A161941 -id:A161941 - cp's OEIS frontend

A164095 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

5, 6, 10, 12, 20, 24, 40, 48, 80, 96, 160, 192, 320, 384, 640, 768, 1280, 1536, 2560, 3072, 5120, 6144, 10240, 12288, 20480, 24576, 40960, 49152, 81920, 98304, 163840, 196608, 327680, 393216, 655360, 786432, 1310720, 1572864, 2621440, 3145728

Offset: 1

Klaus Brockhaus, Aug 10 2009

nonn

Interleaving of A020714 and A007283 without initial term 3.
Partial sums are in A164096.
Binomial transform is A048655 without initial 1, second binomial transform is A161941 without initial 2, third binomial transform is A164037, fourth binomial transform is A161731 without initial 1, fifth binomial transform is A164038, sixth binomial transform is A164110.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Cf. A020714 (5*2^n), A007283 (3*2^n), A164096, A048655, A161941, A164037, A161731, A164038, A164110, A070876.

[ n le 2 select n+4 else 2*Self(n-2): n in [1..40] ];

LinearRecurrence[{0,2},{5,6},50] (* or *) With[{nn=20},Riffle[NestList[ 2#&,5,nn],NestList[2#&,6,nn]]] (* Harvey P. Dale, Aug 15 2020 *)

a(n) = A070876(n)/3.
a(n) = (4-(-1)^n)*2^(1/4*(2*n-1+(-1)^n)).
G.f.: x*(5+6*x)/(1-2*x^2).

A161944 a(n) = ((4+sqrt(2))(3+sqrt(2))^n + (4-sqrt(2))(3-sqrt(2))^n)/4.

Original entry on oeis.org

2, 7, 28, 119, 518, 2275, 10024, 44219, 195146, 861343, 3802036, 16782815, 74082638, 327016123, 1443518272, 6371996771, 28127352722, 124160138935, 548069364556, 2419295214791, 10679285736854, 47140647917587

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

Third binomial transform of A135530.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A135530, A161941 (second binomial transform of A135530).

a:=[2,7];; for n in [3..25] do a[n]:=6*a[n-1]-7*a[n-2]; od; a; # Muniru A Asiru, Apr 04 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

LinearRecurrence[{6,-7}, {2,7}, 50] (* G. C. Greubel, Apr 03 2018 *)
Table[((4+Sqrt[2])(3+Sqrt[2])^n+(4-Sqrt[2])(3-Sqrt[2])^n)/4,{n,0,30}]// Simplify (* Harvey P. Dale, Jun 03 2020 *)

x='x+O('x^30); Vec((2-5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Apr 03 2018

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 2; a(1) = 7.
G.f.: (2-5*x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - G. C. Greubel, Apr 03 2018

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009

A001882 a(2n) = a(2n-1) + 2a(2n-2), a(2n+1) = a(2n) + a(2n-1), with a(1) = 2 and a(2) = 3.

Original entry on oeis.org

2, 3, 5, 11, 16, 38, 54, 130, 184, 444, 628, 1516, 2144, 5176, 7320, 17672, 24992, 60336, 85328, 206000, 291328, 703328, 994656, 2401312, 3395968, 8198592, 11594560, 27991744, 39586304, 95569792, 135156096, 326295680, 461451776, 1114043136, 1575494912

Offset: 1

N. J. A. Sloane

nonn

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).

T. D. Noe, Table of n, a(n) for n = 1..500
Michael Fried et al., Problem E1738, Amer. Math. Monthly, 72 (1965), 1024-1025.
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 linear recurrences with constant coefficients, signature (0, 4, 0, -2).

Cf. A161941 (bisection).

A001882:=-(-2-3*z+3*z**2+z**3)/(1-4*z**2+2*z**4); # [Simon Plouffe in his 1992 dissertation for offset 0.]

a[1] = 2; a[2] = 3; a[n_] := a[n] = If[EvenQ[n], a[n-1] + 2*a[n-2], a[n-1] + a[n-2]]; Table[a[n], {n, 50}] (* T. D. Noe, Aug 10 2012 *)

x='x+O('x^50); Vec((2+3*x-3*x^2-x^3)/(1-4*x^2+2*x^4)) \\ G. C. Greubel, Aug 13 2017

G.f.: see Maple program.

Removed the attribute "conjectured" from the Plouffe g.f. R. J. Mathar, Aug 17 2009

A164037 Expansion of (5-9x)/(1-6x+7*x^2).

Original entry on oeis.org

5, 21, 91, 399, 1757, 7749, 34195, 150927, 666197, 2940693, 12980779, 57299823, 252933485, 1116502149, 4928478499, 21755355951, 96032786213, 423909225621, 1871225850235, 8259990522063, 36461362180733, 160948239429957

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Binomial transform of A161941 without initial 2. Third binomial transform of A164095. Inverse binomial transform of A161731 without initial 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A161941, A164095 (5, 6, 10, 12, 20, 24, ...), A161731.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(3+r)^n+(5-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

CoefficientList[Series[(5-9x)/(1-6x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{5,21},30] (* Harvey P. Dale, Apr 27 2017 *)

Vec((5-9*x)/(1-6*x+7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 5, a(1) = 21.
G.f.: (5-9*x)/(1-6*x+7*x^2).
a(n) = ((5+3*sqrt(2))*(3+sqrt(2))^n+(5-3*sqrt(2))*(3-sqrt(2))^n)/2.
E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009

cp's OEIS Frontend

A164095 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

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A161944 a(n) = ((4+sqrt(2))*(3+sqrt(2))^n + (4-sqrt(2))*(3-sqrt(2))^n)/4.

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A001882 a(2n) = a(2n-1) + 2a(2n-2), a(2n+1) = a(2n) + a(2n-1), with a(1) = 2 and a(2) = 3.

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A164037 Expansion of (5-9*x)/(1-6*x+7*x^2).

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Author

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A161944 a(n) = ((4+sqrt(2))(3+sqrt(2))^n + (4-sqrt(2))(3-sqrt(2))^n)/4.

A164037 Expansion of (5-9x)/(1-6x+7*x^2).