A041090 -id:A041090 - cp's OEIS frontend

A054413 a(n) = 7*a(n-1) + a(n-2), with a(0)=1 and a(1)=7.

1, 7, 50, 357, 2549, 18200, 129949, 927843, 6624850, 47301793, 337737401, 2411463600, 17217982601, 122937341807, 877779375250, 6267392968557, 44749530155149, 319514104054600, 2281348258537349, 16288951913816043, 116304011655249650, 830417033500563593

Offset: 0

Henry Bottomley, May 10 2000

In general, sequences with recurrence a(n) = k*a(n-1) + a(n-2) and a(0)=1 (and a(-1)=0) have the generating function 1/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n) = b(5n), a(3n+1) = b(5n+3), a(3n+2) = 2*b(5n+4) where b(n) is the sequence of denominators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n) is the sequence of denominators of continued fraction convergents to sqrt(k^2/4+1).]
a(p-1) == 53^((p-1)/2) (mod p), for odd primes p. - Gary W. Adamson, Feb 22 2009 [See A087475 for more info about this congruence. - Jason Yuen, Apr 05 2025]
From Johannes W. Meijer, Jun 12 2010: (Start)
For the sequence given above k=7 which implies that it is associated with A041091.
For a similar statement about sequences with recurrence a(n) = k*a(n-1) + a(n-2) but with a(0) = 2, and a(-1) = 0, see A086902; a sequence that is associated with A041090.
For more information follow the Khovanova link and see A087130, A140455 and A178765.
(End)
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 7's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0,1,...,7} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Feb 21 2023: (Start)
Also called the 7-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 7 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Index entries for linear recurrences with constant coefficients, signature (7,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, A000129, A001076, A005668, A006190, A052918, A243399.
Row n=7 of A073133, A172236 and A352361.
Cf. A099367 (squares).

I:=[1, 7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 23 2013

LinearRecurrence[{7, 1}, {1, 7}, 30] (* Vincenzo Librandi, Feb 23 2013 *)

a(n)=([0,1; 1,7]^n*[1;7])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,7,-1) for n in range(1, 19)] # Zerinvary Lajos, Apr 24 2009

a(3n) = A041091(5n), a(3n+1) = A041091(5n+3), a(3n+2) = 2*A041091(5n+4).
G.f.: 1/(1 - 7x - x^2).
a(n) = U(n, 7*i/2)*(-i)^n with i^2=-1 and Chebyshev's U(n, x/2) = S(n, x) polynomials. See A049310.
a(n) = F(n, 7), the n-th Fibonacci polynomial evaluated at x=7. - T. D. Noe, Jan 19 2006
From Sergio Falcon, Sep 24 2007: (Start)
a(n) = (sigma^n - (-sigma)^(-n))/(sqrt(53)) with sigma = (7+sqrt(53))/2;
a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n-1-i,i)*7^(n-1-2i). (End)
a(n) = ((7 + sqrt(53))^n - (7 - sqrt(53))^n)/(2^n*sqrt(53)). Offset 1. a(3)=50. - Al Hakanson (hawkuu(AT)gmail.com), Jan 17 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 7*A097836(n), a(2n) = A097838(n).
Lim_{k->oo} a(n+k)/a(k) = (A086902(n) + A054413(n-1)*sqrt(53))/2.
Lim_{n->oo} A086902(n)/A054413(n-1) = sqrt(53).
(End)
Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = (sqrt(53)-7)/2. - Vladimir Shevelev, Feb 23 2013
From Kai Wang, Feb 24 2020: (Start)
Sum_{m>=0} 1/(a(m)*a(m+2)) = 1/49.
Sum_{m>=0} 1/(a(2*m)*a(2*m+2)) = (sqrt(53)-7)/14.
In general, for sequences with recurrence f(n)= k*f(n-1)+f(n-2) and f(0)=1,
Sum_{m>=0} 1/(f(m)*f(m+2)) = 1/(k^2).
Sum_{m>=0} 1/(f(2*m)*f(2*m+2)) = (sqrt(k^2+4) - k)/(2*k). (End)
E.g.f.: (1/53)*exp(7*x/2)*(53*cosh(sqrt(53)*x/2) + 7*sqrt(53)*sinh(sqrt(53)*x/2)). - Stefano Spezia, Feb 26 2020
G.f.: x/(1 - 7*x - x^2) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (m*k + 7 - m + x)/(1 + m*k*x) ) for arbitrary m (a telescoping series). - Peter Bala, May 08 2024

Formula corrected by Johannes W. Meijer, May 30 2010, Jun 02 2010

Extended by T. D. Noe, May 23 2011

A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

Original entry on oeis.org

2, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, 17555720002, 125348805407, 894997357851, 6390330310364, 45627309530399, 325781497023157, 2326097788692498, 16608466017870643, 118585359913786999, 846705985414379636

