A001076 -id:A001076 - cp's OEIS frontend

A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 1 - x - x^2.

1, 1, 5, 2, 9, 24, 3, 23, 63, 115, 5, 45, 191, 397, 551, 8, 90, 453, 1381, 2358, 2640, 13, 170, 1044, 3807, 9226, 13482, 12649, 21, 317, 2249, 9865, 28785, 58513, 75061, 60605, 34, 579, 4695, 23703, 82485, 202887, 357567, 409779, 290376, 55, 1045, 9501

Offset: 1

Clark Kimberling, Nov 13 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
 1
 1    5
 2    9    24
 3   23    63   115
 5   45   191   397    551
 8   90   453  1381   2358   2640
13  170  1044  3807   9226  13482  12649
21  317  2249  9865  28785  58513  75061  60605
Row 4 represents the polynomial p(4,x) = 3 + 23 x + 63 x^2 + 115 x^3, so  that (T(4,k)) = (3,23,63,115), k-0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000045 (column 1), A004254 (T(n,n-1)), A001109 (row sums p(n,1)), A001076 (alternating row sums, p(n,-1)), A094440, A367208, A367209, A367211, A367297, A367298, A367299, A367300.

p[1, x_] := 1; p[2, x_] := 1 + 5 x; u[x_] := p[2, x]; v[x_] := 1 - x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where p(1,x) = 1, p(2,x) = 1 + 5x, u = p(2,x), and v = 1 - x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/D), b = 1/2 (1 + 5 x - D), c = 1/2 (1 + 5 x + D), where D = sqrt(5 + 6 x + 21 x^2).

A014437 Odd Fibonacci numbers.

Original entry on oeis.org

1, 1, 3, 5, 13, 21, 55, 89, 233, 377, 987, 1597, 4181, 6765, 17711, 28657, 75025, 121393, 317811, 514229, 1346269, 2178309, 5702887, 9227465, 24157817, 39088169, 102334155, 165580141, 433494437, 701408733, 1836311903, 2971215073, 7778742049, 12586269025

Offset: 0

Mohammad K. Azarian

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).

Cf. A001651, A059878, A000045.

Cf. A360957 (sum of reciprocals).

[Fibonacci((3*Floor((n+1)/2)) + (-1)^n): n in [0..50]]; // Vincenzo Librandi, Apr 18 2011

with(combinat):A014437:=proc(n)return fibonacci((3*floor((n+1)/2)) + (-1)^n):end:
seq(A014437(n),n=0..31); # Nathaniel Johnston, Apr 18 2011
# second Maple program:
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|0|4|0>>^n.<<1,1,3,5>>)[1,1]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 22 2025

RecurrenceTable[{a[n] == 4*a[n-2] + a[n-4], a[0]==1, a[1]==1, a[2]==3, a[3]==5},a,{n,0,500}] (* G. C. Greubel, Oct 30 2015 *)
Table[ SeriesCoefficient[(-1 - x + x^2 - x^3)/(-1 + 4*x^2 + x^4), {x, 0, n}], {n, 0, 20}] (* Nikolaos Pantelidis, Feb 01 2023 *)
Select[Fibonacci[Range[50]],OddQ] (* Harvey P. Dale, Sep 01 2023 *)

Vec((-1-x+x^2-x^3)/(-1+4*x^2+x^4) + O(x^200)) \\ Altug Alkan, Oct 31 2015

apply( A014437(n)=fibonacci(n\/2*3+(-1)^n), [0..30]) \\ M. F. Hasler, Nov 18 2018

Fibonacci(3n+1) union Fibonacci(3n+2).
a(n) = Fibonacci(3*floor((n+1)/2) + (-1)^n). - Antti Karttunen, Feb 05 2001
G.f.: ( -1-x+x^2-x^3 ) / ( -1+4*x^2+x^4 ). - R. J. Mathar, Feb 16 2011
a(2n) = v-w, a(2n+1) = v+w, with v=A001076(n+1), w=A001076(n). Therefore, a(2n)+a(2n+1) = 2*A001076(n+1). - Ralf Stephan, Aug 31 2013
From Vladimir Reshetnikov, Oct 30 2015: (Start)
a(n) = ((cos(Pi*n/2)-sqrt(phi)*sin(Pi*n/2))/phi^((3*n+2)/2) + (sqrt(phi)*cos(Pi*n/2)^2+sin(Pi*n/2)^2)*phi^((3*n+1)/2))/sqrt(5), where phi=(1+sqrt(5))/2.
E.g.f.: (cos(x/phi^(3/2))/phi - sin(x/phi^(3/2))/sqrt(phi) + cosh(x*phi^(3/2))*phi + sinh(x*phi^(3/2))*sqrt(phi))/sqrt(5).
(End)

a(30)-a(31) from Vincenzo Librandi, Apr 18 2011

A085449 Horadam sequence (0,1,4,2).

