A030195
a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=1.
Original entry on oeis.org
0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893, 507627, 1924560, 7296561, 27663363, 104879772, 397629405, 1507527531, 5715470808, 21668995017, 82153397475, 311467177476, 1180861724853, 4476986706987, 16973545295520
Offset: 0
G.f. = x + 3*x^2 + 12*x^3 + 45*x^4 + 171*x^5 + 648*x^6 + 2457*x^7 + ...
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Martin Burtscher, Igor Szczyrba, 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=q=3.
- Tanya Khovanova, Recursive Sequences
- W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (39), (41) and (45), rhs, m=3.
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (3,3).
-
a030195 n = a030195_list !! n
a030195_list =
0 : 1 : map (* 3) (zipWith (+) a030195_list (tail a030195_list))
-- Reinhard Zumkeller, Oct 14 2011
-
I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) + 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
-
CoefficientList[Series[1/(1-3x-3x^2), {x, 0, 25}], x] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{3, 3}, {0, 1}, 24] (* Or *)
RecurrenceTable[{a[n] == 3 a[n - 1] + 3 a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 23}] (* Robert G. Wilson v, Aug 18 2012 *)
-
{a(n) = n--; polchebyshev(n, 2, I*sqrt(3)/2) * (-I*sqrt(3))^n};
-
[lucas_number1(n,3,-3) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009
I simplified the definition. As a result the offsets in some of the formulas may need to shifted by 1. -
N. J. A. Sloane, Apr 01 2006
A063727
a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
Original entry on oeis.org
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
Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001
- A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 235.
- John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
The following sequences (and others) belong to the same family:
A001333,
A000129,
A026150,
A002605,
A046717,
A015518,
A084057,
A063727,
A002533,
A002532,
A083098,
A083099,
A083100,
A015519.
-
List([0..25],n->2^n*Fibonacci(n+1)); # Muniru A Asiru, Nov 24 2018
-
[n le 2 select n else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018
-
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 15 2008
-
a[n_]:=(MatrixPower[{{1,5},{1,1}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
CoefficientList[Series[1/(1 - 2 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 31 2014 *)
LinearRecurrence[{2, 4}, {1, 2}, 50] (* G. C. Greubel, Jan 07 2018 *)
-
s(n)=if(n<2,n+1,(s(n-1)+(s(n-2)*2))*2); for(n=0,32,print(s(n)))
-
{ for (n=0, 200, if (n>1, a=2*a1 + 4*a2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b063727.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 28 2009
-
[lucas_number1(n,2,-4) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009
A028859
a(n+2) = 2*a(n+1) + 2*a(n); a(0) = 1, a(1) = 3.
Original entry on oeis.org
1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968
Offset: 0
- S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 73).
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
- Joerg Arndt, Matters Computational (The Fxtbook), section 14.9 "Strings with no two consecutive zeros", pp.318-320.
- C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8.
- Moussa Benoumhani, On the Modes of the Independence Polynomial of the Centipede, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1.
- D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 Example 7.
- 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.
- P. Z. Chinn, R. Grimaldi, and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
- David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.
- Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
- Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
- Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
- 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
- J. Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
- Eric Weisstein's World of Mathematics, Centipede Graph
- Eric Weisstein's World of Mathematics, Independent Vertex Set
- Eric Weisstein's World of Mathematics, Vertex Cover
- Index entries for linear recurrences with constant coefficients, signature (2,2).
Cf.
A155020 (same sequence with term 1 prepended).
-
a028859 n = a028859_list !! n
a028859_list =
1 : 3 : map (* 2) (zipWith (+) a028859_list (tail a028859_list))
-- Reinhard Zumkeller, Oct 15 2011
-
a[0]:=1:a[1]:=3:for n from 2 to 24 do a[n]:=2*a[n-1]+2*a[n-2] od: seq(a[n],n=0..24); # Emeric Deutsch
-
a[n_]:=(MatrixPower[{{1,3},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Table[2^(n - 1) Hypergeometric2F1[(1 - n)/2, -n/2, -n, -2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *)
LinearRecurrence[{2, 2}, {1, 3}, 20] (* Eric W. Weisstein, Jun 14 2017 *)
-
a(n)=([1,3;1,1]^n*[2;1])[2,1] \\ Charles R Greathouse IV, Mar 27 2012
-
A028859(n)=([1,1]*[2,2;1,0]^n)[1] \\ M. F. Hasler, Aug 06 2018
A080040
a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0)=2, a(1)=2.