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 18 2003

a(n+1)/a(n) converges to (7+sqrt(53))/2 = 7.14005... = A176439.
Lim a(n)/a(n+1) as n approaches infinity = 0.1400549... = 2/(7+sqrt(53)) = (sqrt(53)-7)/2 = 1/A176439 = A176439 - 7.
From Johannes W. Meijer, Jun 12 2010: (Start)
In general sequences with recurrence a(n) = k*a(n-1)+a(n-2) with a(0)=2 and a(1)=k [and a(-1)=0] have generating function (2-k*x)/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n+1) = b(5n), a(3n+2)=b(5*n+3), a(3n+3)=2*b(5n+4) where b(n) is the sequence of numerators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n)/2, for n>=1, is the sequence of numerators of continued fraction convergents to sqrt(k^2/4+1).]
For the sequence given above k=7 which implies that it is associated with A041090.
For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0)=1 [and a(-1)=0] see A054413; a sequence that is associated with A041091.
For more information follow the Khovanova link and see A087130, A140455 and A178765.
(End)

			a(4) = 7*a(3) + a(2) = 7*364 + 51 = 2599.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (7,1).

Cf. A058316, A176439.

Cf. A000032 (k=1), A006497 (k=3), A087130 (k=5), A086902 (k=7), A087798 (k=9), A001946 (k=11), A088316 (k=13), A090301 (k=15), A090306 (k=17). - Johannes W. Meijer, Jun 12 2010

I:=[2,7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == 7 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{7,1},{2,7},30] (* Harvey P. Dale, May 25 2023 *)

a(n)=([0,1; 1,7]^n*[2;7])[1,1] \\ Charles R Greathouse IV, Apr 06 2016

a(n) = ((7+sqrt(53))/2)^n + ((7-sqrt(53))/2)^n.
E.g.f. : 2exp(7x/2)cosh(sqrt(53)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)53^k7^(n-2k)}. a(n)=2T(n, 7i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003
G.f.: (2-7x)/(1-7x-x^2). - Philippe Deléham, Nov 16 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 7*A097837(n), a(2n) = A099368(n).
a(3n+1) = A041090(5n), a(3n+2) = A041090(5*n+3), a(3n+3) = 2*A041090(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A086902(n) + A054413(n-1)*sqrt(53))/2.
Limit(A086902(n)/A054413(n-1), n=infinity) = sqrt(53). (End)

A041018 Numerators of continued fraction convergents to sqrt(13).

Original entry on oeis.org

3, 4, 7, 11, 18, 119, 137, 256, 393, 649, 4287, 4936, 9223, 14159, 23382, 154451, 177833, 332284, 510117, 842401, 5564523, 6406924, 11971447, 18378371, 30349818, 200477279, 230827097, 431304376, 662131473

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,36,0,0,0,0,1).

Cf. A010122 (continued fraction for sqrt(13)).

Cf. A010470, A041019 (denominators), A041046, A041090, A041150, A041226, A041318, A041426 and A041550.

a[0]:=3: a[-1]:=1: b(0):=6: b(1):=1; b(2):=1: b(3):=1: b(4):=1:
for n from 1 to 100 do  k:=n mod 5:
   a[n]:=b(k)*a[n-1]+a[n-2]:
   printf("%12d", a[n]):
end do: # Paul Weisenhorn, Aug 17 2018

Numerator[Convergents[Sqrt[13], 30]] (* Vincenzo Librandi, Oct 27 2013 *)
CoefficientList[Series[(3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10),{x,0,50}],x] (* Stefano Spezia, Aug 31 2018 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5*n) = A006497(3*n+1),
a(5*n+1) = (A006497(3*n+2)-A006497(3*n+1))/2,
a(5*n+2) = (A006497(3*n+2)+A006497(3*n+1))/2,
a(5*n+3) = A006497(3*n+2),
a(5*n+4) = A006497(3*n+3)/2.
(End)
G.f.: (3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013
a(n) = A010122(n)*a(n-1)+a(n-2) with a(0)=3, a(-1)=1. - Paul Weisenhorn, Aug 19 2018

A041046 Numerators of continued fraction convergents to sqrt(29).

Original entry on oeis.org

5, 11, 16, 27, 70, 727, 1524, 2251, 3775, 9801, 101785, 213371, 315156, 528527, 1372210, 14250627, 29873464, 44124091, 73997555, 192119201, 1995189565, 4182498331, 6177687896, 10360186227, 26898060350

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The terms of this sequence can be constructed with the terms of sequence A087130.
For the terms of the periodical sequence of the continued fraction for sqrt(29) see A010128. We observe that its period is five. The decimal expansion of sqrt(29) is A010484.  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,140,0,0,0,0,1).

