A026597 -id:A026597 - cp's OEIS frontend

A122994 a(n) = a(n-1)+9*a(n-2) initialized with a(0)=1, a(1)=3.

1, 3, 12, 39, 147, 498, 1821, 6303, 22692, 79419, 283647, 998418, 3551241, 12537003, 44498172, 157331199, 557814747, 1973795538, 6994128261, 24758288103, 87705442452, 310530035379, 1099879017447, 3894649335858, 13793560492881, 48845404515603, 172987448951532

Offset: 0

Roger L. Bagula, Sep 22 2006

The two roots of the denominator of the g.f. (for Binet's formula) are -0.393486... and 0.2823756...

Pisano period lengths: 1, 3, 1, 6, 6, 3, 6, 12, 1, 6, 10, 6, 84, 6, 6, 24,144, 3, 72, 6,... - R. J. Mathar, Aug 10 2012

Index entries for linear recurrences with constant coefficients, signature (1,9).

Cf. A026597.

CoefficientList[Series[(-2 z - 1)/(9 z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

G.f.: -(1+2*x)/(-1+x+9*x^2). a(n) = A015445(n)+2*A015445(n-1). [R. J. Mathar, Aug 12 2009]
a(n) = (1/2+5*sqrt(37)/74) *(1/2+sqrt(37)/2)^(n-1) +(1/2-5*sqrt(37)/74) *(1/2-sqrt(37)/2)^(n-1). [Antonio Alberto Olivares, Jun 07 2011]
a(n) = Sum_{k, 0<=k<=n} A103631(n,k)*3^k. - Philippe Deléham, Dec 17 2011
a(n) = A015445(n) + 2*A015445(n-1), n>0. - Ralf Stephan, Jul 21 2013

Definition replaced with the Deleham recurrence of Mar 2009 by the Assoc. Editors of the OEIS, Mar 12 2010

A158797 a(n) = a(n-1) + 36*a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 42, 258, 1770, 11058, 74778, 472866, 3164874, 20188050, 134123514, 860893314, 5689339818, 36681499122, 241497732570, 1562031700962, 10255950073482, 66489091308114, 435703293953466, 2829310581045570

Offset: 0

Philippe Deléham, Mar 27 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,36).

Cf. A000045, A026597, A122994, A122995, A158608.

I:=[1,6]; [n le 2 select I[n] else Self(n-1) + 36*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 22 2021

LinearRecurrence[{1,36},{1,6},30] (* Harvey P. Dale, Apr 30 2013 *)

Vec((1+5*x)/(1-x-36*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 17 2012

[(6*i)^n*( chebyshev_U(n, -i/12) - (5*i/6)*chebyshev_U(n-1, -i/12) ) for n in (0..40)] # G. C. Greubel, Dec 22 2021

G.f.: (1+5*x)/(1-x-36*x^2).

a(n) = (6*i)^n*( ChebyshevU(n, -i/12) - (5*i/6)*ChebyshevU(n-1, -i/12) ). - G. C. Greubel, Dec 22 2021

A384640 Expansion of (1-3x^2) / (1-x-5x^2+x^3+2*x^4).

Original entry on oeis.org

1, 1, 3, 7, 19, 49, 131, 343, 911, 2397, 6347, 16735, 44251, 116785, 308611, 814815, 2152583, 5684477, 15015355, 39655527, 104742659, 276635985, 730663043, 1929789255, 5096983167, 13461994429, 35555794923, 93909205391, 248032219243, 655098462417, 1730238763395

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 0 in the following graph:
     2
    /|\
 0-1 | 3
    \|/
     4.
Also, for n>=1, the number of walks of length n-1 starting at vertex 1 in the same graph.

			a(3)=7 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4, 0-1-4-1, 0-1-4-2, 0-1-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-2).

Cf. A384641 (vertex 2), A384642 (vertex 3), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1-3*x^2) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]

A122999 G.f.: 1/(1 - x - 25*x^2).

Original entry on oeis.org

1, 1, 26, 51, 701, 1976, 19501, 68901, 556426, 2278951, 16189601, 73163376, 477903401, 2306987801, 14254572826, 71929267851, 428293588501, 2226525284776, 12933864997301, 68596997116701, 391943622049226

Offset: 0

Roger L. Bagula, Sep 22 2006

Index entries for linear recurrences with constant coefficients, signature (1,25).