Original entry on oeis.org
2, 2, 8, 20, 56, 152, 416, 1136, 3104, 8480, 23168, 63296, 172928, 472448, 1290752, 3526400, 9634304, 26321408, 71911424, 196465664, 536754176, 1466439680, 4006387712, 10945654784, 29904084992, 81699479552, 223207129088, 609813217280, 1666040692736, 4551707820032
Offset: 0
Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO], 2015-2017.
- 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.
- D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
- Tanya Khovanova, Recursive Sequences
- D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, Journal of the London Mathematical Society 1.3 (1935): 162-165. See V_n.
- Eric Weisstein's World of Mathematics, Independent Vertex Set
- Eric Weisstein's World of Mathematics, Sunlet Graph
- Eric Weisstein's World of Mathematics, Vertex Cover
- Index entries for linear recurrences with constant coefficients, signature (2,2).
- Index entries for Lucas sequences (2,-2).
-
a080040 n = a080040_list !! n
a080040_list =
2 : 2 : map (* 2) (zipWith (+) a080040_list (tail a080040_list))
-- Reinhard Zumkeller, Oct 15 2011
-
a:=[2,2]; [n le 2 select a[n] else 2*Self(n-1) + 2*Self(n-2):n in [1..27]]; Marius A. Burtea, Jan 20 2020
-
R:=PowerSeriesRing(Rationals(), 27); Coefficients(R!( (2-2*x)/(1-2*x-2*x^2))); // Marius A. Burtea, Jan 20 2020
-
CoefficientList[Series[(2 - 2 t)/(1 - 2 t - 2 t^2), {t, 0, 30}], t]
With[{c = {2, 2}}, LinearRecurrence[c, c, 20]] (* Harvey P. Dale, Apr 24 2016 *)
Round @ Table[LucasL[n, Sqrt[2]] 2^(n/2), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
Table[(1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Sep 27 2017 *)
-
a(n)=([0,1; 2,2]^n*[2;2])[1,1] \\ Charles R Greathouse IV, Apr 08 2016
-
from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(2,2,2,2, lambda n: 0); [next(it) for i in range(27)] # Zerinvary Lajos, Jul 16 2008
-
[lucas_number2(n,2,-2) for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009
A084057
a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=1.
Original entry on oeis.org
1, 1, 6, 16, 56, 176, 576, 1856, 6016, 19456, 62976, 203776, 659456, 2134016, 6905856, 22347776, 72318976, 234029056, 757334016, 2450784256, 7930904576, 25664946176, 83053510656, 268766806016, 869747654656, 2814562533376, 9108115685376, 29474481504256
Offset: 0
- John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
The following sequences (and others) belong to the same family:
A001333,
A000129,
A026150,
A002605,
A046717,
A015518,
A084057,
A063727,
A002533,
A002532,
A083098,
A083099,
A083100,
A015519.
-
I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 31 2016
-
f[n_] := Simplify[((1 + Sqrt[5])^n + (1 - Sqrt[5])^n)/2]; Array[f, 28, 0] (* Or *)
LinearRecurrence[{2, 4}, {1, 1}, 28] (* Robert G. Wilson v, Sep 18 2013 *)
RecurrenceTable[{a[1] == 1, a[2] == 1, a[n] == 2 a[n-1] + 4 a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Jul 31 2016 *)
Table[2^(n-1) LucasL[n], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 19 2016 *)
-
lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=lucas(n)/2*2^n \\ Charles R Greathouse IV, Sep 18 2013
-
from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,1,2,4, lambda n: 0); [next(it) for i in range(1,26)] # Zerinvary Lajos, Jul 09 2008
-
[lucas_number2(n,2,-4)/2 for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009
A002531
a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1); a(0) = a(1) = 1.
Original entry on oeis.org
1, 1, 2, 5, 7, 19, 26, 71, 97, 265, 362, 989, 1351, 3691, 5042, 13775, 18817, 51409, 70226, 191861, 262087, 716035, 978122, 2672279, 3650401, 9973081, 13623482, 37220045, 50843527, 138907099, 189750626, 518408351, 708158977, 1934726305
Offset: 0
1 + 1/(1 + 1/(2 + 1/(1 + 1/2))) = 19/11 so a(5) = 19.
Convergents are 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 19*x^5 + 26*x^6 + 71*x^7 + ... - _Michael Somos_, Mar 22 2022
- I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 181.
- 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).
- A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
- Harry J. Smith, Table of n, a(n) for n = 0..2000
- MacTutor, D'Arcy Thompson on Greek irrationals
- 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
- Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
- D'Arcy Thompson, Excess and Defect: Or the Little More and the Little Less, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 43-55 (13 pages). See page 48.
