A109466 -id:A109466 - cp's OEIS frontend

A057085 a(n) = 9a(n-1) - 9a(n-2) for n>1, with a(0)=0, a(1)=1.

0, 1, 9, 72, 567, 4455, 34992, 274833, 2158569, 16953624, 133155495, 1045816839, 8213952096, 64513217313, 506693386953, 3979621526760, 31256353258263, 245490585583527, 1928108090927376, 15143557548094641, 118939045114505385, 934159388097696696

Offset: 0

Wolfdieter Lang, Aug 11 2000

Scaled Chebyshev U-polynomials evaluated at 3/2.

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=9, q=-9.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=9.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,-9).

Cf. A000045, A000244, A001906, A004187, A030191, A033890, A049310, A109466.

[3^(n-1)*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, May 02 2022

f[n_]:= Fibonacci[2n]*3^(n-1); Table[f@n, {n, 0, 20}] (* or *)
a[0]=0; a[1]=1; a[n_]:= a[n]= 9(a[n-1] -a[n-2]); Table[a[n], {n, 0, 20}] (* or *)
CoefficientList[Series[x/(1-9x +9x^2), {x, 0, 20}], x] (* Robert G. Wilson v Sep 21 2006 *)

a(n)=(1/3)*sum(k=0,n,binomial(n,k)*fibonacci(4*k)) \\ Benoit Cloitre

concat(0, Vec(x/(1-9*x+9*x^2) + O(x^30))) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,9,9) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

a(n) = A001906(n)*3^(n-1).
a(n) = S(n, 3)*3^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(n) = A001906(n)*A000244(n)/3. - Robert G. Wilson v, Sep 21 2006
a(2k) = A004187(k)*9^k/3, a(2k-1) = A033890(k)*9^k.
G.f.: x/(1-9*x+9*x^2).
a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(4*k). - Benoit Cloitre, Jun 21 2003
a(n+1) = Sum_{k=0..n} A109466(n,k)*9^k. - Philippe Deléham, Oct 28 2008
E.g.f.: 2*exp(9*x/2)*sinh(3*sqrt(5)*x/2)/(3*sqrt(5)). - Stefano Spezia, Sep 01 2025

Edited by N. J. A. Sloane, Sep 16 2005

A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.

Original entry on oeis.org

1, 10, 90, 800, 7100, 63000, 559000, 4960000, 44010000, 390500000, 3464900000, 30744000000, 272791000000, 2420470000000, 21476790000000, 190563200000000, 1690864100000000, 15003009000000000, 133121449000000000, 1181184400000000000, 10480629510000000000

Offset: 0

Wolfdieter Lang, Aug 11 2000

This is the m=10 member of the m-family of sequences S(n,sqrt(m))*(sqrt(m))^n; for S(n,x) see Formula. The m=4..9 instances are A001787, A030191, A030192, A030240, A057084-5 and the m=1..3 signed sequences are A010892, A009545, A057083.

The characteristic roots are rp(m) := (m + sqrt(m*(m-4)))/2 and rm(m) := (m-sqrt(m*(m-4)))/2 and a(n,m)= (rp(m)^(n+1) - rm(m)^(n+1))/(rp(m) - rm(m)) is the Binet form of these m-sequences.

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=10, q=-10.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=10.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-10).

Cf. A001090, A001787, A030191, A030192, A030240, A049310.

Cf. A009545, A010892, A057080, A057083, A057084, A057085.

[(10)^n*Evaluate(DicksonSecond(n, 1/10), 1): n in [0..30]]; // G. C. Greubel, May 02 2022

Join[{a=1,b=10},Table[c=10*b-10*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)

Vec(1/(1-10*x+10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,10,10) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009

a(n) = 10*(a(n-1) - a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, sqrt(10))*(sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(2*k) = A057080(k)*10^k, a(2*k+1) = A001090(k)*10^(k+1).
G.f.: 1/(1-10*x+10*x^2).
a(n) = Sum_{k=0..n} A109466(n,k)*10^k. - Philippe Deléham, Oct 28 2008

A106854 Expansion of 1/(1-x(1-5x)).

Original entry on oeis.org

1, 1, -4, -9, 11, 56, 1, -279, -284, 1111, 2531, -3024, -15679, -559, 77836, 80631, -308549, -711704, 831041, 4389561, 234356, -21713449, -22885229, 85682016, 200108161, -228301919, -1228842724, -87333129, 6056880491, 6493546136, -23790856319, -56258586999, 62695694596, 343988629591

Offset: 0

Paul Barry, May 08 2005

Row sums of Riordan array (1,x*(1-5*x)). In general, a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(k,n-k)*r^(n-k), yields the row sums of the Riordan array (1,x*(1-k*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..2859
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 16.
Index entries for linear recurrences with constant coefficients, signature (1,-5).

