A154929 -id:A154929 - cp's OEIS frontend

A154930 Inverse of Fibonacci convolution array A154929.

1, -2, 1, 5, -4, 1, -15, 14, -6, 1, 51, -50, 27, -8, 1, -188, 187, -113, 44, -10, 1, 731, -730, 468, -212, 65, -12, 1, -2950, 2949, -1956, 970, -355, 90, -14, 1, 12235, -12234, 8291, -4356, 1785, -550, 119, -16, 1, -51822, 51821, -35643, 19474, -8612, 3021

Offset: 0

Paul Barry, Jan 17 2009

Alternating sign version of A104259. Row sums are (-1)^n*A033321. First column is (-1)^n*A007317.

			Triangle begins
1,
-2, 1,
5, -4, 1,
-15, 14, -6, 1,
51, -50, 27, -8, 1,
-188, 187, -113, 44, -10, 1,
731, -730, 468, -212, 65, -12, 1,
-2950, 2949, -1956, 970, -355, 90, -14, 1
Production array is
-2, 1,
1, -2, 1,
-1, 1, -2, 1,
1, -1, 1, -2, 1,
-1, 1, -1, 1, -2, 1,
1, -1, 1, -1, 1, -2, 1,
-1, 1, -1, 1, -1, 1, -2, 1
or ((1-x-x^2)/(1+x),x) beheaded.

Cf. A171567, A054336, A171488, A035324.

Riordan array ((1/(1+x))c(-x/(1+x)), (x/(1+x))c(x/(1+x))), c(x) the g.f. of A000108;
Riordan array ((sqrt(1+6x+5x^2)-x-1)/(2x(1+x)),(sqrt(1+6x+5x^2)-x-1)/ (2(1+x)));
Triangle T(n,k) = sum{j=0..n, (-1)^(n-k)*C(n,j)*C(2j-k,j-k)(k+1)/(j+1)}.
T(n,k) = T(n-1,k-1) -2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

A155020 a(n) = 2a(n-1) + 2a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968

Offset: 0

Philippe Deléham, Jan 19 2009

Equals 1 followed by A028859. - Klaus Brockhaus, Jul 18 2009
a(n) is the number of ways to arrange 1- and 2-cent postage stamps (totaling n cents) in a row so that the first stamp is correctly placed and any subsequent stamp may (or not) be placed upside down.
Number of compositions of n into parts k >= 1 where there are F(k+1) = A000045(k+1) sorts of part k. - Joerg Arndt, Sep 30 2012
a(n) is the top-left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3 X 3 matrix [1, 1, 1; 1, 0, 1; 1, 1, 1].
From Tom Copeland, Nov 08 2014: (Start)
(Setting a(0)=0.)
This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t) = x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = (1-sqrt(1-4x))/2; and its inverse Cinv(x) = x*(1-x). (Cf. A091867.)
O.g.f.: G(x) = -P(P(Cinv(-x),1),1) = -P(Cinv(-x),2) = x(1+x)/(1-2x(1+x)) = (x+x^2)/(1-2(x+x^2)) = x + 3*x^2 + 8*x^3 + ... = A155020(x) with a(0)=0.
Ginv(x) = -C(P(P(-x,-1),-1)) = -C(P(-x,-2)) = (-1+sqrt(1+4*x/(1+2*x)))/2 = x*A064613(-x).
G(x) = x*(1+x) + 2*(x*(1+x))^2 + 2^2*(x*(1+x))^3 - ..., and so this array contains the row sums of A030528 * Diag(1, 2^1, 2^2, 2^3, ...). (End)
INVERT transform of Fibonacci(n+1). - Alois P. Heinz, Feb 11 2021

			a(2) = 3 because we have {1,1}, {1,_1} and {2}.
a(3) = 8 because we can order the stamps in eight ways: {1,1,1}  {1,1,_1}  {1,_1,1}  {1,_1,_1}  {2,1}   {2,_1}  {1,2}   {1,_2}, where _1 and _2 are upside down stamps.
a(4) = 22 = 2*3 + 2*8 because we can append 2 or _2 to the a(2) examples and 1 or _1 to the a(3) examples. - _Jon Perry_, Nov 10 2014

Indranil Ghosh, Table of n, a(n) for n = 0..2287
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Jeffrey Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
Index entries for linear recurrences with constant coefficients, signature (2,2).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: this sequence (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).
Cf. A028859 (essentially the same sequence). - Klaus Brockhaus, Jul 18 2009
Cf. A000045, A000108, A030528, A064613, A091867, A154929.
Row sums of A155112.

