cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Paul Barry, May 10 2003

Keywords

Comments

Inverse binomial transform of A001077. Binomial transform of expansion of cosh(sqrt(5)*x) (1,0,5,0,25,...).
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 5 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(5). - Cino Hilliard, Sep 25 2005
Numerators of fractions in the approximation of the square root of 5 satisfying: a(n) = (a(n-1)+c)/(a(n-1)+1), with c=5 and a(1)=1. For denominators see A063727. - Mark Dols, Jul 24 2009
Equals right border of triangle A143969. (1, 6, 16, 56, ...) = row sums of triangle A143969 and INVERT transform of (1, 5, 5, 5, ...). - Gary W. Adamson, Sep 06 2008
a(n) is the number of compositions of n when there are 1 type of 1 and 5 types of other natural numbers. - Milan Janjic, Aug 13 2010
From Gary W. Adamson, Jul 30 2016: (Start)
The sequence is case N=1 in an infinite set obtained by taking powers of the 2 X 2 matrix M = [(1,5); (1,N)], then extracting the upper left terms. The infinite set begins:
N=1 (A084057): 1, 6, 16, 56, 176, 576, 1856, ...
N=2 (A108306): 1, 6, 21, 81, 306, 1161, 4401, ...
N=3 (A164549): 1, 6, 26, 116, 516, 2296, 10216, ...
N=4 (A015449): 1, 6, 31, 161, 836, 4341, 22541, ...
N=5 (A000400): 1, 6, 36, 216, 1296, 7776, 46656, ...
N=6 (A049685): 1, 6, 41, 281, 1926, 13201, 90481, ...
N=7 (.......): 1, 6, 46, 356, 2756, 21336, 222712, ...
...
Sequences in the above set can be obtained by taking INVERT transforms of the following:
N=1 INVERT transform of (1, 5, 5, 5, 5, 5, ...
N=2 ..."......"......". (1, 5, 10, 20, 40, 80, ...
N=3 ..."......"......". (1, 5, 15, 45, 135, 405, ...
N=4 ..."......"......". (1, 5, 20, 80, 320, 1280, ...
...
with the pattern (1, 5, N*5, (N^2)*5, (N^3)*5, ...
It appears that the sequence generated from powers (n>0) of the matrix P = [(1,a); (1,b)], (a,b > 0), then extracting the upper left terms, is equal to the INVERT transform of the sequence starting: (1, a, b*a, (b^2)*a, (b^3)*a, ...). (End)

References

  • John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Links

Crossrefs

Cf. A046717, A002533, A098158, A143969, A269992.
a(n) = A087131(n)/2.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A108306, A164549, A015449, A000400, A049685

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=lucas(n)/2*2^n \\ Charles R Greathouse IV, Sep 18 2013
  • Sage
    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
  • Sage
    [lucas_number2(n,2,-4)/2 for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

Formula

a(n) = ((1+sqrt(5))^n + (1-sqrt(5))^n)/2.
G.f.: (1-x) / (1-2*x-4*x^2).
E.g.f.: exp(x) * cosh(sqrt(5)*x).
a(2n+1) = 2*a(n)*a(n+1) - (-4)^n. - Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*5^k . - Paul Barry, Jul 25 2004
a(n) = Sum_{k=0..n} A098158(n,k)*5^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = 2^(n-1)*A000032(n). - Mark Dols, Jul 24 2009
If p(1)=1, and p(i)=5 for i>1, and if A is the Hessenberg matrix of order n defined by: A(i,j) = p(j-i+1) for 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, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = A063727(n) - A063272(n-1). - R. J. Mathar, Jun 06 2019
a(n) = 1 + 5*A014335(n). - R. J. Mathar, Jun 06 2019
Sum_{n>=1} 1/a(n) = A269992. - Amiram Eldar, Feb 01 2021

A090017 a(n) = 4*a(n-1) + 2*a(n-2) for n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 4, 18, 80, 356, 1584, 7048, 31360, 139536, 620864, 2762528, 12291840, 54692416, 243353344, 1082798208, 4817899520, 21437194496, 95384577024, 424412697088, 1888419942400, 8402505163776, 37386860539904, 166352452487168
Offset: 0

Views

Author

Paul Barry, Nov 19 2003

Keywords

Comments

Starting with "1" = INVERT transform of A007482: (1, 3, 11, 39, 139, ...). - Gary W. Adamson, Aug 06 2010
This is the Lucas sequence U(4,-2). - Bruno Berselli, Jan 09 2013
Lower left term in matrix powers of [(1,5); (1,3)]. Convolved with (1, 2, 0, 0, 0, ...) the result is A164549: (1, 6, 26, 116, ...). - Gary W. Adamson, Aug 10 2016
For n>0, a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

Links

Crossrefs

Cf. A007070, A084059, A007482.
Cf. A164549.