Cf. A106852, A106853, A145934.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Join[{a=1,b=1},Table[c=b-5*a;a=b;b=c,{n,80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x(1-5x)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,-5},{1,1},40] (* Harvey P. Dale, Jan 21 2012 *)

Vec(1/(1-x+5*x^2) + O(x^99)) \\ Altug Alkan, Sep 06 2016

[lucas_number1(n,1,5) for n in range(1,35)] # Zerinvary Lajos, Jul 16 2008

a(n) = ((1+sqrt(-19))^(n+1)-(1-sqrt(-19))^(n+1))/(2^(n+1)sqrt(-19)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k, n-k)*5^(n-k).
a(n) = 5^(n/2)(cos(-n*acot(sqrt(19)/19))-sqrt(19)sin(-n*acot(sqrt(19)/19))/19).
a(n) = a(n-1)-5*a(n-2), a(0)=1, a(1)=1. - Philippe Deléham, Oct 21 2008
a(n) = Sum_{k=0..n} A109466(n,k)*5^(n-k). - Philippe Deléham, Oct 25 2008
G.f.: Q(0)/2, where Q(k) = 1 + 1/( 1 - x*(2*k+1 -5*x)/( x*(2*k+2 -5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2013

A145934 Expansion of 1/(1-x(1-6x)).

Original entry on oeis.org

1, 1, -5, -11, 19, 85, -29, -539, -365, 2869, 5059, -12155, -42509, 30421, 285475, 102949, -1609901, -2227595, 7431811, 20797381, -23793485, -148577771, -5816861, 885649765, 920550931, -4393347659, -9916653245, 16443432709

Offset: 0

Philippe Deléham, Oct 25 2008

Row sums of Riordan array (1, x(1-6x)).

For positive n, a(n) equals the determinant of the n X n tridiagonal matrix with 1's along the main diagonal, 3's along the superdiagonal, and 2's along the subdiagonal (see Mathematica code below). - John M. Campbell, Jul 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..2569
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 17.
Index entries for linear recurrences with constant coefficients, signature (1,-6).

Cf. A010892, A107920, A106852, A106853, A106854.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Table[Det[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2 + 1]*2 + KroneckerDelta[#1, #2 - 1]*3 &, {n, n}]], {n, 1, 40}] (* John M. Campbell, Jul 08 2011 *)
LinearRecurrence[{1,-6}, {1,1}, 30] (* G. C. Greubel, Jan 14 2018 *)

Vec(1/(1-x*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,1,6) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

a(n) = Sum_{k=0..n} A109466(n,k)*6^(n-k).

a(n) = a(n-1) - 6*a(n-2); a(0)=1, a(1)=1. - Philippe Deléham, Oct 25 2008

A146523 Binomial transform of A010685.

Original entry on oeis.org

1, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Philippe Deléham, Oct 30 2008

Linked to A029609 by a Catalan transform.

Hankel transform is (1, -15, 0, 0, 0, 0, 0, 0, 0, ...).

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

Cf. A010685, A020714, A029609, A084215, A109466.

CoefficientList[Series[(1+3x)/(1-2x), {x,0,50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 5*2^(Range[40] -1)] (* G. C. Greubel, Nov 23 2021 *)

a(n)=if(n,5<<(n-1),1) \\ Charles R Greathouse IV, Jan 17 2012

[1]+[5*2^(n-1) for n in (1..50)] # G. C. Greubel, Nov 23 2021

a(n) = 5*2^(n-1) for n >= 1, a(0) = 1.
a(n) = Sum_{k=0..n} A109466(n,k)*A029609(k).
a(n) = A084215(n+1) = A020714(n-1), n > 0. - R. J. Mathar, Nov 02 2008
G.f.: (1 + 3*x)/(1 - 2*x). - Vladimir Joseph Stephan Orlovsky, Jun 21 2011
G.f.: G(0), where G(k)= 1 + 3*x/(1 -  2*x/(2*x + 3*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
E.g.f.: (5*exp(2*x) - 3)/2. - Stefano Spezia, Feb 20 2023

A145976 Expansion of 1/(1-x(1-7x)).

Original entry on oeis.org

1, 1, -6, -13, 29, 120, -83, -923, -342, 6119, 8513, -34320, -93911, 146329, 803706, -220597, -5846539, -4302360, 36623413, 66739933, -189623958, -656803489, 670564217, 5268188640, 574239121, -36303081359, -40322755206, 213798814307

Offset: 0

Philippe Deléham, Oct 26 2008

Row sums of Riordan array (1,x(1-7x)).

