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.

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A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

Original entry on oeis.org

1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360, 35184367894528, 140737496743936, 562949936644096, 2251799847239680
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Binomial transform of expansion of cosh(3*x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry, Apr 05 2003
Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard, Sep 25 2005
This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008
For n > 0, a(n) is term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 3,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010
a(n) = ((1+3)^n+(1-3)^n)/2. In general, if b(0),b(1),... is the k-th binomial transform of the sequence ((3^n+(-3)^n)/2), then b(n) = ((k+3)^n+(k-3)^n)/2. More generally, if b(0),b(1),... is the k-th binomial transform of the sequence ((m^n+(-m)^n)/2), then b(n) = ((k+m)^n+(k-m)^n)/2. See A034494, A081340-A081342, A034659. - Charlie Marion, Jun 25 2011
Pisano period lengths: 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 5, 1, 12, 6, 4, 1, 8, 1, 9, 4, ... - R. J. Mathar, Aug 10 2012
Starting with offset 1 the sequence is the INVERT transform of (1, 9, 9, 9, ...). - Gary W. Adamson, Aug 06 2016

References

  • John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
  • M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

Links

Crossrefs

Cf. A001019, A001045, A014551, A078008, A098158.
Cf. A034494, A081340-A081342, A034659.

Programs

  • GAP
    List([0..30], n-> 2^(n-1)*(2^n +(-1)^n)); # G. C. Greubel, Aug 02 2019
  • Magma
    [2^(n-1)*( 2^n + (-1)^n ): n in [0..30]]; // Vincenzo Librandi, Aug 19 2011
  • Maple
    A003665:=n->2^(n-1)*( 2^n + (-1)^n ): seq(A003665(n), n=0..30); # Wesley Ivan Hurt, Apr 28 2017
  • Mathematica
    CoefficientList[Series[(1+8x)/(1-2x-8x^2), {x,0,30}], x] (* or *)
LinearRecurrence[{2,8}, {1,1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)
  • PARI
    a(n)=2^(n-1)*( 2^n + (-1)^n );
  • Sage
    [2^(n-1)*(2^n +(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 02 2019

Formula

From Paul Barry, Mar 01 2003: (Start)
a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1.
a(n) = (4^n + (-2)^n)/2.
G.f.: (1-x)/((1+2*x)*(1-4*x)). (End)
From Paul Barry, Apr 05 2003: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*9^k.
E.g.f. exp(x)*cosh(3*x). (End)
a(n) = (A078008(n) + A001045(n+1))*2^(n-1) = A014551(n)*2^(n-1). - Paul Barry, Feb 12 2004
Given a(0)=1, b(0)=1 then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard, Sep 25 2005
a(n) = Sum_{k=0..n} A098158(n,k)*9^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = ((1+sqrt(9))^n + (1-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
If p[1]=1, and p[i]=9, (i>1), and if A is 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, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(9*k-1)/(x*(9*k+8) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

Extensions

Entry revised by N. J. A. Sloane, Nov 22 2006

A120612 For n>1, a(n) = 2*a(n-1) + 15*a(n-2); a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 17, 49, 353, 1441, 8177, 37969, 198593, 966721, 4912337, 24325489, 122336033, 609554401, 3054149297, 15251614609, 76315468673, 381405156481, 1907542343057, 9536162033329, 47685459212513, 238413348924961, 1192108586037617, 5960417405949649
Offset: 0

Views

Author

Gary W. Adamson, Jun 17 2006

Keywords

Comments

Characteristic polynomial of matrix M = x^2 - 2x - 15. a(n)/a(n-1) tends to 5, largest eigenvalue of M and a root of the characteristic polynomial.
Binomial transform of [1, 0, 16, 0, 256, 0, 4096, 0, 65536, 0, ...]=: powers of 16 (A001025) with interpolated zeros. - Philippe Deléham, Dec 02 2008
a(n) is the number of compositions of n when there are 1 type of 1 and 16 types of other natural numbers. - Milan Janjic, Aug 13 2010

Examples

			a(4) = 353 = 2*49 + 15*17 = 2*a(3) + 15*a(2).

Links

Crossrefs

Cf. A005059, A081186, A059841.

