.

Third binomial transform of A077957. Inverse binomial transform of A083879. - Philippe Deléham, Dec 01 2008

Michael De Vlieger, Table of n, a(n) for n = 0..1551
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A083879, A056241, A081179, A098158.

f[n_] := Simplify[(3 + Sqrt@2)^n + (3 - Sqrt@2)^n]/2; Array[f, 23, 0] (* Robert G. Wilson v, Oct 31 2010 *)

a(n) = ((3 - sqrt(2))^n + (3 + sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*3^(n-2k)*2^k;
G.f.: (1-3x)/(1-6x+7x^2);
E.g.f.: exp(3x)*cosh(x*sqrt(2)).
a(n) = Sum_{k=0..n} C(n, k)*2^((n-k)/2)(1+(-1)^(n-k))*3^k/2. - Paul Barry, Jan 22 2005
a(n) = Sum_{k=0..n} A098158(n,k)*3^(2k-n)*2^(n-k). - Philippe Deléham, Dec 01 2008
a(n) = A081179(n+1) - 3*A081179(n). - R. J. Mathar, Nov 10 2013
a(n) = Sum_{k=1..n} A056241(n, k) * 2^(k-1). - J. Conrad, Nov 23 2022

cp's OEIS Frontend

A002533 a(n) = 2*a(n-1) + 5*a(n-2), with a(0) = a(1) = 1.

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A083098 a(n) = 2*a(n-1) + 6*a(n-2).

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A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).

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A119900 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.

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A087455 Expansion of (1 - x)/(1 - 2*x + 3*x^2) in powers of x.

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A002533 a(n) = 2a(n-1) + 5a(n-2), with a(0) = a(1) = 1.

A083098 a(n) = 2a(n-1) + 6a(n-2).

A087455 Expansion of (1 - x)/(1 - 2x + 3x^2) in powers of x.

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

A083881 a(n) = 6a(n-1) - 6a(n-2), with a(0)=1, a(1)=3.

A084058 a(n) = 2a(n-1) + 7a(n-2) for n>1, a(0)=1, a(1)=1.

A002535 a(n) = 2a(n-1) + 9a(n-2), with a(0)=a(1)=1.

A083878 a(0)=1, a(1)=3, a(n) = 6a(n-1) - 7a(n-2), n >= 2.