Cf. A041047, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550, A010484.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[29],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[29], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
LinearRecurrence[ {0,0,0,0,140,0,0,0,0,1},{5,11,16,27,70,727,1524,2251,3775,9801},30] (* Harvey P. Dale, Jun 10 2021 *)

a(5*n) = A087130(3*n+1), a(5*n+1) = (A087130(3*n+2) - A087130(3*n+1))/2, a(5*n+2) = ( A087130(3*n+2) + A087130(3*n+1))/2, a(5*n+3) = A087130(3*n+2) and a(5*n+4) = A087130(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: (5 + 11*x + 16*x^2 + 27*x^3 + 70*x^4 + 27*x^5 - 16*x^6 + 11*x^7 - 5*x^8 + x^9)/(1 - 140*x^5 - x^10) - Peter J. C. Moses, Jul 29 2013

A041091 Denominators of continued fraction convergents to sqrt(53).

Original entry on oeis.org

1, 3, 4, 7, 25, 357, 1096, 1453, 2549, 9100, 129949, 398947, 528896, 927843, 3312425, 47301793, 145217804, 192519597, 337737401, 1205731800, 17217982601, 52859679603, 70077662204, 122937341807, 438889687625, 6267392968557, 19241068593296, 25508461561853

Offset: 0

N. J. A. Sloane

The terms of this sequence can be constructed with the terms of sequence A054413. For the terms of the periodic sequence of the continued fraction for sqrt(53) see A010139. We observe that its period is five. The decimal expansion of sqrt(53) is A010506. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,364,0,0,0,0,1).

Cf. A010506, A041090.

Cf. A041019, A041047, A041151, A041227, A041319, A041427 and A041551. - Johannes W. Meijer, Jun 12 2010

convert(sqrt(53), confrac, 30, cvgts): denom(cvgts); # Wesley Ivan Hurt, Dec 17 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[53], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[53], 30]] (* Vincenzo Librandi, Oct 24 2013 *)
LinearRecurrence[{0,0,0,0,364,0,0,0,0,1},{1,3,4,7,25,357,1096,1453,2549,9100},30] (* Harvey P. Dale, Nov 13 2019 *)

a(5*n) = A054413(3*n), a(5*n+1) = (A054413(3*n+1) - A054413(3*n))/2, a(5*n+2)= (A054413(3*n+1) + A054413(3*n))/2, a(5*n+3) = A054413(3*n+1) and a(5*n+4) = A054413(3*n+2)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: -(x^8-3*x^7+4*x^6-7*x^5+25*x^4+7*x^3+4*x^2+3*x+1) / (x^10+364*x^5-1). - Colin Barker, Sep 26 2013

A041150 Numerators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

9, 37, 46, 83, 378, 6887, 27926, 34813, 62739, 285769, 5206581, 21112093, 26318674, 47430767, 216041742, 3936182123, 15960770234, 19896952357, 35857722591, 163327842721, 2975758891569, 12066363408997

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 17 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A087798.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,756,0,0,0,0,1).

Cf. A010536, A041018, A041046, A041090, A041150, A041151, A041226, A041318, A041426,

A041550.

Numerator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

From Johannes W. Meijer, Jun 17 2010: (Start)
a(5*n) = A087798(3*n+1), a(5*n+1) = (A087798(3*n+2) - A087798(3*n+1))/2, a(5*n+2) = (A087798(3*n+2) + A087798(3*n+1))/2, a(5*n+3) = A087798(3*n+2) and a(5*n+4) = A087798(3*n+3)/2. (End)
G.f.: -(x^9-9*x^8+37*x^7-46*x^6+83*x^5+378*x^4+83*x^3+46*x^2+37*x+9) / (x^10+756*x^5-1). - Colin Barker, Nov 04 2013

A041226 Numerators of continued fraction convergents to sqrt(125).

Original entry on oeis.org

11, 56, 67, 123, 682, 15127, 76317, 91444, 167761, 930249, 20633239, 104096444, 124729683, 228826127, 1268860318, 28143753123, 141987625933, 170131379056, 312119004989, 1730726404001, 38388099893011, 193671225869056, 232059325762067, 425730551631123

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A001946.
For the terms of the periodical sequence of the continued fraction for sqrt(125) see A010186. We observe that its period is five. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1364,0,0,0,0,1).

Cf. A041227, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Numerator[Convergents[Sqrt[125], 30]] (* Vincenzo Librandi, Oct 31 2013 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5n)   = A001946(3n+1),
a(5n+1) = (A001946(3n+2) - A001946(3n+1))/2,
a(5n+2) = (A001946(3n+2) + A001946(3n+1))/2,
a(5n+3) = A001946(3n+2),
a(5n+4) = A001946(3n+3)/2. (End)
G.f.: -(x^9 -11*x^8 +56*x^7 -67*x^6 +123*x^5 +682*x^4 +123*x^3 +67*x^2 +56*x +11) / ((x^2 +4*x -1)*(x^4 -7*x^3 +19*x^2 -3*x +1)*(x^4 +3*x^3 +19*x^2 +7*x +1)). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A041318 Numerators of continued fraction convergents to sqrt(173).

