A107920 -id:A107920 - cp's OEIS frontend

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

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

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).

A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

From G. C. Greubel, Jan 18 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
  (p, q) = (1, 1) : 2*A007318(n,k).
  (p, q) = (2, 2) : 2*A038208(n,k).
  (p, q) = (3, 2) : A154692(n,k).
  (p, q) = (3, 3) : 2*A038221(n,k). (End)

			Triangle begins as:
     2;
     3,    3;
     5,    8,     5;
     9,   18,    18,     9;
    17,   40,    48,    40,    17;
    33,   90,   120,   120,    90,    33;
    65,  204,   300,   320,   300,   204,    65;
   129,  462,   756,   840,   840,   756,   462,   129;
   257, 1040,  1904,  2240,  2240,  2240,  1904,  1040,   257;
   513, 2322,  4752,  6048,  6048,  6048,  6048,  4752,  2322,  513;
  1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.
Cf. A215149.
Sums include: A008776 (row), A010673 (alternating sign row).
Columns k: A000051 (k=0).
Main diagonal: A059304.

A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
[A154690(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # R. J. Mathar, Jan 13 2011

T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten

from sage.all import *
def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).
Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).
From G. C. Greubel, Jan 18 2025: (Start)
T(n, n-k) = T(n, k) (symmetry).
T(n, 1) = n + A215149(n), n >= 1.
T(2*n-1, n) = 3*A069720(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)

A078020 Expansion of (1-x)/(1-x+2*x^2).

Original entry on oeis.org

1, 0, -2, -2, 2, 6, 2, -10, -14, 6, 34, 22, -46, -90, 2, 182, 178, -186, -542, -170, 914, 1254, -574, -3082, -1934, 4230, 8098, -362, -16558, -15834, 17282, 48950, 14386, -83514, -112286, 54742, 279314, 169830, -388798, -728458, 49138, 1506054, 1407778, -1604330, -4419886, -1211226, 7628546

Offset: 0

N. J. A. Sloane, Nov 17 2002

Equals the INVERT transform of [1, -1, -1, 1, 1, -1, -1, 1, 1, ...], i.e., 1 followed by repeats of (-1, -1, 1, 1, ...). - Gary W. Adamson, Sep 16 2008

Pisano period lengths: 1, 1, 8, 1, 24, 8, 21, 2, 24, 24, 10, 8, 168, 21, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012

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

Cf. A001607, A107920.

a:=[1,0];; for n in [2..50] do a[n]:=a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Jun 29 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/(1-x+2*x^2) )); // G. C. Greubel, Jun 29 2019

LinearRecurrence[{1,-2}, {1,0}, 50] (* or *) CoefficientList[Series[(1 - x)/(1-x+2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 29 2019 *)

Vec((1-x)/(1-x+2*x^2)+O(x^50)) \\ Charles R Greathouse IV, Sep 25 2012

((1-x)/(1-x+2*x^2)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 29 2019

a(n) = A107920(n+1) - A107920(n). - R. J. Mathar, Mar 14 2011

a(n) = (-1)^n*(A001607(n) + A001607(n-1)). - G. C. Greubel, Jun 29 2019

A146078 Expansion of 1/(1-x(1-9x)).

Original entry on oeis.org

1, 1, -8, -17, 55, 208, -287, -2159, 424, 19855, 16039, -162656, -307007, 1156897, 3919960, -6492113, -41771753, 16657264, 392603041, 242687665, -3290739704, -5474928689, 24141728647, 73416086848, -143859470975, -804604252607

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1, x(1-9x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-9).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

LinearRecurrence[{1, -9}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

x='x+O('x^30); Vec(1/(1-x+9*x^2)) \\ G. C. Greubel, Jan 19 2018

[lucas_number1(n,1,9) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 9*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*9^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+9*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(35)*x/2) + (1/sqrt(35))*sin(sqrt(35)*x/2)). (End)
a(n) = Product_{k=1..n} (1 + 6*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
a(n) = 3^n * U(n, 1/6), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022

A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

Original entry on oeis.org

1, -3, -1, 5, 7, -3, -17, -11, 23, 45, -1, -91, -89, 93, 271, 85, -457, -627, 287, 1541, 967, -2115, -4049, 181, 8279, 7917, -8641, -24475, -7193, 41757, 56143, -27371, -139657, -84915, 194399, 364229, -24569, -753027, -703889, 802165, 2209943

Offset: 0

Paul Barry, Nov 03 2009

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

Variants are A107920 and A001607.

