A097924 -id:A097924 - cp's OEIS frontend

A001077 Numerators of continued fraction convergents to sqrt(5).

1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, 3940598, 16692641, 70711162, 299537289, 1268860318, 5374978561, 22768774562, 96450076809, 408569081798, 1730726404001, 7331474697802, 31056625195209

Offset: 0

N. J. A. Sloane

a(2*n+1) with b(2*n+1) := A001076(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = -1.
a(2*n) with b(2*n) := A001076(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = +1 (see Emerson reference).
Bisection: a(2*n) = T(n,9) = A023039(n), n >= 0 and a(2*n+1) = 2*S(2*n, 2*sqrt(5)) = A075796(n+1), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.
From Greg Dresden, May 21 2023: (Start)
For n >= 2, 8*a(n) is the number of ways to tile this T-shaped figure of length n-1 with four colors of squares and one color of domino; shown here is the figure of length 5 (corresponding to n=6), and it has 8*a(6) = 23112 different tilings.
    _
   || _  
   |||_|||
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(End)

			1  2  9  38  161  (A001077)
-, -, -, --, ---, ...
0  1  4  17   72  (A001076)
1 + 2*x + 9*x^2 + 38*x^3 + 161*x^4 + 682*x^5 + 2889*x^6 + 12238*x^7 + ... - _Michael Somos_, Aug 11 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Thébault, Les Récréations Mathématiques, Gauthier-Villars, Paris, 1952, p. 282.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Ex. 1, pp. 237-238.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A001076, A023039, A049629, A052924, A078343, A164581, A179237, A180148, A329723, A374439.

I:=[1, 2]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 19 2017

A001077:=(-1+2*z)/(-1+4*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat): a:=n->fibonacci(n+1, 4)-2*fibonacci(n, 4): seq(a(n), n=0..30); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4, 1}, {1, 2}, 30]
Join[{1},Numerator[Convergents[Sqrt[5],30]]] (* Harvey P. Dale, Mar 23 2016 *)
CoefficientList[Series[(1-2*x)/(1-4*x-x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 19 2017 *)
LucasL[3*Range[0,30]]/2 (* Rigoberto Florez, Apr 03 2019 *)
a[ n_] := LucasL[n, 4]/2; (* Michael Somos, Nov 02 2021 *)

{a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)}; /* Michael Somos, Aug 11 2009 */

a(n)=if(n<2,n+1,my(t=4);for(i=1,n-2,t=4+1/t);numerator(2+1/t)) \\ Charles R Greathouse IV, Dec 05 2011

