A090017 -id:A090017 - cp's OEIS frontend

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A218984 Power floor sequence of 2+sqrt(6).

Original entry on oeis.org

4, 17, 75, 333, 1481, 6589, 29317, 130445, 580413, 2582541, 11490989, 51129037, 227498125, 1012250573, 4503998541, 20040495309, 89169978317, 396760903885, 1765383572173, 7855056096461, 34950991530189, 155514078313677, 691958296315085, 3078861341887693

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p1(r) = 3.77794213613376987528458445727451673384055973517...

			a(0) = [r] = 4, where r = 2+sqrt(6); a(1) = [4*r] = 17; a(2) = [17*r] = 75.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-2,-2).

Cf. A214992, A090017, A123347, A218985.

x = 2 + Sqrt[6]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A218984 *)
t2 = Table[p2[n], {n, 0, z}]  (* A090017 *)
t3 = Table[p3[n], {n, 0, z}]  (* A123347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218985 *)

Vec((4 - 3*x - 2*x^2) / ((1 - x)*(1 - 4*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].
a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3).
G.f.: (4 - 3*x - 2*x^2)/(1 - 5*x + 2*x^2 + 2*x^3).
a(n) = (1/30)*(6 + (57-23*sqrt(6))*(2-sqrt(6))^n + (2+sqrt(6))^n*(57+23*sqrt(6))). - Colin Barker, Nov 13 2017

A218985 Power ceiling sequence of 2+sqrt(6).

Original entry on oeis.org

5, 23, 103, 459, 2043, 9091, 40451, 179987, 800851, 3563379, 15855219, 70547635, 313900979, 1396699187, 6214598707, 27651793203, 123036370227, 547449067315, 2435869009715, 10838374173491, 48225234713395, 214577687200563, 954761218229043, 4248200247317299

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p4(r) = 5.2127890589687233047437696796862841514303439...

See A218984 for the power floor function, p1(x). For comparison of p4 and p1, limit(p4(r)/p1(r)) = 2*(1+sqrt(6))/5 = 1.379795897113271239278913629882356556786378...

			a(0) = ceiling(r) = 5, where r = 2+sqrt(6).
a(1) = ceiling(5*r) = 23; a(2) = ceiling(23*r) = 103.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-2,-2).

Cf. A214992, A090017, A123347, A218984.

Mathematica
```
(See A218984.)
```

Vec((5 - 2*x - 2*x^2) / ((1 - x)*(1 - 4*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].
a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3).
G.f.: (5 - 2*x - 2*x^2)/(1 - 5*x + 2*x^2 + 2*x^3).
a(n) = (1/15)*(-3 + (39-16*sqrt(6))*(2-sqrt(6))^n + (2+sqrt(6))^n*(39+16*sqrt(6))). - Colin Barker, Nov 13 2017

A335749 a(n) = n![x^n] exp(2x)(ysinh(xy) + cosh(xy)) and y = sqrt(6).

Original entry on oeis.org

1, 8, 34, 152, 676, 3008, 13384, 59552, 264976, 1179008, 5245984, 23341952, 103859776, 462123008, 2056211584, 9149092352, 40708792576, 181133355008, 805951005184, 3586070730752, 15956184933376, 70996881195008, 315899894646784, 1405593340977152, 6254173153202176

Offset: 0

Peter Luschny, Jun 24 2020

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A335312.

aList := proc(len) local H; H := (x, y) -> exp(2*x)*(y*sinh(x*y) + cosh(x*y)):
series(H(x, sqrt(6)), x, len + 1): seq(k!*coeff(%, x, k), k=0..len-1) end:
aList(25);

LinearRecurrence[{4, 2}, {1, 8}, 30] (* Paolo Xausa, Feb 01 2024 *)

Vec((1 + 4*x) / (1 - 4*x - 2*x^2) + O(x^25)) \\ Colin Barker, Jun 25 2020

a(n) = A335312(n, 6).
From Colin Barker, Jun 24 2020: (Start)
G.f.: (1 + 4*x) / (1 - 4*x - 2*x^2) for n>1.
a(n) = 4*a(n-1) + 2*a(n-2). (End)
a(n) = 4*A090017(n)+A090017(n+1). - R. J. Mathar, Mar 10 2022