Programs

  • Mathematica
    Table[(5^n+(-1)^n*3^n)/2,{n,1,30}] (* Alexander Adamchuk, Aug 31 2006 *)
a[n_] := (5^n + (-3)^n)/2; Array[a, 24, 0] (* Or *)
CoefficientList[Series[(1 + 15 x)/(1 - 2 x - 15 x^2), {x, 0, 23}], x] (* Or *)
LinearRecurrence[{2, 15}, {1, 1}, 25] (* Or *)
Table[ MatrixPower[{{1, 2}, {8, 1}}, n][[1, 1]], {n, 0, 30}]  (* Robert G. Wilson v, Sep 18 2013 *)
  • PARI
    a(n)=([1,4; 4,1]^n)[1,1] \\ Charles R Greathouse IV, Oct 16 2013
  • PARI
    concat(1, Vec((15*x+1)/(-15*x^2-2*x+1) + O(x^100))) \\ Colin Barker, Mar 12 2014
  • PARI
    a(n) = ( 5^n + (-1)^n * 3^n ) / 2 \\ Charles R Greathouse IV, May 18 2015

Formula

Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term.
From Alexander Adamchuk, Aug 31 2006: (Start)
a(n) = ( 5^n + (-1)^n * 3^n ) / 2.
a(2n+1) = A005059(2n+1).
a(2n) = A081186(2n). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*16^(n-k). - Philippe Deléham, Dec 26 2007
If p(1)=1, and p(i)=16, (i > 1), and if A is 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, Apr 29 2010

Extensions

More terms from Alexander Adamchuk, Aug 31 2006
Entry revised by Philippe Deléham, Dec 02 2008
More terms from Colin Barker, Mar 12 2014

A090042 a(n) = 2*a(n-1) + 11*a(n-2) for n > 1, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 13, 37, 217, 841, 4069, 17389, 79537, 350353, 1575613, 7005109, 31341961, 139740121, 624241813, 2785624957, 12437909857, 55517694241, 247852396909, 1106399430469, 4939175226937, 22048744189033, 98428415874373, 439393017828109, 1961498610274321, 8756320416657841
Offset: 0

Views

Author

Paul Barry, Nov 20 2003

Keywords

Comments

Binomial transform of A001021 (powers of 12), with interpolated zeros.
For n > 0, a(n) = term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 4,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 12 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

Crossrefs

Cf. A049310, A015520, A084603.

Programs

  • GAP
    a := [1, 1];; for n in [3..30] do a[n] := 2*a[n-1]+ 11*a[n-2]; od; a; # Muniru A Asiru, Feb 18 2018
  • Magma
    I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +11*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019
  • Maple
    a := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n>=2 then 2*procname(n-1) + 11*procname(n-2) fi; end:
seq(a(n), n=0..25); # Muniru A Asiru, Feb 18 2018
  • Mathematica
    a[n_]:= Simplify[((1+Sqrt[12])^n +(1-Sqrt[12])^n)/2]; Array[a, 30, 0] (* or *)
CoefficientList[Series[(x-1)/(11x^2+2x-1), {x,0,30}], x] (* or *)
Table[ MatrixPower[{{1, 2}, {6, 1}}, n][[1, 1]], {n, 0, 30}] (* Robert G. Wilson v, Sep 18 2013 and modified per Wolfdieter Lang Feb 17 2018 *)
LinearRecurrence[{2, 11}, {1, 1}, 30] (* Ray Chandler, Aug 01 2015 *)
  • PARI
    x='x+O('x^30); Vec((1-x)/(1-2*x-11*x^2)) \\ Altug Alkan, Feb 17 2018
  • Sage
    ((1-x)/(1-2*x-11*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

