A122100 -id:A122100 - cp's OEIS frontend

A122099 a(n) = -3*a(n-1) + a(n-3) for n>2, with a(0)=1, a(1)=1, a(2)=0.

1, 1, 0, 1, -2, 6, -17, 49, -141, 406, -1169, 3366, -9692, 27907, -80355, 231373, -666212, 1918281, -5523470, 15904198, -45794313, 131859469, -379674209, 1093228314, -3147825473, 9063802210, -26098178316, 75146709475, -216376326215, 623030800329, -1793945691512, 5165460748321

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 (-3,0,1).

Cf. A122100.

a:=[1,1,0];; for n in [4..40] do a[n]:=-3*a[n-1]+a[n-3]; od; a; # G. C. Greubel, Oct 02 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x+3*x^2)/(1+3*x-x^3) )); // G. C. Greubel, Oct 02 2019

seq(coeff(series((1+4*x+3*x^2)/(1+3*x-x^3), x, n+1), x, n), n = 0..40); # G. C. Greubel, Oct 02 2019

Transpose[NestList[{#[[2]],Last[#],First[#]-3Last[#]}&, {1,1,0},35]][[1]]  (* Harvey P. Dale, Mar 13 2011 *)
LinearRecurrence[{-3,0,1}, {1,1,0}, 40] (* G. C. Greubel, Oct 02 2019 *)

Vec((1+4*x+3*x^2)/(1+3*x-x^3)+O(x^40)) \\ Charles R Greathouse IV, Jan 17 2012

def A122099_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+4*x+3*x^2)/(1+3*x-x^3)).list()
A122099_list(40) # G. C. Greubel, Oct 02 2019

G.f.: (1+4*x+3*x^2)/(1+3*x-x^3).

a(n) = (-1)^n*A122100(n). - R. J. Mathar, Sep 27 2014

A123942 The (1,4)-entry in the 4 X 4 matrix M^n, where M={{3, 2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}} (n>=0).

Original entry on oeis.org

0, 1, 3, 15, 71, 340, 1626, 7778, 37205, 177966, 851280, 4072001, 19477953, 93170570, 445670811, 2131815570, 10197297001, 48777608903, 233322137235, 1116069871981, 5338593130960, 25536552265626, 122151189577128, 584296304368075, 2794914830384226

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

G. C. Greubel, Table of n, a(n) for n = 0..1000
Martin H. Gutknecht and Lloyd N. Trefethen, Real Polynomial Chebyshev Approximation by the Caratheodory-Fejer Method, SIAM Journal on Numerical Analysis, Vol. 19, No. 2 (Apr., 1982), pp. 358-371.
Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

Cf. A122099, A122100.

a:=[0,1,3,15];; for n in [5..30] do a[n]:=4*a[n-1]+4*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Aug 05 2019

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4) )); // G. C. Greubel, Aug 05 2019

with(linalg): M[1]:=matrix(4,4,[3,2,1,1,2,1,1,0,1,1,0,0,1,0,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0, seq(M[n][1,4], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=3: a[3]:=15: for n from 4 to 30 do a[n]:=4*a[n-1] +4*a[n-2]-a[n-3]-a[n-4] od: seq(a[n], n=0..30);

M = {{3,2,1,1}, {2,1,1,0}, {1,1,0,0}, {1,0,0,0}}; v[1] = {0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
LinearRecurrence[{4,4,-1,-1}, {0,1,3,15}, 30] (* G. C. Greubel, Aug 05 2019 *)

concat([0], Vec(x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4) + O(x^30))) \\ Colin Barker, Oct 18 2013

(x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

a(n) = 4*a(n-1) + 4*a(n-2) - a(n-3) - a(n-4) for n>=4 (follows from the minimal polynomial of the matrix M).

G.f.: x*(1-x-x^2)/(1-4*x-4*x^2+x^3+x^4). - Colin Barker, Oct 18 2013

Edited by N. J. A. Sloane, Dec 04 2006

More terms from Colin Barker, Oct 18 2013

A215448 a(0)=1, a(1)=0, a(n) = a(n-1) + a(n-2) + Sum_{i=0...n-1} a(i).

