Rolf Pleisch - cp's OEIS frontend

A190869 a(n) = 10a(n-1) - 2a(n-2), a(0)=0, a(1)=1.

0, 1, 10, 98, 960, 9404, 92120, 902392, 8839680, 86592016, 848240800, 8309223968, 81395758080, 797339132864, 7810599812480, 76511319859072, 749491998965760, 7341897349939456, 71919989501463040, 704516100314751488, 6901321024144588800, 67604178040816385024

Offset: 0

Rolf Pleisch, May 22 2011

a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6,7,8,9} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

Robert Israel, Table of n, a(n) for n = 0..990
Index entries for linear recurrences with constant coefficients, signature (10, -2).

I:=[0,1]; [n le 2 select I[n] else 10*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

f:= gfun:-rectoproc({a(n) = 10*a(n-1) - 2*a(n-2), a(0)=0, a(1)=1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Dec 17 2015

LinearRecurrence[{10, -2}, {0, 1}, 50] (* T. D. Noe, May 23 2011 *)

concat(0, Vec(x/(1-10*x+2*x^2) + O(x^100))) \\ Altug Alkan, Dec 17 2015

a(n) = ((5+sqrt(23))^n-(5-sqrt(23))^n)/(2*sqrt(23)).

G.f.: x/(1-10*x+2*x^2). - Robert Israel, Dec 17 2015

Corrected and extended by T. D. Noe, May 23 2011

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

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

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

Original entry on oeis.org

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

A190871 a(n) = 11a(n-1) - 11a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 11, 110, 1089, 10769, 106480, 1052821, 10409751, 102926230, 1017681269, 10062305429, 99490865760, 983714163641, 9726456276691, 96170163243550, 950880776635449, 9401816747310889, 92960295677429840, 919143268231308461, 9088012698092664831

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,-11).

Cf. A057086, A190872, A190873.

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

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

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

def A190871(n): return (sqrt(11))^(n-1)*chebyshev_U(n-1, sqrt(11)/2)
[A190871(n) for n in range(31)] # G. C. Greubel, Sep 11 2023

a(n) = ((11+sqrt(77))^n-(11-sqrt(77))^n)/(2^n*sqrt(77)).
G.f.: x/(1-11x+11x^2). - Philippe Deléham, Dec 21 2011
E.g.f.: (2/sqrt(77))*exp(11*x/2)*sinh(sqrt(77)*x/2). - G. C. Greubel, Dec 18 2015
a(n) = (sqrt(11))^(n-1)*chebyshev_U(n-1, sqrt(11)/2). - G. C. Greubel, Sep 11 2023

Extended by T. D. Noe, May 23 2011

A164044 a(n+1) = 4*a(n) - n.

Original entry on oeis.org

1, 3, 10, 37, 144, 571, 2278, 9105, 36412, 145639, 582546, 2330173, 9320680, 37282707, 149130814, 596523241, 2386092948, 9544371775, 38177487082, 152709948309, 610839793216, 2443359172843, 9773436691350, 39093746765377

Offset: 0

Rolf Pleisch, Aug 08 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -9, 4).

Table[(5*4^n + 3*n + 4)/9, {n,0,50}] (* G. C. Greubel, Sep 08 2017 *)

x='x+O('x^50); Vec((1-3*x+x^2)/((1-4*x)*(1-x)^2)) \\ G. C. Greubel, Sep 08 2017

a(0)=1; a(n+1) = 4*a(n) - n.
a(n) = (5*4^n + 3*n + 4)/9.
From R. J. Mathar, Aug 09 2009: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3).
G.f.: (1-3*x+x^2)/((1-4*x)*(1-x)^2). (End)
E.g.f.: (1/9)*(5*exp(4*x) + (3*x + 4)*exp(x)). - G. C. Greubel, Sep 08 2017

A164039 a(n+1) = 3*a(n) - n.