E.g.f.: exp(x)*cosh(2*sqrt(3)*x).
a(n) = ((1 + 2*sqrt(3))^n + (1 - 2*sqrt(3))^n)/2.
a(n) = Sum_{k=0..n} A098158(n,k)*12^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=12, (i>1), and if A is 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, Apr 29 2010
From Wolfdieter Lang, Feb 17 2018: (Start)
G.f.: (1-x)/(1 - 2*x - 11*x^2). (See the Mathematica program.)
a(n) = b(n+1) - b(n), with b(n) = A015520(n). This leads to the Binet-de Moivre type formula given in the Mathematica program.
a(n) = (i*sqrt(11))^n*(S(n,-2*i/sqrt(11)) + (i/sqrt(11))*S(n-1,-2*i/sqrt(11))), n >= 0, with Chebyshev S polynomials (coefficients in A049310), with S(-2, x) = -1, S(-1, x) = 0 and i = sqrt(-1). Via Cayley-Hamilton. See the Gary W. Adamson comment above or the Mathematica program of Robert G. Wilson v with another matrix. (End)
From Peter Bala, Jan 07 2022: (Start)
a(n) = [x^n] (x + sqrt(1 + 12*x^2))^n.
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
O.g.f.: 1 + x*d/dx(log(B(x))), where B(x) = 1/sqrt(1 - 2*x - 11*x^2) is the o.g.f. of A084603. (End)

A090139 a(n) = 10*a(n-1) - 20*a(n-2), a(0)=1,a(1)=5.

Original entry on oeis.org

1, 5, 30, 200, 1400, 10000, 72000, 520000, 3760000, 27200000, 196800000, 1424000000, 10304000000, 74560000000, 539520000000, 3904000000000, 28249600000000, 204416000000000, 1479168000000000, 10703360000000000
Offset: 0

Views

Author

Paul Barry, Nov 22 2003

Keywords

Comments

Fifth binomial transform of (1, 0, 5, 0, 25, 0, ...).

Links

Programs

  • GAP
    a:=[1,5];; for n in [3..30] do a[n]:=10*a[n-1]-20*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019
  • Magma
    I:=[1,5]; [n le 2 select I[n] else 10*Self(n-1) -20*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019
  • Mathematica
    LinearRecurrence[{10, -20}, {1,5}, 30] (* G. C. Greubel, Aug 02 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-5*x)/(1-10*x+20*x^2)) \\ G. C. Greubel, Aug 02 2019
  • Sage
    ((1-5*x)/(1-10*x+20*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

a(n) = ((5-sqrt(5))^n + (5+sqrt(5))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k) * 5^(n-k).
a(n) = Sum_{k=0..n} C(n, k) * 5^(n-k/2) * (1+(-1)^k)/2.
a(n) = Sum_{k=0..n} 5^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
G.f.: (1-5*x)/(1-10*x+20*x^2). - G. C. Greubel, Aug 02 2019

A125816 a(n) = ((1+sqrt(13))^n + (1-sqrt(13))^n)/2.

Original entry on oeis.org

1, 1, 14, 40, 248, 976, 4928, 21568, 102272, 463360, 2153984, 9868288, 45584384, 209588224, 966189056, 4447436800, 20489142272, 94347526144, 434564759552, 2001299832832, 9217376780288, 42450351554560, 195509224472576
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2006

Keywords

Comments

Binomial transform of A001022(powers of 13), with interpolated zeros. - Philippe Deléham, Dec 20 2007
a(n-1) is the number of compositions of n when there are 1 type of 1 and 13 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

Crossrefs

Cf. A127262. First differences of A091914.

Programs

  • GAP
    a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+12*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019
  • Magma
    I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +12*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019
  • Mathematica
    Expand[Table[((1+Sqrt[13])^n +(1-Sqrt[13])^n)/(2), {n,0,30}]] (* Artur Jasinski *)
LinearRecurrence[{2,12}, {1,1}, 30] (* G. C. Greubel, Aug 02 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-x)/(1-2*x-12*x^2)) \\ G. C. Greubel, Aug 02 2019
  • Sage
    ((1-x)/(1-2*x-12*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

From Philippe Deléham, Dec 12 2006: (Start)
a(n) = 2*a(n-1) + 12*a(n-2), with a(0)=a(1)=1.
G.f.: (1-x)/(1-2*x-12*x^2). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*13^(n-k). - Philippe Deléham, Dec 20 2007
If p[1]=1, and p[i]=13, (i>1), and if A is 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+1)=det A. - Milan Janjic, Apr 29 2010

A125818 a(n) = ((1 + 3*sqrt(2))^n + (1 - 3*sqrt(2))^n)/2.