Cf. A026597.

m =5; p[x_] := -1 - x/m + x^2; q[x_] := ExpandAll[x^2*p[1/x]]; Table[ SeriesCoefficient[Series[x/q[x], {x, 0, 30}], n]*m^(n - 1), {n, 0, 30}]
Join[{a=1,b=1},Table[c=1*b+25*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/(1-x-25x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {1,25},{1,1},30] (* Harvey P. Dale, Dec 23 2014 *)

makelist(coeff(taylor(1/(1-x-25*x^2), x, 0, n), x, n), n, 0, 20); /* Bruno Berselli, Jun 06 2011 */

Vec(O(x^99)+(1-x-25*x^2)^-1) \\ Charles R Greathouse IV, Jun 06 2011

a(0)=1, a(1)=1, a(n) = a(n-1) + 25*a(n-2) for n>1. - Philippe Deléham, Sep 19 2009

a(n) = (1/2 + sqrt(101)/202)*(1/2 + sqrt(101)/2)^(n-1) + (1/2 - sqrt(101)/202)*(1/2 - sqrt(101)/2)^(n-1). - Antonio Alberto Olivares, Jun 06 2011

Edited by N. J. A. Sloane, Sep 26 2006

Definition corrected by R. J. Mathar, Jan 15 2009

A158798 a(n) = a(n-1) + 64*a(n-2), a(0)=1, a(1)=8.

Original entry on oeis.org

1, 8, 72, 584, 5192, 42568, 374856, 3099208, 27089992, 225439304, 1959198792, 16387314248, 141776036936, 1190564148808, 10264230512712, 86460336036424, 743371088849992, 6276832595181128, 53852582281580616, 455569868373172808, 3902135134394332232

Offset: 0

Philippe Deléham, Mar 27 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,64).

Cf. A000045, A026597, A122994, A122995, A122996, A158608, A158797.

I:=[1,8]; [n le 2 select I[n] else Self(n-1) + 64*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 22 2021

LinearRecurrence[{1,64},{1,8},30] (* Harvey P. Dale, Mar 09 2018 *)

[(8*i)^n*( chebyshev_U(n, -i/16) - (7*i/8)*chebyshev_U(n-1, -i/16) ) for n in (0..40)] # G. C. Greubel, Dec 22 2021

G.f.: (1+7*x)/(1-x-64*x^2).

a(n) = (8*i)^n*( ChebyshevU(n, -i/16) - (7*i/8)*ChebyshevU(n-1, -i/16) ). - G. C. Greubel, Dec 22 2021

Corrected and extended by Harvey P. Dale, Mar 09 2018

A208765 Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 14, 1, 8, 36, 56, 38, 1, 10, 60, 140, 190, 94, 1, 12, 90, 280, 570, 564, 246, 1, 14, 126, 490, 1330, 1974, 1722, 622, 1, 16, 168, 784, 2660, 5264, 6888, 4976, 1606, 1, 18, 216, 1176, 4788, 11844, 20664, 22392, 14454, 4094, 1

Offset: 1

Clark Kimberling, Mar 02 2012

For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

			First five rows:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 18, 14;
  1,  8, 36, 56, 38;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 4x + 6x^2
  1 + 6x + 18x^2 + 14x^3
  1 + 8x + 36x^2 + 56x^3 + 38x^4
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  4,  6,   0;
  1,  6, 18,  14,   0;
  1,  8, 36,  56,  38,  0;
  1, 10, 60, 140, 190, 94, 0. - _Philippe Deléham_, Mar 18 2012

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A208766, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]     (* A208765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A208766 *)
Rest[CoefficientList[CoefficientList[Series[(1-x-y*x+2*y*x^2-4*y^2*x^2)/( 1-2*x-y*x+x^2+y*x^2-4*y^2*x^2), {x,0,20}, {y,0,20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

u(n,x) = u(n-1,x) + 2*x*v(n-1,x),
v(n,x) = 2*x*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 18 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-x-y*x+2*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 4*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n.
T(n,k) = binomial(n-1,k)*A026597(k). (End)

A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 5, 1, 2, 7, 4, 2, 9, 9, 1, 2, 11, 16, 5, 2, 13, 25, 14, 1, 2, 15, 36, 30, 6, 2, 17, 49, 55, 20, 1, 2, 19, 64, 91, 50, 7, 2, 21, 81, 140, 105, 27, 1, 2, 23, 100, 204, 196, 77, 8, 2, 25, 121, 285, 336, 182, 35, 1, 2, 27, 144, 385, 540, 378, 112, 9, 2, 29, 169, 506

Offset: 0

Emeric Deutsch, May 12 2007

Also number of Fibonacci binary words of length n and having k 10 subwords.
Row n has 1+floor(n/2) terms.
Row sums are the Fibonacci numbers (A000045).
T(n,0)=2 for n >= 1.
Sum_{k>=0} k*T(n,k) = A023610(n-2).
Triangle, with zeros omitted, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 14 2012
Riordan array ((1+x)/(1-x), x^2/(1-x)), zeros omitted. - Philippe Deléham, Jan 14 2012

			T(5,2)=4 because we have 10101, 01101, 01010 and 01011.
Triangle starts:
  1;
  2;
  2, 1;
  2, 3;
  2, 5, 1;
  2, 7, 4;
  2, 9, 9, 1;
Triangle (2, -1, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) begins:
  1;
  2, 0;
  2, 1, 0;
  2, 3, 0, 0;
  2, 5, 1, 0, 0;
  2, 7, 4, 0, 0, 0;
  2, 9, 9, 1, 0, 0, 0;

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened.)
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.