Original entry on oeis.org

1, 2, 4, 9, 23, 64, 186, 551, 1645, 4926, 14768, 44293, 132867, 398588, 1195750, 3587235, 10761689, 32285050, 96855132, 290565377, 871696111, 2615088312, 7845264914, 23535794719, 70607384133, 211822152374, 635466457096

Offset: 0

Rolf Pleisch, Aug 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Transpose[NestList[Flatten[{Rest[#],ListCorrelate[#,{3,-7,5}]}]&, {1,2,4},30]][[1]] (* Harvey P. Dale, Mar 24 2011 *)
Table[(3^n + 2*n + 3)/4, {n,0,50}] (* G. C. Greubel, Sep 08 2017 *)

x='x+O('x^50); Vec((1-3*x+x^2)/((1-3*x)*(1-x)^2)) \\ G. C. Greubel, Sep 08 2017

a(0) = 1; a(n+1) = 3*a(n) - n.
a(n) = (3^n + 2*n + 3)/4.
From R. J. Mathar, Aug 09 2009: (Start)
a(n) = 5*a(n-1)-7*a(n-2)+3*a(n-3).
G.f.: (1-3*x+x^2)/((1-3*x)*(1-x)^2). (End)
E.g.f.: (1/4)*((2*x + 3)*exp(x) + exp(3*x)). - G. C. Greubel, Sep 08 2017

A164045 a(n+1) = 5*a(n) - n.

Original entry on oeis.org

1, 4, 18, 87, 431, 2150, 10744, 53713, 268557, 1342776, 6713870, 33569339, 167846683, 839233402, 4196166996, 20980834965, 104904174809, 524520874028, 2622604370122, 13113021850591, 65565109252935, 327825546264654

Offset: 0

Rolf Pleisch, Aug 08 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -11, 5).

Table[(11*5^n + 4*n + 5)/16, {n,0,50}] (* G. C. Greubel, Sep 08 2017 *)
LinearRecurrence[{7,-11,5},{1,4,18},30] (* or *) nxt[{n_,a_}]:={n+1,5a-n-1}; NestList[nxt,{0,1},30][[;;,2]] (* Harvey P. Dale, Sep 29 2023 *)

a(n) = (11*5^n + 4*n + 5)/16 \\ Michel Marcus, Jul 18 2013

a(0)=1; a(n+1) = 5*a(n) - n.
a(n) = (11*5^n + 4*n + 5)/16.
From R. J. Mathar, Aug 09 2009: (Start)
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
G.f.: (1-3*x+x^2)/((1-5*x)*(1-x)^2). (End)
E.g.f.: (1/16)*(11*exp(5*x) + (4*x + 5)*exp(x)). - G. C. Greubel, Sep 08 2017

A140479 n^2 - number of digits of n^2.

Original entry on oeis.org

0, 0, 3, 8, 14, 23, 34, 47, 62, 79, 97, 118, 141, 166, 193, 222, 253, 286, 321, 358, 397, 438, 481, 526, 573, 622, 673, 726, 781, 838, 897, 958, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597

Offset: 0

Rolf Pleisch, Jun 29 2008

a(n) = n^2 - length(digits(n^2)) \\ Michel Marcus, Jul 18 2013

a(n) = A000290(n) - A055642(A000290(n)). Michel Marcus, Jul 18 2013

Corrected and extended by N. J. A. Sloane, Jun 29 2008

A139817 2^n - number of digits of 2^n.

Original entry on oeis.org

0, 1, 3, 7, 14, 30, 62, 125, 253, 509, 1020, 2044, 4092, 8188, 16379, 32763, 65531, 131066, 262138, 524282, 1048569, 2097145, 4194297, 8388601, 16777208, 33554424, 67108856, 134217719, 268435447, 536870903, 1073741814, 2147483638, 4294967286, 8589934582

Offset: 0

Rolf Pleisch, May 23 2008

Cf. A000079, A055642.

Table[2^n-IntegerLength[2^n],{n,0,40}] (* Harvey P. Dale, Apr 05 2015 *)

a(n) = 2^n - length(digits(2^n)) \\ Michel Marcus, Jul 18 2013

a(n) = A000079(n) - A055642(A000079(n)). - Michel Marcus, Jul 18 2013

More terms from N. J. A. Sloane, Jun 29 2008

cp's OEIS Frontend

User: Rolf Pleisch

A190869 a(n) = 10*a(n-1) - 2*a(n-2), a(0)=0, a(1)=1.

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

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

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A190871 a(n) = 11*a(n-1) - 11*a(n-2), a(0)=0, a(1)=1.

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A164044 a(n+1) = 4*a(n) - n.

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A164039 a(n+1) = 3*a(n) - n.

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A190869 a(n) = 10a(n-1) - 2a(n-2), a(0)=0, a(1)=1.

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

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

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

A190871 a(n) = 11a(n-1) - 11a(n-2), a(0)=0, a(1)=1.