Original entry on oeis.org

1, 1, 19, 55, 433, 1801, 10963, 52543, 291457, 1476145, 7907059, 40908583, 216237169, 1127920249, 5931872371, 31038388975, 162918608257, 853489829089, 4476595998547, 23462519091607, 123027170158513, 644917164874345, 3381296222443411, 17726184247750687
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2006

Keywords

Comments

Binomial transform of [1, 0, 18, 0, 324, 0, 5832, 0, 104976, 0, ...] =: powers of 18 (A001027) with interpolated zeros. - Philippe Deléham, Dec 02 2008
a(n-1) is the number of compositions of n when there are 1 type of 1 and 18 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

Crossrefs

Cf. A125817.

Programs

  • GAP
    a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+17*a[n-2]; od; a; # G. C. Greubel, Aug 03 2019
  • Magma
    I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +17*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 03 2019
  • Mathematica
    Expand[Table[((1+3*Sqrt[2])^n +(1-3*Sqrt[2])^n)/2, {n,0,30}]]
(* alternate program *)
LinearRecurrence[{2, 17}, {1, 1}, 30] (* T. D. Noe, Mar 28 2012 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-x)/(1-2*x-17*x^2)) \\ G. C. Greubel, Aug 03 2019
  • Sage
    ((1-x)/(1-2*x-17*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

Formula

From Philippe Deléham, Dec 12 2006: (Start)
a(n) = 2*a(n-1) + 17*a(n-2), with a(0)=a(1)=1.
G.f.: (1-x)/(1-2*x-17*x^2). (End)
a(n) = Sum_{k=0..n} A098158(n,k)*18^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=18, (i>1), and if A is Hessenberg matrix of order n If p[1]=1, and p[i]=18, (i>1), and if A is 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+1)=det A. - Milan Janjic, Apr 29 2010

A133294 a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 12, 34, 188, 716, 3312, 13784, 60688, 259216, 1125312, 4842784, 20938688, 90305216, 389997312, 1683046784, 7266066688, 31362601216, 135385869312, 584397750784, 2522654194688, 10889285897216, 47005113741312
Offset: 0

Views

Author

Philippe Deléham, Dec 20 2007

Keywords

Comments

Binomial transform of [1, 0, 11, 0, 121, 0, 1331, 0, 14641, 0, ...]=: powers of 11 (A001020) with interpolated zeros. - Philippe Deléham, Dec 02 2008
A083101 is an essentially identical sequence (with a different start). - N. J. A. Sloane, Dec 31 2012

Links

Crossrefs

Cf. A083101, A090042.

Programs

  • GAP
    a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+10*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019
  • Magma
    I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019
  • Mathematica
    a[n_]:= Simplify[((1+Sqrt[11])^n + (1-Sqrt[11])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-10x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 10}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-x)/(1-2*x-10*x^2)) \\ G. C. Greubel, Aug 02 2019
  • Sage
    ((1-x)/(1-2*x-10*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

a(n) = Sum_{k=0..n} A098158(n,k)*11^(n-k).
G.f.: (1-x)/(1-2*x-10*x^2).
a(n) = A083101(n-1) for n >= 1.
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

Extensions

Terms a(23) onward added by G. C. Greubel, Aug 02 2019

A133343 a(n) = 2*a(n-1) + 13*a(n-2), for n>1, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 15, 43, 281, 1121, 5895, 26363, 129361, 601441, 2884575, 13587883, 64675241, 305992961, 1452764055, 6883436603, 32652805921, 154790287681, 734067052335, 3480407844523, 16503687369401, 78252676717601
Offset: 0

Views

Author

Philippe Deléham, Dec 21 2007

Keywords

Comments

Binomial transform of A001023 (powers of 14), with interpolated zeros.
a(n) is the number of compositions of n when there are 1 type of 1 and 14 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

Crossrefs

Cf. A001023, A098158.