Original entry on oeis.org

0, 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136, 294912, 954368, 3088384, 9994240, 32342016, 104660992, 338690048, 1096024064, 3546808320, 11477712896, 37142659072, 120196169728, 388962975744, 1258710630400

Offset: 0

Ross La Haye, Aug 18 2003

a(n) / a(n-1) converges to sqrt(5) + 1 as n approaches infinity. sqrt(5) + 1 can also be written as Phi^3 - 1, 2 * Phi, Phi^2 + Phi - 1 and (L(n) / F(n)) + 1, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity.

Binomial transform is A001076. - Paul Barry, Aug 25 2003

			a(4) = 24 because a(3) = 8, a(2) = 2, s = 2, r = 4 and (2 * 8) + (4 * 2) = 24.
G.f. = x + 2*x^2 + 8*x^3 + 24*x^4 + 80*x^5 + 256*x^6 + 832*x^7 + ... - _Michael Somos_, Mar 07 2021

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Eric Weisstein, Horadam Sequence.
Eric Weisstein, Fibonacci Number.
Eric Weisstein, Pell Number.
Eric Weisstein, Lucas Number.
Eric Weisstein, Lucas Sequence.
Index entries for linear recurrences with constant coefficients, signature (2, 4).

Cf. A024318, A000032, A000129, A001076, A085939, A269991.

Essentially the same as A063727.

a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]+4*a[n-2]; od; a; # Muniru A Asiru, Oct 09 2018

[2^(n-1)*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Oct 08 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*(a[n-1]+2*a[n-2]) od: seq(a[n], n=0..26); # Zerinvary Lajos, Mar 17 2008

Table[2^(n-1)*Fibonacci[n], {n,0,50}] (* G. C. Greubel, Oct 08 2018 *)

vector(50, n, n--; 2^(n-1)*fibonacci(n)) \\ G. C. Greubel, Oct 08 2018

a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 2, r = 4.
From Paul Barry, Aug 25 2003: (Start)
G.f.: x/(1-2*x-4*x^2).
a(n) = sqrt(5)*((1+sqrt(5))^n - (1-sqrt(5))^n)/10.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)5^k . (End)
The signed version 0, 1, -2, ... has a(n)=sqrt(5)((sqrt(5)-1)^n-(-sqrt(5)-1)^n)/10. It is the second inverse binomial transform of A085449. - Paul Barry, Aug 25 2003
a(n) = 2^(n-1)*Fib(n). - Paul Barry, Mar 22 2004
Sum_{n>=1} 1/a(n) = A269991. - Amiram Eldar, Feb 01 2021
a(n) = -(-4)^n*a(-n) for all integer n. - Michael Somos, Mar 07 2021

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

Original entry on oeis.org

1, 2, 3, 5, 10, 8, 12, 34, 38, 21, 29, 104, 161, 130, 55, 70, 305, 592, 654, 420, 144, 169, 866, 2023, 2788, 2436, 1308, 377, 408, 2404, 6556, 10810, 11756, 8574, 3970, 987, 985, 6560, 20446, 39164, 50779, 46064, 28987, 11822, 2584, 2378, 17663, 61912, 134960, 202630, 218717, 171232, 95078, 34690, 6765

Offset: 1

Clark Kimberling, Nov 26 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
    1
    2    3
    5   10    8
   12   34   38    21
   29  104  161   130    55
   70  305  592   654   420  144
  169  866 2023  2788  2436 1308  377
  408 2404 6556 10810 11756 8574 3970 987
Row 4 represents the polynomial p(4,x) = 12 + 34*x + 38*x^2 + 21*x^3, so (T(4,k)) = (12,34,38,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000129 (column 1), A001906 (p(n,n-1)), A107839 (row sums, p(n,1)), A077925 (alternating row sums, p(n,-1)), A023000 (p(n,2)), A001076 (p(n,-2)), A186446 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367298, A367299, A367300, A367301.

p[1, x_] := 1; p[2, x_] := 2 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 2 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 4*x + 5*x^2)), b = (1/2)*(3*x + 2 + 1/k), c = (1/2)*(3*x + 2 - 1/k).

A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.

Original entry on oeis.org

1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=6.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=6.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,6).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.