Original entry on oeis.org

1, 0, 2, 5, 15, 43, 124, 357, 1028, 2960, 8523, 24541, 70663, 203466, 585857, 1686908, 4857258, 13985917, 40270843, 115955271, 333879896, 961368845, 2768151264, 7970573896, 22950352843, 66082907265, 190278147899, 547884090854, 1577569365297, 4542429947992

Offset: 0

Alex Ratushnyak, Aug 10 2012

For the general recurrence X(n) = 3*X(n-1) - X(n-3) we get Sum_{k=3..n} X(k) = 3*Sum_{k=2..n-1} X(k) - Sum_{k=0..n-3} X(k), which implies the following summation formula: X(n) - X(n-1) - X(n-2) - X(2) + X(1) + X(0) = Sum_{k=2..n-1} X(k). Similarly from the formula X(n) + X(n-3) = 3*X(n-1) we deduce the following relations: Sum_{k=0..2*n-1} X(3*k) = 3*Sum_{k=0..n-1} X(6*k+2), Sum_{k=0..2*n-1} X(3*k+1) = 3*Sum_{k=1..n} X(6*k), and Sum_{k=0..2*n-1} X(3*k+2) = 3*Sum_{k=1..n} X(6*k-2). Lastly from the formula X(n)-X(n-1)=(X(n-1)-X(n-3))+X(n-1) we obtain the relations: Sum_{k=2..2*n+1} (-1)^(k-1)*X(k) = X(2*n) - X(0) + Sum_{k=1..n} X(2*k) and Sum_{k=3..2n} (-1)^k*X(k) = X(2*n-1) - X(1) + Sum_{k=2..n} X(2*k-1). - Roman Witula, Aug 27 2012

Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A052536: same formula, seed {0, 1}, first term removed.
Cf. A122100: same formula, seed {0,-1}, first two terms removed.
Cf. A052545: same formula, seed {1, 1}.

LinearRecurrence[{3,0,-1},{1,0,2},30] (* Harvey P. Dale, Jan 26 2017 *)

a = [1]*33
a[1]=0
sum = a[0]+a[1]
for n in range(2,33):
    print(a[n-2], end=', ')
    a[n] = a[n-1] + a[n-2] + sum
    sum += a[n]

a(0)=1, a(1)=0, for n>=2, a(n) = a(n-1) + a(n-2) + (a(0)+...+a(n-1)).
Conjecture: a(n) = +3*a(n-1) -a(n-3) = A076264(n) -3 *A076264(n-1) +2*A076264(n-2). G.f. (2*x-1)*(x-1) / ( 1-3*x+x^3 ). - R. J. Mathar, Aug 11 2012
Proof of the above conjecture: we have a(n) - a(n-1) = a(n-1) + a(n-2) + (a(0) + ... + a(n-1)) - a(n-2) - a(n-3) - (a(0) + ... + a(n-2)), which after simple algebra implies a(n) - a(n-1) = 2*a(n-1) - a(n-3), so the Mathar's formula holds true (see also Witula's comment above). - Roman Witula, Aug 27 2012

A123941 The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}.

Original entry on oeis.org

0, 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639, 954538564968, 2748484256480

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Essentially the same as A076264. - Tom Edgar, May 12 2015

Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A018919, A122099, A122100.

a:=[0,1,3];; for n in [4..30] do a[n]:=3*a[n-1]-a[n-3]; od; a; # Muniru A Asiru, Oct 28 2018

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/(1-3*x+x^3) )); // G. C. Greubel, Aug 05 2019

with(linalg): M[1]:=matrix(3,3,[2,1,1,1,1,0,1,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,2], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=3: for n from 3 to 30 do a[n]:=3*a[n-1]-a[n-3] od: seq(a[n], n=0..30);

M = {{2,1,1}, {1,1,0}, {1,0,0}}; v[1] = {0,0,1}; v[n_]:= v[n] =M.v[n-1];Table[v[n][[2]], {n, 30}]
LinearRecurrence[{3,0,-1}, {0,1,3}, 30] (* G. C. Greubel, Aug 05 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+x^3))) \\ G. C. Greubel, Aug 05 2019

(x/(1-3*x+x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

a(n) = 3*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=3 (follows from the minimal polynomial x^3-3x^2+1 of the matrix M).
a(n) = A076264(n-1). - R. J. Mathar, Jun 18 2008
G.f.: x/(1 - 3*x + x^3). - Arkadiusz Wesolowski, Oct 29 2012
a(n) = A018919(n-2) for n >= 2. - Georg Fischer, Oct 28 2018

Edited by N. J. A. Sloane, Nov 07 2006

A123943 The (1,5)-entry in the 5 X 5 matrix M^n, where M={{5, 3, 2, 1, 1}, {3, 2, 1, 1, 0}, {2, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}.

