A190871 -id:A190871 - cp's OEIS frontend

A190872 a(n) = 11a(n-1) - 9a(n-2), a(0)=0, a(1)=1.

0, 1, 11, 112, 1133, 11455, 115808, 1170793, 11836451, 119663824, 1209774005, 12230539639, 123647969984, 1250052813073, 12637749213947, 127764766035760, 1291672683467837, 13058516623824367, 132018628710857504, 1334678266205013241, 13493293269857428115

Offset: 0

Rolf Pleisch, May 22 2011

a(k) is Heuberger and Wagner's G_k at lemma 6.2 (2). They show (theorem 3.3 (1)) that the largest number of maximum matchings in a tree of 7k+1 vertices is a(k+1) and there is a unique free tree with this many maximum matchings. (See A333347 for all tree sizes.) - Kevin Ryde, Apr 11 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Clemens Heuberger and Stephan Wagner, The Number of Maximum Matchings in a Tree, Discrete Mathematics, volume 311, issue 21, November 2011, pages 2512-2542; arXiv preprint, arXiv:1011.6554 [math.CO], 2010.
Index entries for linear recurrences with constant coefficients, signature (11,-9).

Cf. A333345 (growth power), A190871, A190873.

I:=[0,1]; [n le 2 select I[n] else 11*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 19 2015

LinearRecurrence[{11, -9}, {0, 1}, 50] (* T. D. Noe, May 23 2011 *)

concat(0, Vec(x/(1-11*x+9*x^2) + O(x^100))) \\ Altug Alkan, Dec 18 2015

a(n) = polcoeff(lift(Mod('x,'x^2-11*'x+9)^n), 1); \\ Kevin Ryde, Apr 11 2020

a(n) = ((11+sqrt(85))^n-(11-sqrt(85))^n)/(2^n*sqrt(85)).
G.f.: x/(1-11*x+9*x^2). - Philippe Deléham, Feb 12 2012
E.g.f.: (2/sqrt(85))*exp(11*x/2)*sinh(sqrt(85)*x/2). - G. C. Greubel, Dec 18 2015
a(n) = (L^n - H^n)/(L-H) where L = (11+sqrt(85))/2 and H = (11-sqrt(85))/2. [Heuberger and Wagner lemma 6.2 (2)] - Kevin Ryde, Apr 11 2020

Extended by T. D. Noe, May 23 2011

A190873 a(n) = 12a(n-1) - 12a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 12, 132, 1440, 15696, 171072, 1864512, 20321280, 221481216, 2413919232, 26309256192, 286744043520, 3125217447936, 34061680852992, 371237560860672, 4046110560092160, 44098475990777856, 480628385168228352, 5238358910129405952, 57092766299534131200

Offset: 0

Rolf Pleisch, May 22 2011

nonn

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

Cf. A057086, A190871, A190872.

[n le 2 select n-1 else 12*(Self(n-1) - Self(n-2)): n in [1..31]]; // G. C. Greubel, Sep 11 2023

LinearRecurrence[{12,-12}, {0,1}, 50] (* T. D. Noe, May 23 2011 *)

concat(0, Vec(x/(1-12*x+12*x^2) + O(x^100))) \\ Altug Alkan, Dec 18 2015

def A190873(n): return (2*sqrt(3))^(n-1)*chebyshev_U(n-1, sqrt(3))
[A190873(n) for n in range(31)] # G. C. Greubel, Sep 11 2023

a(n) = 2^(n-2)*((3+sqrt(6))^n - (3-sqrt(6))^n)/sqrt(6).
G.f.: x/(1 - 12*x + 12*x^2). - Philippe Deléham, Dec 21 2011
E.g.f.: (1/(2*sqrt(6)))*exp(6*x)*sinh(2*sqrt(6)*x). - G. C. Greubel, Dec 18 2015
a(n) = (2*sqrt(3))^(n-1)*chebyshev_U(n-1, sqrt(3)). - G. C. Greubel, Sep 11 2023

Extended by T. D. Noe, May 23 2011

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

A190870 a(n) = 11a(n-1) - 22a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 11, 99, 847, 7139, 59895, 501787, 4201967, 35182323, 294562279, 2466173963, 20647543455, 172867150819, 1447292702999, 12117142414971, 101448127098703, 849352264956371, 7111016118348615, 59535427472794603, 498447347597071103, 4173141419166300867

Offset: 0

Rolf Pleisch, May 22 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -22).

Cf. A190871, A190872, A190873

LinearRecurrence[{11, -22}, {0, 1}, 50] (* T. D. Noe, May 23 2011 *)

concat(0, Vec(x/(1-11*x+22*x^2) + O(x^100))) \\ Altug Alkan, Dec 18 2015

a(n) = ((11+sqrt(33))^n-(11-sqrt(33))^n)/(2^n*sqrt(33)).
E.g.f.: (2/sqrt(33))*exp(11*x/2)*sinh(sqrt(33)*x/2). - G. C. Greubel, Dec 18 2015
G.f.: x/(1-11*x+22*x^2). - G. C. Greubel, Dec 18 2015

Extended by T. D. Noe, May 23 2011

A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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

Offset: 0

Philippe Deléham, Dec 21 2011

Riordan array (1/(1-x+x^2), x*(x-1)/(1-x+x^2)).

			Triangle begins :
1
1, -1
0, -1, 1
-1, 1, 1, -1
-1, 3, -2, -1, 1
0, 2, -5, 3, 1, -1

Cf. A129267, A199324

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1+(y-1)*x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190873(n+1), A190871(n+1), A057086(n), A057085(n+1), A057084(n), A030240(n), A030192(n), A030191(n), A001787(n+1), A057083(n), A099087(n), A010892(n), A000007(n), (-1)^n*A000045(n+1) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2 respectively.

cp's OEIS Frontend

A190872 a(n) = 11*a(n-1) - 9*a(n-2), a(0)=0, a(1)=1.

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A190873 a(n) = 12*a(n-1) - 12*a(n-2), a(0)=0, a(1)=1.

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A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

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A190870 a(n) = 11*a(n-1) - 22*a(n-2), a(0)=0, a(1)=1.

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Mathematica

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Extensions

A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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Author

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Comments

Examples

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Formula

A190872 a(n) = 11a(n-1) - 9a(n-2), a(0)=0, a(1)=1.

A190873 a(n) = 12a(n-1) - 12a(n-2), a(0)=0, a(1)=1.

A190870 a(n) = 11a(n-1) - 22a(n-2), a(0)=0, a(1)=1.