I:=[1,6]; [n le 2 select I[n] else 6*Self(n-1)+6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

Join[{a=0,b=1},Table[c=6*b+6*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,6},{1,6},40] (* Harvey P. Dale, Nov 05 2011 *)

x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,6,-6) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 6*a(n-2); a(0)=1, a(1)=6.
a(n) = S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-6*x-6*x^2).
a(n) = Sum_{k=0..n} 5^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6a(n-1)-4a(n-2).

Original entry on oeis.org

0, 1, 6, 32, 168, 880, 4608, 24128, 126336, 661504, 3463680, 18136064, 94961664, 497225728, 2603507712, 13632143360, 71378829312, 373744402432, 1956951097344, 10246728974336, 53652569456640, 280928500842496, 1470960727228416, 7702050360000512, 40328459251089408

Offset: 0

Benoit Cloitre, Jun 21 2003

Binomial transform of A001076. - Paul Barry, Aug 25 2003

The ratio a(n+1)/(a(n+1)-4*a(n)) converges to 2 + sqrt(5). - Karl V. Keller, Jr., May 17 2015

			a(5) = 6 * a(4) - 4 * a(3) = 6*168 - 4*32 = 880.

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
Yuhan Jiang, The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model, arXiv:2312.09427 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A030191.

[n le 2 select (n-1) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // Vincenzo Librandi, May 15 2015

Join[{a = 0, b = 1}, Table[c = 6 * b - 4 * a; a = b; b = c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
LinearRecurrence[{6, -4}, {0, 1}, 30] (* Vincenzo Librandi, May 15 2015 *)

a(n)=(1/2)*sum(k=0,n,binomial(n,k)*fibonacci(3*k))

a(n)={2^(n-1)*fibonacci(2*n)} \\ Andrew Howroyd, Oct 27 2020

[lucas_number1(n,6,4) for n in range(0, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = (1/2)*sum(k = 0, n, binomial(n, k)*F(3*k)) where F(k) denotes the k-th Fibonacci number.
a(n) = sqrt(5)((3+sqrt(5))^n - (3-sqrt(5))^n)/10. - Paul Barry, Aug 25 2003
a(n) = Sum(C(n, 2k+1)5^k 3^(n-2k-1), k = 0, .., Floor[(n-1)/2]). a(n) = 2^(n-1)F(2n). - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
a(n) is the rightmost term in M^n * [1 0] where M is the 2X2 matrix [5 1 / 1 1]. The characteristic polynomial of M = x^2 - 6x + 4. a(n)/a(n-1) tends to (3 + sqrt(5)), a root of the polynomial and an eigenvalue of M. - Gary W. Adamson, Dec 16 2004
a(n) = sum{k = 0..n, sum{j = 0..n, C(n, j)C(j, k)F(j+k)/2}}. - Paul Barry, Feb 14 2005
G.f.: x/(1 - 6x + 4x^2). - R. J. Mathar, Sep 09 2008
If p[i] = (4^i-1)/3, and if A is the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det A. - Milan Janjic, May 08 2010
a(n) = 5a(n - 1) + a(n - 2) + a(n - 3) + ... + a(1) + 1. - Gary W. Adamson, Feb 18 2011
a(n) = 2^(n-1)*A001906(n). - R. J. Mathar, Apr 03 2011

A084844 Denominators of the continued fraction n + 1/(n + 1/...) [n times].

Original entry on oeis.org

1, 2, 10, 72, 701, 8658, 129949, 2298912, 46866034, 1082120050, 27916772489, 795910114440, 24851643870041, 843458630403298, 30918112619119426, 1217359297034666112, 51240457936070359069, 2296067756927144738850, 109127748348241605689981

Offset: 1

Hollie L. Buchanan II, Jun 08 2003

The (n-1)-th term of the Lucas sequence U(n,-1). The numerator is the n-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - T. D. Noe, Aug 19 2004
From Flávio V. Fernandes, Mar 05 2021: (Start)
Also, the n-th term of the n-th metallic sequence (the diagonal through the array A073133, and its equivalents, which is rows formed by sequences beginning with A000045, A000129, A006190, A001076, A052918) as shown below (for n>=1):
   0    1    0    1     0      1  ... A000035
   0   [1]   1    2     3      5  ... A000045
   0    1   [2]   5    12     29  ... A000129
   0    1    3  [10]   33    109  ... A006190
   0    1    4   17   [72]   305  ... A001076
   0    1    5   26   135   [701] ... A052918. (End)

			a(4) = 72 since 4 + 1/(4 + 1/(4 + 1/4)) = 305/72.