Original entry on oeis.org

13, 79, 92, 171, 1118, 29239, 176552, 205791, 382343, 2499849, 65378417, 394770351, 460148768, 854919119, 5589663482, 146186169651, 882706681388, 1028892851039, 1911599532427, 12498490045601, 326872340718053, 1973732534353919, 2300604875071972

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 2236, 0, 0, 0, 0, 1).

Cf. A041319, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Cf. A010217 (continued fraction).

Numerator[Convergents[Sqrt[173], 30]] (* Vincenzo Librandi, Nov 01 2013 *)
LinearRecurrence[{0,0,0,0,2236,0,0,0,0,1},{13,79,92,171,1118,29239,176552,205791,382343,2499849},30] (* Harvey P. Dale, Jul 28 2018 *)

a(5*n) = A088316(3*n+1), a(5*n+1) = (A088316(3*n+2) - A088316(3*n+1))/2, a(5*n+2) = (A088316(3*n+2)+A088316(3*n+1))/2, a(5*n+3) = A088316(3*n+2) and a(5*n+4) = A088316(3*n+3)/2. [Johannes W. Meijer, Jun 12 2010]

G.f.: -(x^9-13*x^8+79*x^7-92*x^6+171*x^5+1118*x^4+171*x^3+92*x^2+79*x+13) / (x^10+2236*x^5-1). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A041426 Numerators of continued fraction convergents to sqrt(229).

Original entry on oeis.org

15, 106, 121, 227, 1710, 51527, 362399, 413926, 776325, 5848201, 176222355, 1239404686, 1415627041, 2655031727, 20000849130, 602680505627, 4238764388519, 4841444894146, 9080209282665, 68402909872801, 2061167505466695, 14496575448139666, 16557742953606361

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A090301.
For the terms of the periodical sequence of the continued fraction for sqrt(229) see A040213. We observe that its period is five. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 3420, 0, 0, 0, 0, 1).

Cf. A041427, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Numerator[Convergents[Sqrt[229], 30]] (* Vincenzo Librandi, Nov 01 2013 *)
LinearRecurrence[{0,0,0,0,3420,0,0,0,0,1},{15,106,121,227,1710,51527,362399,413926,776325,5848201},30] (* Harvey P. Dale, Dec 19 2016 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5n) = A090301(3n+1), a(5n+1) = (A090301(3n+2) - A090301(3n+1))/2, a(5n+2) = (A090301(3n+2) + A090301(3n+1))/2, a(5n+3) = A090301(3n+2) and a(5n+4) = A090301(3n+3)/2. (End)
G.f.: -(x^9-15*x^8+106*x^7-121*x^6+227*x^5+1710*x^4+227*x^3+121*x^2+106*x+15) / (x^10+3420*x^5-1). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A041550 Numerators of continued fraction convergents to sqrt(293).

Original entry on oeis.org

17, 137, 154, 291, 2482, 84679, 679914, 764593, 1444507, 12320649, 420346573, 3375093233, 3795439806, 7170533039, 61159704118, 2086600473051, 16753963488526, 18840563961577, 35594527450103, 303596783562401, 10357885168571737, 83166678132136297

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A090306.
For the terms of the periodical sequence of the continued fraction for sqrt(293) see A040275. We observe that its period is five. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 4964, 0, 0, 0, 0, 1).

Cf. A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550, A041551.

Numerator[Convergents[Sqrt[293], 30]] (* Vincenzo Librandi, Nov 04 2013 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5n) = A090306(3n+1), a(5n+1) = (A090306(3n+2) - A090306(3n+1))/2, a(5n+2) = (A090306(3n+2) + A090306(3n+1))/2, a(5n+3) = A090306(3n+2) and a(5n+4) = A090306(3n+3)/2. (End)
G.f.: -(x^9-17*x^8+137*x^7-154*x^6+291*x^5+2482*x^4+291*x^3+154*x^2+137*x+17) / (x^10+4964*x^5-1). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

cp's OEIS Frontend

A054413 a(n) = 7*a(n-1) + a(n-2), with a(0)=1 and a(1)=7.

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A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

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A041018 Numerators of continued fraction convergents to sqrt(13).

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A041046 Numerators of continued fraction convergents to sqrt(29).

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A041091 Denominators of continued fraction convergents to sqrt(53).

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A041150 Numerators of continued fraction convergents to sqrt(85).

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A041226 Numerators of continued fraction convergents to sqrt(125).

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