x='x+O('x^30); Vec((1-2*x)/(1-4*x-x^2)) \\ G. C. Greubel, Dec 19 2017

[lucas_number2(n,4,-1)/2 for n in range(0, 30)] # Zerinvary Lajos, May 14 2009

G.f.: (1-2*x)/(1-4*x-x^2).
a(n) = 4*a(n-1) + a(n-2), a(0)=1, a(1)=2.
a(n) = ((2 + sqrt(5))^n + (2 - sqrt(5))^n)/2.
a(n) = A014448(n)/2.
Limit_{n->infinity} a(n)/a(n-1) = phi^3 = 2 + sqrt(5). - Gregory V. Richardson, Oct 13 2002
a(n) = ((-i)^n)*T(n, 2*i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.
Binomial transform of A084057. - Paul Barry, May 10 2003
E.g.f.: exp(2x)cosh(sqrt(5)x). - Paul Barry, May 10 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*5^k*2^(n-2k). - Paul Barry, Nov 15 2003
a(n) = 4*a(n-1) + a(n-2) when n > 2; a(1) = 1, a(2) = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 25 2004
a(n) = A001076(n+1) - 2*A001076(n) = A097924(n) - A015448(n+1); a(n+1) = A097924(n) + 2*A001076(n) = A097924(n) + 2(A048876(n) - A048875(n)). - Creighton Dement, Mar 19 2005
a(n) = F(3*n)/2 + F(3*n-1) where F() = Fibonacci numbers A000045. - Gerald McGarvey, Apr 28 2007
a(n) = A000032(3*n)/2.
For n >= 1: a(n) = (1/2)*Fibonacci(6*n)/Fibonacci(3*n) and a(n) = integer part of (2 + sqrt(5))^n. - Artur Jasinski, Nov 28 2011
a(n) = Sum_{k=0..n} A201730(n,k)*4^k. - Philippe Deléham, Dec 06 2011
a(n) = A001076(n) + A015448(n). - R. J. Mathar, Jul 06 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) is the (1,1)-entry of the matrix W^n with W=[2, sqrt(5); sqrt(5), 2]. - Carmine Suriano, Mar 21 2014
From Rigoberto Florez, Apr 03 2019: (Start)
a(n) = A099919(n) + A049651(n) if n > 0.
a(n) = 1 + Sum_{k=0..n-1} L(3*k + 1) if n >= 0, L(n) = n-th Lucas number (A000032). (End)
From Christopher Hohl, Aug 22 2021: (Start)
For n >= 2, a(2n-1) = A079962(6n-9) + A079962(6n-3).
For n >= 1, a(2n) = sqrt(20*A079962(6n-3)^2 + 1). (End)
a(n) = Sum_{k=0..n-2} A168561(n-2,k)*4^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*4^k, n>0. - R. J. Mathar, Feb 14 2024
a(n) = 4^n*Sum_{k=0..n} A374439(n, k)*(-1/4)^k. - Peter Luschny, Jul 26 2024
From Peter Bala, Jul 08 2025: (Start)
The following series telescope:
Sum_{n >= 1} 1/(a(n) + 5*(-1)^(n+1)/a(n)) = 3/8, since 1/(a(n) + 5*(-1)^(n+1)/a(n)) = b(n) - b(n+1), where b(n) = (1/4) * (a(n) + a(n-1)) / (a(n)*a(n-1)).
Sum_{n >= 1} (-1)^(n+1)/(a(n) + 5*(-1)^(n+1)/a(n)) = 1/8, since 1/(a(n) + 5*(-1)^(n+1)/a(n)) = c(n) + c(n+1), where c(n) = (1/4) * (a(n) - a(n-1)) / (a(n)*a(n-1)). (End)

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

A048875 Generalized Pellian with second term of 6.

Original entry on oeis.org

1, 6, 25, 106, 449, 1902, 8057, 34130, 144577, 612438, 2594329, 10989754, 46553345, 197203134, 835365881, 3538666658, 14990032513, 63498796710, 268985219353, 1139439674122, 4826743915841, 20446415337486, 86612405265785, 366896036400626, 1554196550868289

Offset: 0

Barry E. Williams

			G.f. = 1 + 6*x + 25*x^2 + 106*x^3 + 449*x^4 + 1902*x^5 + 8057*x^6 + 34130*x^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. A015448, A001076, A001077, A033887, A134418, A097924, A374439.

with(combinat): a:=n->2*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..17); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4,1},{1,6},40] (* Harvey P. Dale, Nov 30 2011 *)
a[ n_] := (4 I ChebyshevT[ n + 1, -2 I] - 3 ChebyshevT[ n, -2 I]) I^n / 5; (* Michael Somos, Feb 23 2014 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ (1 + 6 x) / (1 + 4 x - x^2), {x, 0, -n}], SeriesCoefficient[ (1 + 2 x) / (1 - 4 x - x^2), {x, 0, n}]]; (* Michael Somos, Feb 23 2014 *)

a[0]:1$ a[1]:6$ a[n]:=4*a[n-1]+a[n-2]$ makelist(a[n], n, 0, 30); /* Martin Ettl, Nov 03 2012 */

{a(n) = ( 4*I*polchebyshev( n+1, 1, -2*I) - 3*polchebyshev( n, 1, -2*I)) * I^n / 5}; /* Michael Somos, Feb 23 2014 */