- Hein van Winkel, Q-quadrangles inscribed in a circle, 2014. See Table 1. [Reference from Antreas Hatzipolakis, Jul 14 2014]
- Index entries for "core" sequences
- Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).
- Index entries for sequences related to Chebyshev polynomials.
-
a:=[1,1,2,5];; for n in [5..40] do a[n]:=4*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Nov 16 2018
-
m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1 +x-2*x^2+x^3)/(1-4*x^2+x^4))); // G. C. Greubel, Nov 16 2018
-
A002531 := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif type(n,odd) then A002531(n-1)+A002531(n-2) else 2*A002531(n-1)+A002531(n-2) fi; end; [ seq(A002531(n), n=0..50) ];
with(numtheory): tp := cfrac (tan(Pi/3),100): seq(nthnumer(tp,i), i=-1..32 ); # Zerinvary Lajos, Feb 07 2007
A002531:=(1+z-2*z**2+z**3)/(1-4*z**2+z**4); # Simon Plouffe; see his 1992 dissertation
-
Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 40}], 1, 1] (* Stefan Steinerberger, Apr 01 2006 *)
Join[{1},Numerator[Convergents[Sqrt[3],40]]] (* Harvey P. Dale, Jan 23 2012 *)
CoefficientList[Series[(1 + x - 2 x^2 + x^3)/(1 - 4 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 01 2014 *)
LinearRecurrence[{0, 4, 0, -1}, {1, 1, 2, 5}, 35] (* Robert G. Wilson v, Feb 11 2018 *)
a[ n_] := ChebyshevT[n, Sqrt[-1/2]]*Sqrt[2]^Mod[n,2]/I^n //Simplify; (* Michael Somos, Mar 22 2022 *)
a[ n_] := If[n<0, (-1)^n*a[-n], SeriesCoefficient[ (1 + x - 2*x^2 + x^3) / (1 - 4*x^2 + x^4), {x, 0, n}]]; (* Michael Somos, Sep 23 2024 *)
-
a(n)=contfracpnqn(vector(n,i,1+(i>1)*(i%2)))[1,1]
-
apply( {A002531(n,w=quadgen(12))=real((2+w)^(n\/2)*if(bittest(n, 0), w-1, 1))}, [0..30]) \\ M. F. Hasler, Nov 04 2019
-
{a(n) = if(n<0, (-1)^n*a(-n), polcoeff( (1 + x - 2*x^2 + x^3) / (1 - 4*x^2 + x^4) + x*O(x^n), n))}; /* Michael Somos, Sep 23 2024 */
-
s=((1+x-2*x^2+x^3)/(1-4*x^2+x^4)).series(x,40); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018
A002532
a(n) = 2*a(n-1) + 5*a(n-2), a(0) = 0, a(1) = 1.
Original entry on oeis.org
0, 1, 2, 9, 28, 101, 342, 1189, 4088, 14121, 48682, 167969, 579348, 1998541, 6893822, 23780349, 82029808, 282961361, 976071762, 3366950329, 11614259468, 40063270581, 138197838502, 476712029909, 1644413252328, 5672386654201, 19566839570042, 67495612411089
Offset: 0
G.f. = x + 2*x^2 + 9*x^3 + 28*x^4 + 101*x^5 + 342*x^6 + 1189*x^7 + ...
- John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
- 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).
- A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- 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
- Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12 [Annotated scanned copy]
- Index entries for linear recurrences with constant coefficients, signature (2,5).
Cf.
A015581 (similar application, but no distinguishing identical vs. fraternal twins).
The following sequences (and others) belong to the same family:
A001333,
A000129,
A026150,
A002605,
A046717,
A015518,
A084057,
A063727,
A002533,
A002532,
A083098,
A083099,
A083100,
A015519.
-
[Floor(((1+Sqrt(6))^n-(1-Sqrt(6))^n)/(2*Sqrt(6))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011
-
[n le 2 select n-1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018
-
A002532:=-z/(-1+2*z+5*z**2); # Conjectured by Simon Plouffe in his 1992 dissertation
# second program
seq(simplify(2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5)), n = 2..25); # Peter Bala, Jul 06 2025
-
Expand[Table[((1 + Sqrt[6])^n - (1 - Sqrt[6])^n)/(2Sqrt[6]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
a[n_]:=(MatrixPower[{{1,2},{1,-3}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{2,5},{0,1},30] (* Harvey P. Dale, Nov 03 2011 *)
-
Vec(1/(1-2*x-5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 17 2012
-
from sage.combinat.sloane_functions import recur_gen2; it = recur_gen2(0,1,2,5); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008
-
[lucas_number1(n,2,-5) for n in range(0, 26)] # Zerinvary Lajos, Apr 22 2009
A083099
a(n) = 2*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.