G. C. Greubel, Table of n, a(n) for n = 0..1500
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 17.
Index entries for linear recurrences with constant coefficients, signature (1,-7).

Cf. A010892, A107920, A106852, A106853, A106854, A145934.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

Join[{a=1,b=1},Table[c=b-7*a;a=b;b=c,{n,80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x(1-7x)),{x,0,50}],x] (* or *) LinearRecurrence[{1,-7},{1,1},50] (* Harvey P. Dale, May 11 2011 *)

Vec(1/(1-x*(1-7*x)) + O(x^40)) \\ Michel Marcus, Jan 29 2016

[lucas_number1(n,1,7) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 7*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k=0..n} A109466(n,k)*7^(n-k).

Corrected by Zerinvary Lajos, Apr 22 2009

Corrected by D. S. McNeil, Aug 20 2010

A100334 An inverse Catalan transform of Fibonacci(2n).

Original entry on oeis.org

0, 1, 2, 2, 0, -5, -13, -21, -21, 0, 55, 144, 233, 233, 0, -610, -1597, -2584, -2584, 0, 6765, 17711, 28657, 28657, 0, -75025, -196418, -317811, -317811, 0, 832040, 2178309, 3524578, 3524578, 0, -9227465, -24157817, -39088169, -39088169, 0, 102334155, 267914296, 433494437, 433494437, 0, -1134903170

Offset: 0

Paul Barry, Nov 17 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,2,-1).

Cf. A001906, A109466.

Cf. A102312 (F(5n)), A134489 (F(5n+2)), A134490 (F(5n+3)).

I:=[0,1,2,2]; [n le 4 select I[n] else 3*Self(n-1) -4*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..41]]; // G. C. Greubel, Jan 30 2023

Table[FullSimplify[GoldenRatio^n*Sqrt[2/5 + 2*Sqrt[5]/25]*Sin[Pi*n/5 + Pi/5] - (1/GoldenRatio)^n*Sqrt[2/5 - 2*Sqrt[5]/25]*Sin[2*Pi*n/5 + 2*Pi/5]], {n, 0, 41}] (* Arkadiusz Wesolowski, Oct 26 2012 *)
LinearRecurrence[{3,-4,2,-1}, {0,1,2,2}, 41] (* G. C. Greubel, Jan 30 2023 *)

def A100334(n): return sum((-1)^k*binomial(n-k,k)*fibonacci(2*n-2*k) for k in range(1+(n//2)))
[A100334(n) for n in range(41)] # G. C. Greubel, Jan 30 2023

G.f.: x*(1-x)/(1-3*x+4*x^2-2*x^3+x^4).
a(n) = (phi)^n*sqrt(2/5+2*sqrt(5)/25)*sin(Pi*(n+1)/5) -(1/phi)^n*sqrt(2/5-2*sqrt(5)/25)*sin(2*Pi*(n+1)/5), where phi=(1+sqrt(5))/2;
a(n) = Sum_{k=0..floor(n/2)} (C(n-k, k)*(-1)^k*Sum_{j=0..n-k} C(n-k, j)*F(j));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*Fibonacci(2n-2k).
a(n) = 3*a(n-1)-4*a(n-2)+2*a(n-3)-a(n-4). - Paul Curtz, May 13 2008
a(n) = Sum_{k=0..n} A109466(n,k)*A001906(k). - Philippe Deléham, Oct 30 2008
a(5*n) = -F(-5*n), a(5*n+1) = -F(-5*n-2), a(5*n+2) = a(5*n+3) = F(-5*n-3), a(5*n+4) = 0. - Ehren Metcalfe, Apr 04 2019

A110509 Riordan array (1, x(1-2x)).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 0, -4, 1, 0, 0, 4, -6, 1, 0, 0, 0, 12, -8, 1, 0, 0, 0, -8, 24, -10, 1, 0, 0, 0, 0, -32, 40, -12, 1, 0, 0, 0, 0, 16, -80, 60, -14, 1, 0, 0, 0, 0, 0, 80, -160, 84, -16, 1, 0, 0, 0, 0, 0, -32, 240, -280, 112, -18, 1, 0, 0, 0, 0, 0, 0, -192, 560, -448, 144, -20, 1, 0, 0, 0, 0, 0, 0, 64, -672, 1120, -672, 180, -22, 1

Offset: 0

Paul Barry, Jul 24 2005

Inverse is Riordan array (1,xc(2x)) [A110510]. Row sums are A107920(n+1). Diagonal sums are (-1)^n*A052947(n).