I:=[1,1,3,8]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*combinat[fibonacci](1+i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

CoefficientList[Series[(1 -x -x^2)/(1 -2x -2x^2), {x,0,20}], x]
With[{m=2}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

makelist(sum(binomial(n-k,k)*2^(n-k-1),k,0,floor(n/2)),n,1,12); /* Emanuele Munarini, Feb 04 2014 */

Vec( (1-x-x^2)/(1-2*x-2*x^2) + O(x^66) )  /* Joerg Arndt, Sep 30 2012 */

[1]+[(-1)*(sqrt(2)*i)^(n-2)*chebyshev_U(n, -sqrt(2)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1 - x - x^2)/(1 - 2*x - 2*x^2).
G.f.: 1/( 1 - Sum_{k>=1} (x+x^2)^k ) - 1/( 1 - Sum_{k>=1} F(k+1)*x^k ) where F(k) = A000045(k). - Joerg Arndt, Sep 30 2012
a(n+1) = Sum_{k=0..n} A154929(n,k) = A028859(n).
a(n) = Sum_{k=0..floor(n/2)} ( binomial(n-k,k)*2^(n-k-1) ) for n > 0. - Emanuele Munarini, Feb 04 2014
a(n) = (1/2)*[n=0] - (sqrt(2)*i)^(n-2)*ChebyshevU(n, -sqrt(2)*i/2). - G. C. Greubel, Mar 25 2021
E.g.f.: (3 + exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)))/6. - Stefano Spezia, Mar 02 2024

A154964 a(n) = 3a(n-1) + 6a(n-2), n>2, a(0)=1, a(1)=1, a(2)=5.

Original entry on oeis.org

1, 1, 5, 21, 93, 405, 1773, 7749, 33885, 148149, 647757, 2832165, 12383037, 54142101, 236724525, 1035026181, 4525425693, 19786434165, 86511856653, 378254174949, 1653833664765, 7231026043989, 31616080120557, 138234396625605, 604399670600157, 2642605391554101

Offset: 0

Philippe Deléham, Jan 18 2009

For n>=1, a(n) is the number of words of length n-1 over the alphabet {1,2,3,4,5} such that no two even numbers appear consecutively. - Armend Shabani, Mar 01 2017

Index entries for linear recurrences with constant coefficients, signature (3,6).

{1}~Join~LinearRecurrence[{3, 6}, {1, 5}, 25] (* or *)
CoefficientList[Series[(1 - 2 x - 4 x^2)/(1 - 3 x - 6 x^2), {x, 0, 25}], x] (* Michael De Vlieger, Mar 02 2017 *)

Vec((1-2*x-4*x^2)/(1-3*x-6*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

G.f.: (1 - 2*x - 4*x^2)/(1 - 3*x - 6*x^2).
a(n+1) = Sum_{k=0..n} A154929(n,k)*2^(n-k).
G.f.: Q(0)/6 +2/3 , where Q(k) = 1 + 1/(1 - x*(6*k+3 + 6*x )/( x*(6*k+6 + 6*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013
a(n) = A083858(n+1)/3, n>=1. - R. J. Mathar, Feb 06 2020

A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 10, 6, 1, 0, 8, 22, 21, 8, 1, 0, 13, 45, 59, 36, 10, 1, 0, 21, 88, 147, 124, 55, 12, 1, 0, 34, 167, 339, 366, 225, 78, 14, 1, 0, 55, 310, 741, 976, 770, 370, 105, 16, 1, 0, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 0, 144, 1020, 3174, 5696, 6505, 4920, 2485, 824, 171, 20, 1

Offset: 0

Philippe Deléham, Jan 20 2009

A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  4,  1;
  0,  5, 10,  6,  1;
  0,  8, 22, 21,  8,  1;
  0, 13, 45, 59, 36, 10, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.
Cf. A000045, A154929.
T(2n,n) gives A262910.
Cf. A291385.

T:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n-j,j)*Binomial(n-j,k)*k/(n-j): j in [0..Floor(n/2)]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 26 2021

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n+1)); # Peter Luschny, Oct 19 2022

T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 26 2021 *)

def T(n,k): return 1 if n==0 else sum( binomial(n-j,j)*binomial(n-j,k)*k/(n-j) for j in (0..n//2) )
flatten([[T(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 26 2021

Recurrence: T(n+2,k+1) = T(n+1,k+1) + T(n+1,k) + T(n,k+1) + T(n,k).
Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.
G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - G. C. Greubel, Mar 26 2021
Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - Alois P. Heinz, Sep 29 2022

Typos in two terms corrected by Alois P. Heinz, Aug 08 2015

A154996 a(n) = 5a(n-1) + 20a(n-2), n>2 ; a(0)=1, a(1)=1, a(2)=9.