Alois P. Heinz, Table of n, a(n) for n = 1..387
Eric Weisstein's World of Mathematics, Lucas Sequence

Cf. A084845 (numerators).

Cf. A000045, A097690, A097691, A117715, A290864 (primes in this sequence).

A084844 :=proc(n) combinat[fibonacci](n, n) end:
seq(A084844(n), n=1..30); # Zerinvary Lajos, Jan 03 2007

myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Denominator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Denominator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)
Table[Fibonacci[n, n], {n, 1, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
Table[DifferenceRoot[Function[{y,m},{y[2+m]==n*y[1+m]+y[m],y[0]==0,y[1]==1}]][n],{n,1,20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

from sympy import fibonacci
def a(n):
    return fibonacci(n, n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 12 2017

a(n) = (s^n - (-s)^(-n))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016
a(n) = y(n,n), where y(m+2,n) = n*y(m+1,n) + y(m,n), with y(0,n)=0, y(1,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 03 2017
a(n) = A117715(n,n). - Bobby Jacobs, Aug 12 2017
a(n) = [x^n] x/(1 - n*x - x^2). - Ilya Gutkovskiy, Oct 10 2017
a(n) == 0 (mod n) for even n and 1 (mod n) for odd n. - Flávio V. Fernandes, Dec 08 2020
a(n) == 0 (mod n) for even n and 1 (mod n^2) for odd n; see A065599. - Flávio V. Fernandes, Dec 25 2020
a(n) == 0 (mod 2*(n/2)^2) for even n and 1 (mod n^2) for odd n; see A129194. - Flávio V. Fernandes, Feb 06 2021

A243399 a(0) = 1, a(1) = 19; for n > 1, a(n) = 19*a(n-1) + a(n-2).

Original entry on oeis.org

1, 19, 362, 6897, 131405, 2503592, 47699653, 908796999, 17314842634, 329890807045, 6285240176489, 119749454160336, 2281524869222873, 43468721969394923, 828187242287726410, 15779026325436196713, 300629687425575463957, 5727743087411370011896

Offset: 0

Bruno Berselli, Jun 04 2014

a(n+1)/a(n) tends to (19 + sqrt(365))/2.
a(n) equals the number of words of length n on alphabet {0,1,...,19} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, May 03 2023: (Start)
Also called the 19-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 19 kinds of squares available. (End)

Bruno Berselli, Table of n, a(n) for n = 0..200
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Index entries for linear recurrences with constant coefficients, signature (19,1).

Row n=19 of A073133, A172236 and A352361 and column k=19 of A157103.

Sequences with g.f. 1/(1-k*x-x^2) or x/(1-k*x-x^2): A000045 (k=1), A000129 (k=2), A006190 (k=3), A001076 (k=4), A052918 (k=5), A005668 (k=6), A054413 (k=7), A041025 (k=8), A099371 (k=9), A041041 (k=10), A049666 (k=11), A041061 (k=12), A140455 (k=13), A041085 (k=14), A154597 (k=15), A041113 (k=16), A178765 (k=17), A041145 (k=18), this sequence (k=19), A041181 (k=20). Also, many other sequences are in the OEIS with even k greater than 20 (denominators of continued fraction convergents to sqrt((k/2)^2+1)).

[n le 2 select 19^(n-1) else 19*Self(n-1)+Self(n-2): n in [1..20]];

RecurrenceTable[{a[n] == 19 a[n - 1] + a[n - 2], a[0] == 1, a[1] == 19}, a, {n, 0, 20}]

a[0]:1$ a[1]:19$ a[n]:=19*a[n-1]+a[n-2]$ makelist(a[n], n, 0, 20);

v=vector(20); v[1]=1; v[2]=19; for(i=3, #v, v[i]=19*v[i-1]+v[i-2]); v

from sage.combinat.sloane_functions import recur_gen2
a = recur_gen2(1,19,19,1)
[next(a) for i in (0..20)]

G.f.: 1/(1 - 19*x - x^2).
a(n) = (-1)^n*a(-n-2) = ((19 + sqrt(365))^(n+1)-(19 - sqrt(365))^(n+1))/(2^(n+1)*sqrt(365)).
a(n) = F(n+1, 19), the (n+1)-th Fibonacci polynomial evaluated at x = 19.
a(n)*a(n-2) - a(n-1)^2 = (-1)^n, with a(-2)=1, a(-1)=0.

A015537 Expansion of x/(1 - 5x - 4x^2).