Original entry on oeis.org

0, 1, 5, 40, 315, 2490, 19681, 155563, 1229604, 9719061, 76821600, 607214857, 4799560053, 37936780428, 299860673343, 2370164848026, 18734305316497, 148080078051971, 1170457572108040, 9251554605638681, 73126326541645648, 578006601205833441

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

See A123942 for references.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,0,-6,0,1).

Cf. A122099, A122100.

a:=[0,1,5,40,315];; for n in [6..30] do a[n]:=8*a[n-1]-6*a[n-3] +a[n-5]; od; a; # G. C. Greubel, Aug 05 2019

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) )); // G. C. Greubel, Aug 05 2019

with(linalg): M[1]:=matrix(5,5,[5,3,2,1,1,3,2,1,1,0,2,1,1,0,0,1,1,0,0,0,1,0,0,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,5],n=1..30);
a[0]:=0: a[1]:=1: a[2]:=5: a[3]:=40: a[4]:=315: for n from 5 to 30 do a[n]:=8*a[n-1]-6*a[n-3]+a[n-5] od: seq(a[n],n=0..30);
# third Maple program:
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>,
         <1|0|-6|0|8>>^n. <<0, 1, 5, 40, 315>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 05 2019

M = {{5,3,2,1,1}, {3,2,1,1,0}, {2,1,1,0,0}, {1,1,0,0,0}, {1,0,0,0,0}}; v[1] = {0,0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
LinearRecurrence[{8,0,-6,0,1}, {0,1,5,40,315}, 30] (* G. C. Greubel, Aug 05 2019 *)

concat(0, Vec(x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) + O(x^30))) \\ Colin Barker, Mar 03 2017

(x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

a(n) = 8*a(n-1) - 6*a(n-3) + a(n-5) for n>=5 (follows from the minimal polynomial of the matrix M).

G.f.: x*(1 - 3*x + x^3) / (1 - 8*x + 6*x^3 - x^5). - Colin Barker, Mar 03 2017

Edited by N. J. A. Sloane, Dec 04 2006

A049126 Revert transform of ((x - 1)(3x - 1))/(1 - 3x + x^3).

Original entry on oeis.org

1, 1, 2, 6, 22, 90, 393, 1789, 8378, 40050, 194492, 956448, 4752519, 23822571, 120315345, 611644737, 3127389106, 16072642890, 82981119252, 430187414196, 2238469102212, 11687227631892, 61208286479382, 321465732705594

Offset: 1

Olivier Gérard

nonn

			G.f. = x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 90*x^6 + 393*x^7 + 1789*x^8 + ...

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for reversions of series

Cf. A122100.

Rest[CoefficientList[InverseSeries[Series[x*((x - 1)(3x - 1))/(1 - 3x + x^3), {x, 0, 40}], x], x]] (* Vaclav Kotesovec, Jan 02 2021 *)

{a(n) = if( n<1, 0, polcoeff( serreverse(x * (1 - 4*x + 3*x^2) / (1 - 3*x + x^3) + x * O(x^n)), n))}; /* Michael Somos, May 11 2012 */

Revert transform of A122100 offset 1. - Michael Somos, May 11 2012
Recurrence: 12*(n-1)*n*(117*n - 604)*a(n) = 4*(n-1)*(4329*n^2 - 29719*n + 39384)*a(n-1) - 3*(23985*n^3 - 230056*n^2 + 679659*n - 630308)*a(n-2) + 3*(38961*n^3 - 437004*n^2 + 1609875*n - 1955812)*a(n-3) - 27*(n-4)*(2223*n^2 - 18496*n + 38133)*a(n-4) + 81*(n-5)*(n-4)*(117*n - 487)*a(n-5). - Vaclav Kotesovec, Jan 02 2021
a(n) ~ 3^(n - 1/4) * (2 + sqrt(3))^(n - 3/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, Jan 02 2021

cp's OEIS Frontend

A122099 a(n) = -3*a(n-1) + a(n-3) for n>2, with a(0)=1, a(1)=1, a(2)=0.

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A123942 The (1,4)-entry in the 4 X 4 matrix M^n, where M={{3, 2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}} (n>=0).

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A215448 a(0)=1, a(1)=0, a(n) = a(n-1) + a(n-2) + Sum_{i=0...n-1} a(i).

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A123941 The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}.

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A123943 The (1,5)-entry in the 5 X 5 matrix M^n, where M={{5, 3, 2, 1, 1}, {3, 2, 1, 1, 0}, {2, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}.

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GAP

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Extensions

A049126 Revert transform of ((x - 1)(3x - 1))/(1 - 3x + x^3).

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