A123348 Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

Original entry on oeis.org

0, 3, 12, 54, 240, 1068, 4752, 21144, 94080, 418608, 1862592, 8287584, 36875520, 164077248, 730060032, 3248394624, 14453698560, 64311583488, 286153731072, 1273238091264, 5665259827200, 25207515491328, 112160581619712, 499057357461504, 2220550593085440, 9880317087264768, 43962369535229952

Offset: 0

N. J. A. Sloane, Oct 10 2006

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78, Binet formula page 77).
Index entries for linear recurrences with constant coefficients, signature (4,2).

A123348 := proc(n)
    3*((2+sqrt(6))^n-(2-sqrt(6))^n)/2/sqrt(6) ;
    expand(%) ;
    simplify(%) ;
end proc:
seq( A123348(n),n=0..30) ; # R. J. Mathar, Jul 26 2019

Conjectured g.f.: 1/(1 - Q(0)) - 1, where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Apr 13 2013

G.f.: -3*x / (2*x^2+4*x-1). a(n)=3*A090017(n). - Colin Barker, Aug 29 2013

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, -1, 3, 5, 1, -1, -1, 10, 7, 1, 0, -6, 7, 21, 9, 1, 1, -6, -10, 31, 36, 11, 1, 1, 1, -29, 7, 79, 55, 13, 1, 0, 9, -24, -63, 81, 159, 78, 15, 1, -1, 9, 15, -123, -54, 264, 279, 105, 17, 1, -1, -1, 57, -69, -321, 132, 624, 447, 136, 19, 1, 0, -12, 50, 126, -459, -507, 741, 1245, 671, 171, 21, 1, 1, -12, -20, 302, -81, -1419, -258, 2163, 2227, 959, 210, 23, 1

Offset: 1

Ilya Gutkovskiy, Apr 08 2016

The polynomials Q_n(x) have generating function G(x,t) = t/(1 - (x + 1)*t - (x - 1)*t^2) = t + (x + 1)*t^2 + x*(x + 3)*t^3 + (x^3 + 5*x^2 + 3*x - 1)*t^4 + ...
Q_n(x) can be defined by the recurrence relation Q_n(x) = (x + 1)*Q_(n-1)(x) + (x - 1)*Q_(n-2)(x), Q_0(x)=0, Q_1(x)=1.
Discriminants of Q_n(x) gives the sequence: 0, 1, 1, 9, 320, 35600, 10948608, 8664190976, 16836271800320, 77757312009240576, 833309554769920000000, 20346889104219547132493824,...
Q_n(0)  = A128834(n).
Q_n(1)  = A000079(n-1), n>0.
Q_n(2)  = A006190(n).
Q_n(3)  = A090017(n).
Q_n(4)  = A015536(n).
Q_n(5)  = A135032(n).
Q_n(6)  = A015562(n).
Q_n(7)  = A190560(n).
Q_n(8)  = A015583(n).
Q_n(9)  = A190957(n).
Q_n(10) = A015603(n).

			Triangle begins:
   1;
   1,  1;
   0,  3,  1;
  -1,  3,  5,  1;
  -1, -1, 10,  7,  1;
...
The first few polynomials are:
Q_0(x) = 0;
Q_1(x) = 1;
Q_2(x) = x + 1;
Q_3(x) = x^2 + 3*x;
Q_4(x) = x^3 + 5*x^2 + 3*x - 1;
Q_5(x) = x^4 + 7*x^3 + 10*x^2 - x - 1,
...

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Ilya Gutkovskiy, Polynomials Q_n(x)
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A049310.

Flatten[Table[CoefficientList[((x + Sqrt[x (x + 6) - 3] + 1)^n - (x - Sqrt[x (x + 6) - 3] + 1)^n)/2^n/Sqrt[x (x + 6) - 3], x], {n, 0, 13}]]

A108012 a(n)= 8a(n-1) -16a(n-2) +4*a(n-4).