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2011
  • Mathematica
    a[n_Integer] := (-I Sqrt[2])^(n - 1) ChebyshevU[ n - 1, I Sqrt[2] ]
a[n_]:=(MatrixPower[{{1,5},{1,3}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
t={0,1};Do[AppendTo[t,4*t[[-1]]+2*t[[-2]]],{n,2,23}];t (* or *) LinearRecurrence[{4,2},{0,1},24] (* Indranil Ghosh, Feb 21 2017 *)
  • PARI
    Vec(x/(1-4*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Oct 12 2011
  • Sage
    [lucas_number1(n, 4, -2) for n in range(0, 23)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1 - 4*x - 2*x^2).
a(n) = (-i*sqrt(2))^(n-1) U(n-1, i*sqrt(2)) where U is the Chebyshev polynomial of the second kind and i^2 = -1.
a(n) = ((2+sqrt(6))^n - (2-sqrt(6))^n)/(2 sqrt(6)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009, Jan 07 2009
a(n+1) = Sum_{k=0..n} A099089(n,k)*2^k. - Philippe Deléham, Nov 21 2011
From Ilya Gutkovskiy, Aug 22 2016: (Start)
E.g.f.: sinh(sqrt(6)*x)*exp(2*x)/sqrt(6).
Number of zeros in substitution system {0 -> 11, 1 -> 11011} at step n from initial string "1" (1 -> 11011 -> 1101111011111101111011 -> ...). (End)

Extensions

Edited by Stuart Clary, Oct 25 2009

A015449 Expansion of (1-4*x)/(1-5*x-x^2).

Original entry on oeis.org

1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686, 11913527111, 61861971241, 321223383316, 1667978887821, 8661117822421, 44973567999926, 233528957822051
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Row m=5 of A135597.
Binomial transform of A152187. - Johannes W. Meijer, Aug 01 2010
For n>=1, row sums of triangle
m/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....5
.2..|..1.....5....25
.3..|..1....10....25.....125
.4..|..1....10....75.....125....625
.5..|..1....15....75.....500....625....3125
.6..|..1....15...150.....500...3125....3125...15625
.7..|..1....20...150....1250...3125...18750...15625...78125
which is triangle for numbers 5^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
a(n+1) is (for n>=0) the number of length-n strings of 6 letters {0,1,2,3,4,5} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012
With offset 1, the sequence is the INVERT transform (1, 5, 5*4, 5*4^2, 5*4^3, ...); i.e., of A003947. The sequence can also be obtained by taking powers of the matrix [(1,5); (1,4)] and extracting the upper left terms. - Gary W. Adamson, Jul 31 2016

Links

Crossrefs

Cf. A084057, A108306, A164549.

Programs

  • GAP
    a:=[1,1];; for n in [3..30] do a[n]:=5*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Oct 23 2019
  • Magma
    [n le 2 select 1 else 5*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012
  • Maple
    a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..21); # Zerinvary Lajos, Jul 26 2006
  • Mathematica
    Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{1,5},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{5,1}, {1,1}, 30] (* Vincenzo Librandi, Nov 06 2012 *)
CoefficientList[Series[(1-4*x)/(1-5*x-x^2), {x,0,30}], x] (* G. C. Greubel, Dec 19 2017 *)
Sum[Fibonacci[Range[30] +k-2, 5], {k,0,1}] (* G. C. Greubel, Oct 23 2019 *)
  • PARI
    Vec((1-4*x)/(1-5*x-x^2) +O('x^30)) \\ _G. C. Greubel, Dec 19 2017
  • Sage
    def A015449_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/(1-5*x-x^2)).list()
A015449_list(30) # G. C. Greubel, Oct 23 2019

Formula

a(n) = 5*a(n-1) + a(n-2).
a(n) = Sum_{k=0..n} 4^k*A055830(n,k). - Philippe Deléham, Oct 18 2006
G.f.: (1-4*x)/(1-5*x-x^2). - Philippe Deléham, Nov 20 2008
For n >= 2, a(n) = F_n(5) + F_(n+1)(5), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
a(n) = Sum_{k=0..n} A046854(n-1,k)*5^k. - R. J. Mathar, Feb 10 2024

A123011 a(n) = 2*a(n-1) + 5*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 15, 55, 185, 645, 2215, 7655, 26385, 91045, 314015, 1083255, 3736585, 12889445, 44461815, 153370855, 529050785, 1824955845, 6295165615, 21715110455, 74906048985, 258387650245, 891305545415, 3074549342055
Offset: 0

Views

Author

Roger L. Bagula, Sep 23 2006

Keywords

Links

Crossrefs

Cf. A002532, A164532, A164549.