Cf. A000045, A023610.
Cf. A029635, A029653.
Columns: A040000, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

T:=proc(n,k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k,k)+binomial(n-k-1,k) else 0 fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

MapAt[# - 1 &, #, 1] &@ Table[Binomial[n - k, k] + Binomial[n - k - 1, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k) for n >= 1 and 0 <= k <= floor(n/2).
G.f. = G(t,z) = (1+z)/(1-z-tz^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A078050(n), A057079(n), A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Jan 14 2012
T(n,k) = T(n-1,k) + T(n-2,k-1) with T(0,0)=1, T(1,0)=2, T(1,1)=0 and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Jan 14 2012

A384641 Expansion of (1+2x-x^3) / (1-x-5x^2+x^3+2*x^4).

Original entry on oeis.org

1, 3, 8, 21, 56, 147, 390, 1027, 2718, 7169, 18952, 50025, 132180, 349015, 921986, 2434831, 6431386, 16985525, 44863652, 118490229, 312960192, 826576635, 2183160062, 5766102587, 15229405878, 40223605481, 106238212160, 280594628513, 741103272076, 1957390991519

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 2 in the following graph:
     2
    /|\
 0-1 | 3
    \|/
     4.

			a(2)=8 because we have the walks 2-1-0, 2-1-2, 2-1-4, 2-3-2, 2-3-4, 2-4-1, 2-4-2, 2-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-2).

Cf. A384640 (vertices 0, 1), A384642 (vertex 3), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+2*x-x^3) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]

A384642 Expansion of (1+x-x^2+x^3) / (1-x-5x^2+x^3+2x^4).

Original entry on oeis.org

1, 2, 6, 16, 42, 112, 294, 780, 2054, 5436, 14338, 37904, 100050, 264360, 698030, 1843972, 4869662, 12862772, 33971050, 89727304, 236980458, 625920384, 1653153270, 4366320124, 11532205174, 30458811756, 80447210962, 212476424320, 561189257026, 1482206544152

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 3 in the following graph:
     2
    /|\
 0-1 | 3
    \|/
     4.

			a(2)=6 because we have the walks 3-2-1, 3-2-3, 3-2-4, 3-4-1, 3-4-2, 3-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-2).

Cf. A384640 (vertices 0, 1), A384641 (vertex 2), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).

a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[4,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]

A122112 a(n) = 4*a(n-2) - a(n-1), with a(0)=1, a(1)=-2.

Original entry on oeis.org

1, -2, 6, -14, 38, -94, 246, -622, 1606, -4094, 10518, -26894, 68966, -176542, 452406, -1158574, 2968198, -7602494, 19475286, -49885262, 127786406, -327327454, 838473078, -2147782894, 5501675206, -14092806782, 36099507606, -92470734734

Offset: 0

Philippe Deléham, Oct 18 2006

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

Cf. A026597, A055830.

a:=[1,-2];; for n in [3..30] do a[n]:=-a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Oct 02 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1+x-4*x^2) )); // G. C. Greubel, Oct 02 2019

seq(coeff(series((1-x)/(1+x-4*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 02 2019

LinearRecurrence[{-1,4}, {1,-2}, 30] (* G. C. Greubel, Oct 02 2019 *)

my(x='x+O('x^30)); Vec((1-x)/(1+x-4*x^2)) \\ G. C. Greubel, Oct 02 2019

def A122112_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1+x-4*x^2)).list()
A122112_list(30) # G. C. Greubel, Oct 02 2019

a(n) = (-1)^n * A026597(n).
a(n) = Sum_{k=0..n} (-2)^(n-k) * A055830(n,k).
G.f.: (1-x)/(1+x-4*x^2).
a(n) = (-2*i)^n*( ChebyshevU(n, -i/4) - (i/2)*ChebyshevU(n-1, -i/4) ). - G. C. Greubel, Dec 23 2021
E.g.f.: exp(-x/2)*(17*cosh(sqrt(17)*x/2) - 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Apr 03 2023

Corrected by T. D. Noe, Nov 07 2006

cp's OEIS Frontend

A122994 a(n) = a(n-1)+9*a(n-2) initialized with a(0)=1, a(1)=3.

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A158797 a(n) = a(n-1) + 36*a(n-2), a(0)=1, a(1)=6.

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A384640 Expansion of (1-3*x^2) / (1-x-5*x^2+x^3+2*x^4).

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A122999 G.f.: 1/(1 - x - 25*x^2).

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A158798 a(n) = a(n-1) + 64*a(n-2), a(0)=1, a(1)=8.

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A208765 Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section.

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A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

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A384640 Expansion of (1-3x^2) / (1-x-5x^2+x^3+2*x^4).

A384641 Expansion of (1+2x-x^3) / (1-x-5x^2+x^3+2*x^4).

A384642 Expansion of (1+x-x^2+x^3) / (1-x-5x^2+x^3+2x^4).