Original entry on oeis.org
0, 1, 2, 10, 32, 124, 440, 1624, 5888, 21520, 78368, 285856, 1041920, 3798976, 13849472, 50492800, 184082432, 671121664, 2446737920, 8920205824, 32520839168, 118562913280, 432250861568, 1575879202816, 5745263575040
Offset: 0
Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
- John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
The following sequences (and others) belong to the same family:
A000129,
A001333,
A002532,
A002533,
A002605,
A015518,
A015519,
A026150,
A046717,
A063727,
A083098,
A083099,
A083100,
A084057.
-
[n le 2 select n-1 else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
-
A083099 := proc(n)
option remember;
if n <= 1 then
n;
else
2*procname(n-1)+6*procname(n-2) ;
end if;
end proc: # R. J. Mathar, Sep 23 2016
-
CoefficientList[Series[x/(1-2x-6x^2), {x, 0, 25}], x] (* Adapted for offset 0 by Vincenzo Librandi, Feb 07 2014 *)
Expand[Table[((1 + Sqrt[7])^n - (1 - Sqrt[7])^n)7/(14Sqrt[7]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{2,6}, {0,1}, 25] (* Sture Sjöstedt, Dec 06 2011 *)
-
a(n)=([0,1; 6,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016
-
my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x-6*x^2))) \\ G. C. Greubel, Jan 24 2018
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[lucas_number1(n,2,-6) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009
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A083099=BinaryRecurrenceSequence(2,6,0,1)
[A083099(n) for n in range(41)] # G. C. Greubel, Jun 01 2023
A015519
a(n) = 2*a(n-1) + 7*a(n-2), with a(0) = 0, a(1) = 1.
Original entry on oeis.org
0, 1, 2, 11, 36, 149, 550, 2143, 8136, 31273, 119498, 457907, 1752300, 6709949, 25685998, 98341639, 376485264, 1441362001, 5518120850, 21125775707, 80878397364, 309637224677, 1185423230902, 4538307034543, 17374576685400
Offset: 0
- John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
The following sequences (and others) belong to the same family:
A000129,
A001333,
A002532,
A002533,
A002605,
A015518,
A015519,
A026150,
A046717,
A063727,
A083098,
A083099,
A083100,
A084057.
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[ n eq 1 select 0 else n eq 2 select 1 else 2*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 23 2011
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LinearRecurrence[{2,7},{0,1},30] (* Harvey P. Dale, Oct 09 2017 *)
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a(n)=([0,1; 7,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, May 10 2016
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[lucas_number1(n,2,-7) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009
A002533
a(n) = 2*a(n-1) + 5*a(n-2), with a(0) = a(1) = 1.
Original entry on oeis.org
1, 1, 7, 19, 73, 241, 847, 2899, 10033, 34561, 119287, 411379, 1419193, 4895281, 16886527, 58249459, 200931553, 693110401, 2390878567, 8247309139, 28449011113, 98134567921, 338514191407, 1167701222419, 4027973401873, 13894452915841, 47928772841047, 165329810261299
Offset: 0
- John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
- 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).
- A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- 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
- Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
- Index entries for linear recurrences with constant coefficients, signature (2,5).
The following sequences (and others) belong to the same family:
A001333,
A000129,
A026150,
A002605,
A046717,
A015518,
A084057,
A063727,
A002533,
A002532,
A083098,
A083099,
A083100,
A015519.
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[(1/2)*Floor((1+Sqrt(6))^n+(1-Sqrt(6))^n): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011
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[n le 2 select 1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018
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A002533:=(-1+z)/(-1+2*z+5*z**2); # conjectured by Simon Plouffe in his 1992 dissertation
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f[n_] := Simplify[((1 + Sqrt[6])^n + (1 - Sqrt[6])^n)/2]; Array[f, 28, 0] (* Or *)
LinearRecurrence[{2, 5}, {1, 1}, 28] (* Or *)
Table[ MatrixPower[{{1, 2}, {3, 1}}, n][[1, 1]], {n, 0, 25}]
(* Robert G. Wilson v, Sep 18 2013 *)
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a(n)=([0,1; 5,2]^n*[1;1])[1,1] \\ Charles R Greathouse IV, May 10 2016
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x='x+O('x^30); Vec((1-x)/(1-2*x-5*x^2)) \\ G. C. Greubel, Jan 08 2018
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[lucas_number2(n,2,-5)/2 for n in range(0, 21)] # Zerinvary Lajos, Apr 30 2009
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