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

a[n_] := Sin[n*ArcTan[Sqrt[7]]]; FullSimplify[Join[{1}, Table[- (2^(n/2 + 1)/Sqrt[7])*(2*a[n] + Sqrt[2]*a[n + 1]), {n, 1, 100}]]] (* or *) Join[{1}, LinearRecurrence[{1,-2},{-3,-1},100]] (* G. C. Greubel, Jun 13 2016 *)

G.f.: (1-4x+4x^2)/(1-x+2x^2).
From G. C. Greubel, Jun 13 2016: (Start)
a(n) = a(n-1) - 2*a(n-2).
a(n) = -(2^((n+2)/2)/sqrt(7))*( 2*sin(n*arctan(sqrt(7))) + sqrt(2)*sin((n+1)*arctan(sqrt(7))) ), n>=1, and a(0)=1. (End)

A105578 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, -1, 0, 3, 4, -1, -8, -5, 12, 23, 0, -45, -44, 47, 136, 43, -228, -313, 144, 771, 484, -1057, -2024, 91, 4140, 3959, -4320, -12237, -3596, 20879, 28072, -13685, -69828, -42457, 97200, 182115, -12284, -376513, -351944, 401083, 1104972, 302807, -1907136, -2512749, 1301524, 6327023, 3723976

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: ibaseiseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2).

Cf. A002249, A014551, A078020, A105577, A105579.

Equals (A107920(n) + 1)/2.

-Join[{-1,-1,a=0,b=1},Table[c=1*b-2*a-1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
LinearRecurrence[{2,-3,2},{1,1,0},50] (* Harvey P. Dale, Mar 28 2019 *)

Vec(-(x^2-x+1)/((x-1)*(2*x^2-x+1)) + O(x^100)) \\ Colin Barker, Feb 08 2015

a(n) - a(n+1) = A001607(n); a(n+2) - 2a(n+1) + a(n) = - A078020(n).

G.f.: -(x^2-x+1) / ((x-1)*(2*x^2-x+1)). - Colin Barker, Feb 08 2015

A169998 a(0)=1, a(1)=1; thereafter a(n) = -a(n-1) - 2*a(n-2).

Original entry on oeis.org

1, 1, -3, 1, 5, -7, -3, 17, -11, -23, 45, 1, -91, 89, 93, -271, 85, 457, -627, -287, 1541, -967, -2115, 4049, 181, -8279, 7917, 8641, -24475, 7193, 41757, -56143, -27371, 139657, -84915, -194399, 364229, 24569, -753027, 703889, 802165, -2209943, 605613, 3814273, -5025499, -2603047

Offset: 0

N. J. A. Sloane, Aug 29 2010

Cassels, following Nagell, shows that a(n) = +- 1 only for n = 1, 2, 3, 5, 13.

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

J. W. S. Cassels, Local Fields, Cambridge, 1986, see p. 67.

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.
Index entries for linear recurrences with constant coefficients, signature (-1,-2).

f:=proc(n) option remember; if n <= 1 then 1 else -f(n-1)-2*f(n-2); fi; end;

LinearRecurrence[{-1, -2}, {1, 1}, 46] (* Jean-François Alcover, Feb 23 2024 *)

a(n)=([0,1;-2,-1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

G.f.: ( 1+2*x ) / ( 1+x+2*x^2 ). - R. J. Mathar, Jul 14 2011

A105577 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 1, a(1) = 5, a(2) = 6.

Original entry on oeis.org

1, 5, 6, -1, -10, -5, 18, 31, -2, -61, -54, 71, 182, 43, -318, -401, 238, 1043, 570, -1513, -2650, 379, 5682, 4927, -6434, -16285, -3414, 29159, 35990, -22325, -94302, -49649, 138958, 238259, -39654, -516169, -436858, 595483, 1469202, 278239, -2660162, -3216637, 2103690, 8536967

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 2lesseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Index entries for linear recurrences with constant coefficients, signature (2,-3,2).

Cf. A002249, A014551, A105578.

Equals (1/4) [A107920(n+4) + 2*A107920(n-1) + 3].

LinearRecurrence[{2,-3,2},{1,5,6},50] (* Harvey P. Dale, Apr 13 2019 *)

G.f.: (1+3*x-x^2)/((1-x)*(1-x+2*x^2)). - Colin Barker, Mar 26 2012

E.g.f.: exp(x/2)*(21*exp(x/2) - 7*cos(sqrt(7)*x/2) + 15*sqrt(7)*sin(sqrt(7)*x/2))/14. - Stefano Spezia, May 22 2025

A146083 Expansion of 1/(1 - x(1 - 11x)).

Original entry on oeis.org

1, 1, -10, -21, 89, 320, -659, -4179, 3070, 49039, 15269, -524160, -692119, 5073641, 12686950, -43123101, -182679551, 291674560, 2301149621, -907270539, -26219916370, -16239940441, 272179139629, 450818484480, -2543152051439

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-11x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-11).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978, A146078, A146080.

[n le 2 select 1 else Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 31 2016

LinearRecurrence[{1,-11},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

Vec(1/(1-x*(1-11*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,11) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-11*a(n-2) ; a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*11^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+11*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(43)*x/2) + (1/sqrt(43))*sin(sqrt(43)*x/2)).
(End)

cp's OEIS Frontend

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

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

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A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

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A078020 Expansion of (1-x)/(1-x+2*x^2).

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A146078 Expansion of 1/(1-x*(1-9*x)).

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A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

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A105578 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.

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

A146078 Expansion of 1/(1-x(1-9x)).

A105577 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 1, a(1) = 5, a(2) = 6.

A146083 Expansion of 1/(1 - x(1 - 11x)).