Original entry on oeis.org

1, 1, 9, 65, 505, 3825, 29225, 222625, 1697625, 12940625, 98655625, 752090625, 5733565625, 43709640625, 333219515625, 2540290390625, 19365842265625, 147635019140625, 1125491941015625, 8580160087890625, 65410639259765625

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,20).

Cf. A154997, A154999, A155000, A155001.

I:=[1,9]; [1] cat [n le 2 select I[n] else 5*(Self(n-1) +4*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 21 2021

m:=30; S:=series( (1-4*x-16*x^2)/(1-5*x-20*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 21 2021

Join[{1},LinearRecurrence[{5,20},{1,9},20]] (* Harvey P. Dale, Jan 19 2012 *)

def A154996_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-4*x-16*x^2)/(1-5*x-20*x^2) ).list()
A154996_list(30) # G. C. Greubel, Apr 21 2021

G.f.: (1 -4*x -16*x^2)/(1 -5*x -20*x^2).

a(n+1) = Sum_{k=0..n} A154929(n,k)*4^(n-k).

A154997 a(n) = 6a(n-1) + 30a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

Original entry on oeis.org

1, 1, 11, 96, 906, 8316, 77076, 711936, 6583896, 60861456, 562685616, 5201957376, 48092312736, 444612597696, 4110444968256, 38001047740416, 351319635490176, 3247949245153536, 30027284535626496, 277602184568365056

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,30).

Cf. A154996, A154999, A155000, A155001.

I:=[1,11]; [1] cat [n le 2 select I[n] else 6*(Self(n-1) +5*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 21 2021

m:=30; S:=series( (1-5*x-25*x^2)/(1-6*x-30*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 21 2021

Join[{1},LinearRecurrence[{6,30},{1,11},20]] (* Harvey P. Dale, Feb 07 2012 *)

def A154996_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x-25*x^2)/(1-6*x-30*x^2) ).list()
A154996_list(30) # G. C. Greubel, Apr 21 2021

G.f.: (1 -5*x -25*x^2)/(1 -6*x -30*x^2).

a(n+1) = Sum_{k=0..n} A154929(n,k)*5^(n-k).

A154999 a(n) = 7a(n-1) + 42a(n-2), n>2; a(0)=1, a(1)=1, a(2)=13.

Original entry on oeis.org

1, 1, 13, 133, 1477, 15925, 173509, 1883413, 20471269, 222402229, 2416608901, 26257155925, 285297665317, 3099884206069, 33681691385797, 365966976355477, 3976399872691813, 43205412115772725, 469446679463465221

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (7,42).

Cf. A154929, A154996, A154997, A155000, A155001.

I:=[1,13]; [1] cat [n le 2 select I[n] else 7*(Self(n-1) +6*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

LinearRecurrence[{7,42}, {1,1,13}, 31] (* G. C. Greubel, Apr 20 2021 *)
CoefficientList[Series[(1-6x-36x^2)/(1-7x-42x^2),{x,0,20}],x] (* Harvey P. Dale, Jan 14 2022 *)

def A154999_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-6*x-36*x^2)/(1-7*x-42*x^2) ).list()
A154999_list(30) # G. C. Greubel, Apr 20 2021

a(n+1) = Sum_{k=0..n} A154929(n,k)*6^(n-k).

G.f.: (1 - 6*x - 36*x^2)/(1 - 7*x - 42*x^2). - G. C. Greubel, Apr 20 2021

More terms from Philippe Deléham, Jan 27 2009

Corrected by D. S. McNeil, Aug 20 2010

A155001 a(n) = 9a(n-1) + 72a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17.

Original entry on oeis.org

1, 1, 17, 225, 3249, 45441, 642897, 9057825, 127809009, 1802444481, 25424248977, 358594243425, 5057894117169, 71339832581121, 1006226869666257, 14192509772837025, 200180922571503729, 2823489006787799361

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..850
Index entries for linear recurrences with constant coefficients, signature (9,72).

Cf. A154929, A154996, A154997, A154999, A155000.

I:=[1,17]; [1] cat [n le 2 select I[n] else 9*(Self(n-1) +8*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

a[0] := 1: a[1] := 1: a[2] := 17: for n from 3 to 25 do a[n] := 9*a[n-1]+72*a[n-2] end do: seq(a[n], n = 0 .. 17); # Emeric Deutsch, Jan 21 2009

LinearRecurrence[{9,72},{1,1,17},20] (* Harvey P. Dale, Apr 26 2016 *)

def A155001_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-8*x-64*x^2)/(1-9*x-72*x^2) ).list()
A155001_list(30) # G. C. Greubel, Apr 20 2021

a(n+1) = Sum_{k=0..n} A154929(n,k)*8^(n-k).