{a(n) = if( n<0, polcoeff( (1 + 6*x) / (1 + 4*x - x^2) + x * O(x^-n), -n), polcoeff( (1 + 2*x) / (1 - 4*x - x^2) + x * O(x^n), n))}; \\ Michael Somos, Feb 23 2014

a(n) = ((4+sqrt(5))*(2+sqrt(5))^n - (4-sqrt(5))*(2-sqrt(5))^n)*sqrt(5)/2.
a(n) = 4a(n-1) + a(n-2); a(0)=1, a(1)=6.
Binomial transform of A134418: (1, 5, 14, 48, 152, ...). - Gary W. Adamson, Nov 23 2007
G.f.: (1+2*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(-1 - n) = (-1)^n * A097924(n) for all n in Z. - Michael Somos, Feb 23 2014
a(n) = A001076(n+1) + 2*A001076(n). - R. J. Mathar, Sep 11 2019
a(n) = 4^n*Sum_{k=0..n} A374439(n, k)*(1/4)^k. - Peter Luschny, Jul 26 2024

Corrected by T. D. Noe, Nov 07 2006

A115032 Expansion of (5-14x+x^2)/((1-x)(x^2-18*x+1)).

Original entry on oeis.org

5, 81, 1445, 25921, 465125, 8346321, 149768645, 2687489281, 48225038405, 865363202001, 15528312597605, 278644263554881, 5000068431390245, 89722587501469521, 1610006506595061125, 28890394531209630721, 518417095055178291845, 9302617316461999622481

Offset: 0

Creighton Dement, Feb 26 2006

Relates squares of numerators and denominators of continued fraction convergents to sqrt(5).
Sequence is generated by the floretion A*B*C with A = + 'i - 'k + i' - k' - 'jj' - 'ij' - 'ji' - 'jk' - 'kj' ; B = - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' ; C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' (apart from a factor (-1)^n)
Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B*C].
The sequence a(n-1), n >= 0, with a(-1) = 1, is also the curvature of circles inscribed in a special way in the larger segment of a circle of radius 5/4 (in some units) cut by a chord of length 2. For the smaller segment, the analogous curvature sequence is given in A240926. For more details see comments on A240926. See also the illustration in the present sequence, and the proof of the coincidence of the curvatures with a(n-1) in part I of the W. Lang link. - Kival Ngaokrajang, Aug 23 2014

			G.f. = 5 + 81*x + 1445*x^2 + 25921*x^3 + 465125*x^4 + 8346321*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..795
Creighton Dement, Floretions associated with A115032.
Wolfdieter Lang, A proof for the touching circle problem (part I).
Giovanni Lucca, Circle chains inside the arbelos and integer sequences, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A001076, A001077, A007805, A023039, A097924.

Cf. also A000045, A049660, A049310, A023039. - Wolfdieter Lang, Aug 22 2014

seq((9*combinat:-fibonacci(6*(n+1)) - combinat:-fibonacci(6*n) + 8)/16, n = 0 .. 20); # Robert Israel, Aug 25 2014

LinearRecurrence[{19,-19,1},{5,81,1445},30] (* Harvey P. Dale, Nov 14 2014 *)
CoefficientList[Series[(5 - 14*x + x^2)/((1 - x)*(x^2 - 18*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