Original entry on oeis.org

0, 1, 5, 29, 165, 941, 5365, 30589, 174405, 994381, 5669525, 32325149, 184303845, 1050819821, 5991314485, 34159851709, 194764516485, 1110461989261, 6331368012245, 36098688018269, 205818912140325, 1173489312774701, 6690722212434805

Offset: 0

Olivier Gérard

First differences give A122690(n) = {1, 4, 24, 136, 776, 4424, 25224, ...}. Partial sums of a(n) are {0, 1, 6, 35, 200, ...} = (A123270(n) - 1)/8. - Alexander Adamchuk, Nov 03 2006
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 5's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011
Pisano period lengths:  1, 1, 8, 1, 4, 8, 48, 1, 24, 4, 40, 8, 42, 48, 8, 2, 72, 24, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Index entries for linear recurrences with constant coefficients, signature (5,4).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A122690, A123270, A180222, A180226.

a:=[0,1];; for n in [3..30] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

[n le 2 select n-1 else 5*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012

seq( simplify((2/I)^(n-1)*ChebyshevU(n-1, 5*I/4)), n=0..20); # G. C. Greubel, Dec 26 2019

LinearRecurrence[{5,4}, {0,1}, 30] (* Vincenzo Librandi, Nov 12 2012 *)
Table[2^(n-1)*Fibonacci[n, 5/2], {n, 0, 30}] (* G. C. Greubel, Dec 26 2019 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-4*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-4) for n in range(0, 22)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 4*a(n-2).
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*4^k*5^(n-2*k-1). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..(n-1)} A122690(k). - Alexander Adamchuk, Nov 03 2006
a(n) = 2^(n-1)*Fibonacci(n, 5/2) = (2/i)^(n-1)*ChebyshevU(n-1, 5*i/4). - G. C. Greubel, Dec 26 2019

A048876 a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 29, 123, 521, 2207, 9349, 39603, 167761, 710647, 3010349, 12752043, 54018521, 228826127, 969323029, 4106118243, 17393796001, 73681302247, 312119004989, 1322157322203, 5600748293801, 23725150497407, 100501350283429, 425730551631123, 1803423556807921

Offset: 0

Barry E. Williams

Generalized Pell equation with second term of 7.

T. D. Noe, Table of n, a(n) for n = 0..200
M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.
Tanya Khovanova, Recursive Sequences
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A001076, A001077, A015448, A033887.

with(combinat): a:=n->3*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(2 + s)^n + (1 - s)(2 - s)^n)/2]]; (* Or *)
f[n_] := Fibonacci[3 n + 3] - Fibonacci[3 n - 1]; (* Or *)
f[n_] := Mod[ Fibonacci[3n + 7], Fibonacci[3n + 3]]; Array[f, 22, 0]
a[n_] := 4a[n - 1] + a[n - 2]; a[0] = 1; a[1] = 7; Array[a, 22, 0] (* Or *)
CoefficientList[ Series[(1 + 3x)/(1 - 4x - x^2), {x, 0, 21}], x] (* Robert G. Wilson v *)
LinearRecurrence[{4,1},{1,7},30] (* Harvey P. Dale, Jun 13 2015 *)
Table[LucasL[3*n + 1], {n, 0, 20}] (* Rigoberto Florez, Apr 04 2019 *)

Vec((1+3*x)/(1-4*x-x^2) + O(x^30)) \\ Altug Alkan, Oct 07 2015

G.f.: (1+3*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = ((1+sqrt(5))*(2+sqrt(5))^n + (1-sqrt(5))*(2-sqrt(5))^n )/2.
a(n) = A000032(3*n+1). - Thomas Baruchel, Nov 26 2003
From Gary Detlefs, Mar 06 2011: (Start)
a(n) = Fibonacci(3*n+7) mod Fibonacci(3*n+3), n > 0.
a(n) = Fibonacci(3*n+3) - Fibonacci(3*n-1). (End)
a(n) = A001076(n+1)+3*A001076(n). - R. J. Mathar, Oct 22 2013
a(n) = 5*F(2*n)*F(n+1) - L(n-1)*(-1)^n. - J. M. Bergot, Mar 22 2016
a(n) = Sum_{k=0..n} binomial(n,k)*5^floor((k+1)/2)*2^(n-k). - Tony Foster III, Sep 03 2017

cp's OEIS Frontend

A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 1 - x - x^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A014437 Odd Fibonacci numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A085449 Horadam sequence (0,1,4,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6*a(n-1)-4*a(n-2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 1 - x - x^2.

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

A084326 a(0)=0, a(1)=1; for n>1, a(n) = 6a(n-1)-4a(n-2).

A015537 Expansion of x/(1 - 5x - 4x^2).