Programs

  • Magma
    [ n le 2 select 4*n-3 else 2*Self(n-1)+5*Self(n-2): n in [1..24] ]; // Klaus Brockhaus, Aug 15 2009
  • Mathematica
    LinearRecurrence[{2,5}, {1,5}, 31] (* G. C. Greubel, Jul 13 2021 *)
  • Sage
    [1]+[(sqrt(5)*i)^(n-1)*(sqrt(5)*i*chebyshev_U(n, -i/sqrt(5)) + 3*chebyshev_U(n-1, -i/sqrt(5))) for n in (1..30)] # G. C. Greubel, Jul 13 2021

Formula

a(n) = ((3+2*sqrt(6))*(1+sqrt(6))^n + (3-2*sqrt(6))*(1-sqrt(6))^n)/6. - Klaus Brockhaus, Aug 15 2009
From Klaus Brockhaus, Aug 15 2009: (Start)
G.f.: (1+3*x)/(1-2*x-5*x^2).
Binomial transform of A164532.
Inverse binomial transform of A164549. (End)
a(n) = (sqrt(5)*i)^(n-1)*(sqrt(5)*i*ChebyshevU(n, -i/sqrt(5)) + 3*ChebyshevU(n-1, -i/sqrt(5))) for n > 0 with a(0) = 1. - G. C. Greubel, Jul 13 2021

Extensions

Edited by N. J. A. Sloane, Aug 27 2009, using simpler definition suggested by Klaus Brockhaus, Aug 15 2009

A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 8, 54, 352, 2276, 14688, 94744, 611072, 3941136, 25418368, 163935584, 1057300992, 6819052096, 43979406848, 283644733824, 1829363802112, 11798463078656, 76094066608128, 490767902078464, 3165202550546432
Offset: 1

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

Comments

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(6) = 6.4494897427....
Binomial transform of A164550, second binomial transform of A164549, third binomial transform of A123011, fourth binomial transform of A164532.
Binomial transform is A164551, second binomial transform is A164552, third binomial transform is A164553.

Links

Crossrefs

Cf. A010464 (decimal expansion of square root of 6), A123011, A164532, A164549, A164550, A164551, A164552, A164553.

Programs

  • GAP
    a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-10*a[n-2]; od; a; # G. C. Greubel, May 21 2019
  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
  • Mathematica
    LinearRecurrence[{8, -10}, {1, 8}, 30] (* or *) Table[Simplify[((4 + Sqrt[6])^n -(4-Sqrt[6])^n)/(2*Sqrt[6])], {n, 30}] (* G. C. Greubel, Sep 06 2016 *)
  • PARI
    a(n)=([0,1; -10,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 07 2016
  • PARI
    my(x='x+O('x^30)); Vec(x/(1-8*x+10*x^2)) \\ G. C. Greubel, May 21 2019
  • Sage
    [lucas_number1(n,8,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 8*a(n-1) - 10*a(n-2) for n > 1, where a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x + 10*x^2). (End)

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009
Edited by Klaus Brockhaus, Oct 04 2009

A164550 a(n) = 6*a(n-1) - 3*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 39, 213, 1161, 6327, 34479, 187893, 1023921, 5579847, 30407319, 165704373, 903004281, 4920912567, 26816462559, 146136037653, 796366838241, 4339792916487, 23649656984199, 128878563155733, 702322407981801
Offset: 0

Views

Author

Klaus Brockhaus, Aug 15 2009

Keywords

Comments

Binomial transform of A164549.
Inverse binomial transform of A154235.

Links

Crossrefs

Cf. A154235, A164549.

Programs

  • Magma
    [ n le 2 select 6*n-5 else 6*Self(n-1)-3*Self(n-2): n in [1..21] ];
  • Mathematica
    LinearRecurrence[{6,-3}, {1,7}, 31] (* G. C. Greubel, Jul 16 2021 *)
  • Sage
    [3^((n-1)/2)*(sqrt(3)*chebyshev_U(n, sqrt(3)) + chebyshev_U(n-1, sqrt(3))) for n in (0..30)] # G. C. Greubel, Jul 16 2021

Formula

a(n) = ((3+2*sqrt(6))*(3+sqrt(6))^n + (3-2*sqrt(6))*(3-sqrt(6))^n)/6.
G.f.: (1+x)/(1-6*x+3*x^2).
a(n) = 3^((n-1)/2)*(sqrt(3)*ChebyshevU(n, sqrt(3)) + ChebyshevU(n-1, sqrt(3))). - G. C. Greubel, Jul 16 2021

A268409 a(n) = 4*a(n - 1) + 2*a(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

Views

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Mathematica
    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}]
  • PARI
    Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

Formula

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
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