Vec((5-14*x+x^2)/((1-x)*(x^2-18*x+1)) + O(x^20)) \\ Michel Marcus, Aug 23 2014

sqrt(a(2*n)) = sqrt(5)*A007805(n) = sqrt(5)*Fibonacci(6*n+3)/2 = sqrt(5)*A001076(2*n+1); sqrt(a(2*n+1)) = A023039(2*n+1) = A001077(2*n).
From Wolfdieter Lang, Aug 22 2014: (Start)
O.g.f.: (5-14*x+x^2)/((1-x)*(x^2-18*x+1)) (see the name).
a(n) = (9*F(6*(n+1)) - F(6*n) + 8)/16, n >= 0 with F(n) = A000045(n) (Fibonacci). From the partial fraction decomposition of the o.g.f.: (1/2)*((9 - x)/(1 - 18*x + x^2) + 1/(1 - x)). For F(6*n)/8 see A049660(n). a(-1) = 1 with F(-6) = -F(6) = -8.
a(n) = (9*S(n, 18) - S(n-1, 18) + 1)/2, with the Chebyshev S-polynomials (see A049310). From A049660.
a(n) = (A023039(n+1) + 1)/2.
(End)
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - Colin Barker, Aug 23 2014
From Wolfdieter Lang, Aug 24 2014: (Start)
a(n) = 18*a(n-1) - a(n-2) - 8, n >= 1, a(-1) = 1, a(0) = 5. See the Chebyshev S-polynomial formula above.
The o.g.f. for the sequence a(n-1) with a(-1) = 1, n >= 0, [1, 5,  81, 1445, ..] is (1-14*x+5*x^2)/((1-x)*(1-18*x+x^2)).
(See the Colin Barker formula from Aug 04 2014 in the history of A240926.) (End)

More terms from Michel Marcus, Aug 23 2014
Edited (comment by Kival Ngaokrajang rewritten, Chebyshev index link added) by Wolfdieter Lang, Aug 26 2014
Partially edited by Jon E. Schoenfield and N. J. A. Sloane, Mar 15 2024

A100545 Expansion of (7-2x) / (1-3x+x^2).

Original entry on oeis.org

7, 19, 50, 131, 343, 898, 2351, 6155, 16114, 42187, 110447, 289154, 757015, 1981891, 5188658, 13584083, 35563591, 93106690, 243756479, 638162747, 1670731762, 4374032539, 11451365855, 29980065026, 78488829223, 205486422643, 537970438706, 1408424893475, 3687304241719, 9653487831682, 25273159253327

Offset: 0

Creighton Dement, Dec 31 2004

A Floretion integer sequence relating to Fibonacci numbers.

Inverse binomial transform of A013655; inversion of A097924.

Colin Barker, Table of n, a(n) for n = 0..1000
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A000045, A001906, A005248, A013655, A097924.

List([0..30], n-> Fibonacci(2*n+4) +Lucas(1,-1,2*n+3)[2] ); # G. C. Greubel, Jan 17 2020

[Fibonacci(2*n+4) +Lucas(2*n+3): n in [0..30]]; // G. C. Greubel, Jan 17 2020

F := proc(n) combinat[fibonacci](n) ; end: A100545 := proc(n) 4*F(2*(n+1)) + F(2*n+1)+F(2*n+3) ; end: for n from 0 to 30 do printf("%d,",A100545(n)) ; od ; # R. J. Mathar, Oct 26 2006

Table[Fibonacci[2*(n+2)] + LucasL[2*n+3], {n,0,30}] (* G. C. Greubel, Jan 17 2020 *)

Vec((7-2*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 11 2015

[fibonacci(2*n+4) +lucas_number2(2*n+3,1,-1) for n in (0..30)] # G. C. Greubel, Jan 17 2020

a(n-1) = 4*Fibonacci(2*n) + Fibonacci(2*n-1) + Fibonacci(2*n+1).
a(n) + a(n+1) = A055849(n+2).
a(n) = 3*a(n-1) - a(n-2) with a(0)=7 and a(1)=19. - Philippe Deléham, Nov 16 2008
a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-17+7*sqrt(5)) + (3+sqrt(5))^n*(17+7*sqrt(5)))) / sqrt(5). - Colin Barker, Oct 14 2015
From G. C. Greubel, Jan 17 2020: (Start)
a(n) = Fibonacci(2*n+4) + Lucas(2*n+3).
E.g.f.: 2*exp(3*t/2)*(cosh(sqrt(5)*t/2) + (4/sqrt(5))*sinh(sqrt(5)*t/2)). (End)

Corrected and extended by T. D. Noe and R. J. Mathar, Oct 26 2006

A328695 Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.