			Rows begin
1;
0,  1;
0, -2,  1;
0,  0, -4,  1;
0,  0,  4, -6,  1;
0,  0,  0, 12, -8,   1;
0,  0,  0, -8, 24, -10, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, k_] := (-2)^(n - k)*Binomial[k, n - k]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

for(n=0,25, for(k=0,n, print1((-2)^(n-k)*binomial(k, n-k), ", "))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(n, k) = (-2)^(n-k)*binomial(k, n-k).

T(n,k) = A109466(n,k)*2^(n-k). - Philippe Deléham, Oct 26 2008

A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -3, -2, 1, 1, 0, -2, -5, -3, 1, 1, 1, 2, -2, -7, -4, 1, 1, 1, 5, 7, -1, -9, -5, 1, 1, 0, 3, 12, 15, 1, -11, -6, 1, 1, -1, -3, 3, 21, 26, 4, -13, -7, 1, 1, -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1

Offset: 0

Philippe Deléham, Jun 08 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x+x^2),(x*(1-x))/(1-x+x^2)); inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 .

Row sums are ( with the addition of a first row {0}): 0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,... (see A009545). - Roger L. Bagula, Nov 15 2009

			Triangle begins:
   1;
   1,  1;
   0,  1,   1;
  -1, -1,   1,  1;
  -1, -3,  -2,  1,  1;
   0, -2,  -5, -3,  1,   1;
   1,  2,  -2, -7, -4,   1,   1;
   1,  5,   7, -1, -9,  -5,   1,   1;
   0,  3,  12, 15,  1, -11,  -6,   1,  1;
  -1, -3,   3, 21, 26,   4, -13,  -7,  1, 1;
  -1, -7, -15, -3, 31,  40,   8, -15, -8, 1, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A063967, A167925, A199324, A202503, A202551.

T:= proc(n, k) option remember;
      if k<0 or  k>n  then 0
    elif n=0 and k=0 then 1
    else T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2020

m = {{a, 1}, {-1, 1}}; v[0]:= {0, 1}; v[n_]:= v[n] = m.v[n-1]; Table[CoefficientList[v[n][[1]], a], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 15 2009 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0 && k==0, 1, T[n-1, k-1] + T[n-1, k] - T[n-2, k-1] - T[n-2, k] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2020 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    else: return T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 14 2020

Sum{k=0..n} T(n,k)*x^k = { (-1)^n*A057093(n), (-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) } for x=-11, -10, ..., 8, 9, respectively .
Sum{k=0..n} T(n,k)*A000045(k) = A100334(n).
Sum{k=0..floor(n/2)} T(n-k,k) = A050935(n+2).
T(n,k)= Sum{j>=0} A109466(n,j)*binomial(j,k).
T(n,k) = (-1)^(n-k)*A199324(n,k) = (-1)^k*A202551(n,k) = A202503(n,n-k). - Philippe Deléham, Mar 26 2013
G.f.: 1/(1-x*y+x^2*y-x+x^2). - R. J. Mathar, Aug 11 2015

Riordan array definition corrected by Ralf Stephan, Jan 02 2014

A145978 Expansion of 1/(1-x(1-8x)).

Original entry on oeis.org

1, 1, -7, -15, 41, 161, -167, -1455, -119, 11521, 12473, -79695, -179479, 458081, 1893913, -1770735, -16922039, -2756159, 132620153, 154669425, -906291799, -2143647199, 5106687193, 22255864785, -18597632759, -196644551039, -47863488967, 1525292919345

Offset: 0

Philippe Deléham, Oct 26 2008

Row sums of Riordan array (1,1(1-8x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-8).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

LinearRecurrence[{1,-8},{1,1},50] (* G. C. Greubel, Jan 29 2016 *)

Vec(1/(1-x*(1-8*x)) + O(x^40)) \\ Michel Marcus, Jan 29 2016

[lucas_number1(n,1,8) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 8*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k=0..n} A109466(n,k)*8^(n-k).

cp's OEIS Frontend

A057085 a(n) = 9*a(n-1) - 9*a(n-2) for n>1, with a(0)=0, a(1)=1.

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A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.

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A106854 Expansion of 1/(1-x*(1-5*x)).

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A145934 Expansion of 1/(1-x*(1-6*x)).

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A146523 Binomial transform of A010685.

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A145976 Expansion of 1/(1-x*(1-7*x)).

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A057085 a(n) = 9a(n-1) - 9a(n-2) for n>1, with a(0)=0, a(1)=1.

A106854 Expansion of 1/(1-x(1-5x)).

A145934 Expansion of 1/(1-x(1-6x)).

A145976 Expansion of 1/(1-x(1-7x)).

A145978 Expansion of 1/(1-x(1-8x)).