Programs

  • Magma
    [n le 2 select 1 else 2*Self(n-1) +13*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 15 2022
  • Mathematica
    f[n_]:= Simplify[((1+Sqrt[14])^n + (1-Sqrt[14])^n)/2]; Array[f, 25, 0] (* Or *)
CoefficientList[Series[(1+13x)/(1-2x-13x^2), {x, 0, 23}], x] (* Or *)
LinearRecurrence[{2, 13}, {1, 1}, 25] (* Or *)
Table[ MatrixPower[{{1, 2}, {7, 1}}, n][[1, 1]], {n, 0, 30}]  (* Robert G. Wilson v, Sep 18 2013 *)
  • PARI
    Vec((1-x)/(1-2*x-13*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012
  • SageMath
    A133343=BinaryRecurrenceSequence(2,13,1,1)
[A133343(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

Formula

G.f.: (1-x)/(1-2*x-13*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*14^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=14, (i>1), and if A is 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, Apr 29 2010
a(n) = (b*i)^(n-1)*(b*i*ChebyshevU(n, -i/b) - ChebyshevU(n-1, -i/b)), with b = sqrt(13). - G. C. Greubel, Oct 15 2022

A139011 Real part of (2 + i)^n, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882
Offset: 0

Views

Author

Gary W. Adamson, Apr 05 2008

Keywords

Comments

Imaginary part of (2 + i)^n gives A099456.
Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495.
Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008

Examples

			1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ...
a(5) = -38 since (2 + i)^5 = (-38 + 41*i).
a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5).
a(5) = -38 = A006496(5).

Links

Crossrefs

Cf. A099456, A006495, A006496, A056594, A146559 (inv bin. transf.).

Programs

  • Magma
    [ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011
  • Maple
    restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..29 ); # Zerinvary Lajos, Apr 06 2009
  • Mathematica
    Re[(2+I)^Range[0,30]] (* or *) LinearRecurrence[{4,-5},{1,2},30] (* Harvey P. Dale, Nov 02 2022 *)
  • PARI
    a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */
  • PARI
    Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017
  • Sage
    [lucas_number2(n,4,5)/2 for n in range(0,31)] # Zerinvary Lajos, Jul 08 2008

Formula

Real part of (2 + i)^n, i^2 = -1.
Term (1,1) of matrix [2,-1; 1,2]^n.
(a(n))^2 + (A099456(n))^2 = 5^n.
From R. J. Mathar, Apr 06 2008: (Start)
O.g.f.: (1-2x) /(1-4x+5x^2).
a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End)
E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009
a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010
a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008
2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011
a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*binomial(n,2*k). - Gerry Martens, Sep 18 2022

Extensions

Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009
Added a(0)=1 by Michael Somos, Dec 26 2010
Edited by Franklin T. Adams-Watters, Apr 10 2011

A143079 a(n) = ((9+sqrt(9))^n + (9-sqrt(9))^n)/2.

Original entry on oeis.org

1, 9, 90, 972, 11016, 128304, 1516320, 18055872, 215830656, 2584929024, 30988915200, 371685583872, 4459138615296, 53503133036544, 641998414356480, 7703745879785472, 92443540002471936, 1109314016699940864
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Oct 15 2008

Keywords

Comments

Equals A007318^9 * [1, 0, 9, 0, 81, 0, 729, ...]. - Gary W. Adamson, Oct 23 2008

Links

Programs

  • PARI
    A143079(n)=3^n*(1+1<M. F. Hasler, Oct 22 2008

Formula

From Philippe Deléham, Oct 18 2008: (Start)
a(n) = 18*a(n-1) - 72*a(n-2).
a(n) = 6^n*(2^n+1)/2.
G.f.: (1-9x)/(1-18x+72x^2). (End)
From R. J. Mathar, Oct 21 2008: (Start)
a(n) = (12^n + 6^n)/2.
G.f.: (1-9x)/((1-12x)(1-6x)). (End)
a(n) = 3^n*A007582(n) = (6^n+12^n)/2 = A000051(n)*A000079(n)*A000244(n)/2. - M. F. Hasler, Oct 22 2008
A007318^9 * [1, 0, 9, 0, 81, 0, 729, ...] produces this sequence. - Gary W. Adamson, Oct 23 2008
a(n) = Sum_{k=0..n} A098158(n,k)*9^k. - Philippe Deléham, Oct 23 2008

Extensions

Extended by R. J. Mathar and M. F. Hasler, Oct 21 2008
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