Original entry on oeis.org

1, 4, 2, 17, 9, 3, 72, 38, 5, 12, 305, 161, 8, 51, 6, 1292, 682, 13, 216, 10, 7, 5473, 2889, 21, 915, 16, 30, 14, 23184, 12238, 34, 3876, 26, 127, 59, 25, 98209, 51841, 55, 16419, 42, 538, 250, 106, 11, 416020, 219602, 89, 69552, 68, 2279, 1059, 449, 18, 33

Offset: 1

Clark Kimberling, Oct 26 2019

Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.  The appearance of a sequence s(r) below means that corresponding row of R is the same as s(r) except possibly for one or more initial terms of s(r).
Row 1 of R: A001076
Row 2 of R: A001077
Row 3 of R: A000045
Row 4 of R: A115179
Row 5 of R: A006355
Row 6 of R: A097924
Row 8 of R: A048875
Row 9 of R: A000032

			Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,4,17,72,...).
_______________
Northwest corner of R:
   1   4   17   72  305  1292   5473
   2   9   38  161  682  2889  12238
   3   5    8   13   21    34     55
  12  51  216  915 3876 16419  69552
   6  10   16   26   42    68    110
   7  30  127  538 2279  9654  40895

Cf. A035513, A001076, A001077, A000045, A115179, A006355, A097924, A048875, A000032, A328696, A328697.

w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 0, w[n, k]/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
Table[u[n], {n, 1, 10}] (* A328695 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328695 sequence *)

A291660 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=2, a(1)=3, a(2)=5, a(3)=7, a sequence related to Lucas numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 18, 30, 49, 79, 127, 205, 332, 538, 871, 1409, 2279, 3687, 5966, 9654, 15621, 25275, 40895, 66169, 107064, 173234, 280299, 453533, 733831, 1187363, 1921194, 3108558, 5029753, 8138311, 13168063, 21306373, 34474436, 55780810, 90255247, 146036057, 236291303

Offset: 0

Jean-François Alcover and Paul Curtz, Aug 31 2017

nonn

The array of successive differences begins:
2,  3,  5,  7, 11, 18, 30, 49, 79, 127, ... = a(n)
1,  2,  2,  4,  7, 12, 19, 30, 48,  78, ... = b(n)
1,  0,  2,  3,  5,  7, 11, 18, 30,  49, ... = a(n-2)
-1, 2,  1,  2,  2,  4,  7, 12, 19,  30, ... = b(n-2)
3, -1,  1,  0,  2,  3,  5,  7, 11,  18, ... = a(n-4)
...
Main diagonal is 2,2,2,... = A007395.
Adding a(n) and first column with alternating signs, one gets two autosequences: 2*Lucas numbers A000032 (2, 1, 3, 4, 7, 11, 18, ...) or 2*A286350 (0, 2, 2, 3, 4, 7, 12, ...) according to signs.

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 1).

Cf. A000032, A097924, A286350.

L:=[2,3,5,7];; for i in [5..10^3] do L[i]:=2*L[i-1]-L[i-2]+L[i-4]; od; L;  #  Muniru A Asiru, Sep 02 2017

LinearRecurrence[{2, -1, 0, 1}, {2, 3, 5, 7}, 40]

G.f.: (2 - x + x^2)/(1 - 2*x + x^2 - x^4).
a(3n) = A097924(n).
a(3n) + a(3n+1) = a(3n+2).
a(n) = (1/15)*2^(-n-1)*((30-9*sqrt(5))*(1-sqrt(5))^n + (1+sqrt(5))^n*(30 + 9*sqrt(5)) + 5*2^(n+1)*sqrt(3)*sin(n*Pi/3)).

cp's OEIS Frontend

A001077 Numerators of continued fraction convergents to sqrt(5).

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A048875 Generalized Pellian with second term of 6.

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A115032 Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)).

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A100545 Expansion of (7-2*x) / (1-3*x+x^2).

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A328695 Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.

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A291660 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=2, a(1)=3, a(2)=5, a(3)=7, a sequence related to Lucas numbers.

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A115032 Expansion of (5-14x+x^2)/((1-x)(x^2-18*x+1)).

A100545 Expansion of (7-2x) / (1-3x+x^2).