G.f.: (1 - 8*x - 64*x^2)/(1 - 9*x - 72*x^2). - G. C. Greubel, Apr 20 2021

Corrected by Philippe Deléham, Jan 21 2009

Corrected and extended by Emeric Deutsch and R. J. Mathar, Jan 21 2009

A154968 a(n) = 4a(n-1) + 12a(n-2), n>2 with a(0)=1, a(1)=1, a(2)=7.

Original entry on oeis.org

1, 1, 7, 40, 244, 1456, 8752, 52480, 314944, 1889536, 11337472, 68024320, 408146944, 2448879616, 14693281792, 88159682560, 528958111744, 3173748637696, 19042491891712, 114254951219200, 685529707577344

Offset: 0

Philippe Deléham, Jan 18 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,12).

Cf. A154929.

[n eq 0 select 1 else (6^(n+1) -(-2)^(n+1))/32: n in [0..40]]; // G. C. Greubel, Mar 01 2021

LinearRecurrence[{4,12}, {1,1,7}, 40] (* G. C. Greubel, Mar 01 2021 *)

[1]+[(6^(n+1) - (-2)^(n+1))/32 for n in [1..40]] # G. C. Greubel, Mar 01 2021

16*a(n) = 3*6^n +(-1)^n*2^n, n>0. - R. J. Mathar, Sep 03 2013
From G. C. Greubel, Mar 01 2021: (Start)
a(n) = (6^(n+1) - (-2)^(n+1))/32 + (3/4)*[n=0].
E.g.f.: (exp(-2*x) + 3*exp(6*x))/16. (End)

A155017 a(n) = 10a(n-1) + 90a(n-2), n>2 ; a(0)=1, a(1)=1, a(2)=19 .

Original entry on oeis.org

1, 1, 19, 280, 4510, 70300, 1108900, 17416000, 273961000, 4307050000, 67726990000, 1064904400000, 16744473100000, 263286127000000, 4139863849000000, 65094389920000000, 1023531645610000000

Offset: 0

Philippe Deléham, Jan 19 2009

nonn

10^(floor((n - 2)/2)) | a(n) for n>=1. - G. C. Greubel, Dec 30 2017

G. C. Greubel, Table of n, a(n) for n = 0..825
Index entries for linear recurrences with constant coefficients, signature (10, 90).

I:=[1,1,19]; [1] cat [n le 2 select I[n] else 10*Self(n-1) + 90*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 30 2017

Join[{1},LinearRecurrence[{10,90},{1,19},20]] (* Harvey P. Dale, Oct 10 2012 *)
CoefficientList[Series[(1 - 9*x - 81*x^2)/(1 - 10*x - 90*x^2), {x,0,50}], x] (* G. C. Greubel, Dec 30 2017 *)

x='x+O('x^30); Vec((1-9*x-81*x^2)/(1-10*x-90*x^2)) \\ G. C. Greubel, Dec 30 2017

G.f.: (1-9*x-81*x^2)/(1-10*x-90*x^2).

a(n+1) = Sum_{k=0..n} A154929(n,k)*9^(n-k).

cp's OEIS Frontend

A154930 Inverse of Fibonacci convolution array A154929.

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A155020 a(n) = 2*a(n-1) + 2*a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

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A154964 a(n) = 3*a(n-1) + 6*a(n-2), n>2, a(0)=1, a(1)=1, a(2)=5.

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A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A154996 a(n) = 5*a(n-1) + 20*a(n-2), n>2 ; a(0)=1, a(1)=1, a(2)=9.

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A154997 a(n) = 6*a(n-1) + 30*a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

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A154999 a(n) = 7*a(n-1) + 42*a(n-2), n>2; a(0)=1, a(1)=1, a(2)=13.

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A155020 a(n) = 2a(n-1) + 2a(n-2) for n > 2, a(0)=1, a(1)=1, a(2)=3.

A154964 a(n) = 3a(n-1) + 6a(n-2), n>2, a(0)=1, a(1)=1, a(2)=5.

A154996 a(n) = 5a(n-1) + 20a(n-2), n>2 ; a(0)=1, a(1)=1, a(2)=9.

A154997 a(n) = 6a(n-1) + 30a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

A154999 a(n) = 7a(n-1) + 42a(n-2), n>2; a(0)=1, a(1)=1, a(2)=13.

A155001 a(n) = 9a(n-1) + 72a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17.

A154968 a(n) = 4a(n-1) + 12a(n-2), n>2 with a(0)=1, a(1)=1, a(2)=7.

A155017 a(n) = 10a(n-1) + 90a(n-2), n>2 ; a(0)=1, a(1)=1, a(2)=19 .