Original entry on oeis.org

0, 4, 18, 76, 320, 1360, 5832, 25200, 109568, 478784, 2100512, 9244352, 40784896, 180284672, 798121088, 3537391360, 15692333056, 69661541376, 309407486464, 1374824795136, 6110847909888, 27168232722432, 120809925167104

Offset: 0

Roger L. Bagula, May 30 2005

Index entries for linear recurrences with constant coefficients, signature (8,-16,0,4)

M = {{0, 0, 0, 2}, {1, 4, 0, 0}, {0, 2, 0, 0}, {0, 0, 1, 4}} v[1] = {0, 1, 1, 2}; v[n_] := v[n] = M.v[n - 1]; digits = 50; a = Table[v[n][[1]], {n, 1, digits}]
LinearRecurrence[{8,-16,0,4},{0,4,18,76},30] (* Harvey P. Dale, Aug 11 2017 *)

G.f.: 2*x*(-2+7*x+2*x^2)/( (2*x^2+4*x-1) * (2*x^2-4*x+1)). [Sep 28 2009]

a(n) = A111567(n-1) + A090017(n+1) - A090017(n) - A090017(n-1), n>0.

Definition replaced by recurrence by the Associate Editors of the OEIS, Sep 28 2009

A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.

Original entry on oeis.org

3, 5, 26, 114, 508, 2260, 10056, 44744, 199088, 885840, 3941536, 17537824, 78034368, 347213120, 1544921216, 6874111104, 30586286848, 136093369600, 605546052096, 2694370947584, 11988575894528, 53343045473280, 237349333682176, 1056083425675264

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m, is (k - (k*r - m)*x)/(1 - r*x - s*x^2). This recurrence gives the closed form b(n) = (2^(-n - 1)*((k*r - 2*m)*(r - sqrt(r^2 + 4*s))^n + (2*m - k*r)*(sqrt(r^2 + 4*s) + r)^n + k*sqrt(r^2 + 4*s)*(r - sqrt(r^2 + 4*s))^n + k*sqrt(r^2 + 4*s)*(sqrt(r^2 + 4*s) + r)^n))/sqrt(r^2 + 4*s).

Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A021001, A084059, A090017, A107979, A164549, A176213, A189741.

[n le 2 select 2*n+1 else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == 4 a[n - 1] + 2 a[n - 2]}, a, {n, 23}]
LinearRecurrence[{4, 2}, {3, 5}, 24]
Table[((18 + Sqrt[6]) (2 - Sqrt[6])^n - (Sqrt[6] - 18) (2 + Sqrt[6])^n)/12, {n, 0, 23}]

Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

G.f.: (3 - 7*x)/(1 - 4*x - 2*x^2).
a(n) = ((18 + sqrt(6))*(2 - sqrt(6))^n - (sqrt(6) - 18)*(2 + sqrt(6))^n)/12.
Lim_{n -> infinity} a(n + 1)/a(n) = 2 + sqrt(6) = A176213.
a(n) = 3*A090017(n+1) -7*A090017(n). - R. J. Mathar, Mar 12 2017

cp's OEIS Frontend

A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

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A218984 Power floor sequence of 2+sqrt(6).

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A218985 Power ceiling sequence of 2+sqrt(6).

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A335749 a(n) = n!*[x^n] exp(2*x)*(y*sinh(x*y) + cosh(x*y)) and y = sqrt(6).

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A123348 Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

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A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)*((x + sqrt(x*(x + 6) - 3) + 1)^n - (x - sqrt(x*(x + 6) - 3) + 1)^n)/sqrt(x*(x + 6) - 3).

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A108012 a(n)= 8*a(n-1) -16*a(n-2) +4*a(n-4).

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A268409 a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5.

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A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A335749 a(n) = n![x^n] exp(2x)(ysinh(xy) + cosh(xy)) and y = sqrt(6).

A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)((x + sqrt(x(x + 6) - 3) + 1)^n - (x - sqrt(x(x + 6) - 3) + 1)^n)/sqrt(x(x + 6) - 3).

A108012 a(n)= 8a(n-1) -16a(n-2) +4*a(n-4).

A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.