A077943 -id:A077943 - cp's OEIS frontend

A075115 Binomial transform of A073145: a(n)=Sum(binomial(n,k)*A073145(k),(k=0,..,n)).

3, 2, 0, 2, 8, 12, 12, 16, 32, 56, 80, 112, 176, 288, 448, 672, 1024, 1600, 2496, 3840, 5888, 9088, 14080, 21760, 33536, 51712, 79872, 123392, 190464, 293888, 453632, 700416, 1081344, 1669120, 2576384, 3977216, 6139904, 9478144, 14630912

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 02 2002

a(n) is nonnegative since the real root of x^3-2*x^2+2*x-2 is dominant. - Michael Somos, Feb 28 2007

Michael De Vlieger, Table of n, a(n) for n = 0..5303
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 21.
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (2,-2,2).

Cf. A073145, A073498.

CoefficientList[Series[(3-4*x+2*x^2)/(1-2*x+2*x^2-2*x^3), {x, 0, 40}], x]
LinearRecurrence[{2,-2,2},{3,2,0},40] (* Harvey P. Dale, Jan 24 2019 *)

{a(n)= if(n<0, 0, polsym( x^3 -2*x^2 +2*x -2, n) [n+1])} /* Michael Somos, Feb 28 2007 */

a(n)=2a(n-1)-2a(n-2)+2a(n-3), a(0)=3, a(1)=2, a(2)=0.
G.f.: (3 - 4*x + 2*x^2)/(1 - 2*x + 2*x^2 - 2*x^3).
a(n) = 3*A077943(n) -4*A077943(n-1) +2*A077943(n-2). - R. J. Mathar, Mar 13 2021

A077993 Expansion of 1/(1+2x+2x^2+2*x^3).

Original entry on oeis.org

1, -2, 2, -2, 4, -8, 12, -16, 24, -40, 64, -96, 144, -224, 352, -544, 832, -1280, 1984, -3072, 4736, -7296, 11264, -17408, 26880, -41472, 64000, -98816, 152576, -235520, 363520, -561152, 866304, -1337344, 2064384, -3186688, 4919296, -7593984, 11722752, -18096128, 27934720, -43122688

Offset: 0

N. J. A. Sloane, Nov 17 2002

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-2).

Cf. A077943.

a:=[1,-2,2];; for n in [4..50] do a[n]:=-2*(a[n-1]+a[n-2]+a[n-3]); od; a; # G. C. Greubel, Jun 27 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+2*x+2*x^2+2*x^3) )); // G. C. Greubel, Jun 27 2019

LinearRecurrence[{-2,-2,-2}, {1,-2,2}, 50] (* or *) CoefficientList[ Series[1/(1+2*x+2*x^2+2*x^3), {x,0,50}], x] (* G. C. Greubel, Jun 27 2019 *)

my(x='x+O('x^50)); Vec(1/(1+2*x+2*x^2+2*x^3)) \\ G. C. Greubel, Jun 27 2019

(1/(1+2*x+2*x^2+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(n) = (-1)^n * A077943(n). - R. J. Mathar, Aug 04 2008

A078003 Expansion of (1-x)/(1-2x+2x^2-2*x^3).

Original entry on oeis.org

1, 1, 0, 0, 2, 4, 4, 4, 8, 16, 24, 32, 48, 80, 128, 192, 288, 448, 704, 1088, 1664, 2560, 3968, 6144, 9472, 14592, 22528, 34816, 53760, 82944, 128000, 197632, 305152, 471040, 727040, 1122304, 1732608, 2674688, 4128768, 6373376, 9838592, 15187968, 23445504, 36192256

Offset: 0

N. J. A. Sloane, Nov 17 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2).

Cf. A077943.

a:=[1,1,0];; for n in [4..50] do a[n]:=2*(a[n-1]-a[n-2]+a[n-3]); od; a; # G. C. Greubel, Jun 27 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x)/( 1-2*x+2*x^2-2*x^3) )); // G. C. Greubel, Jun 27 2019

LinearRecurrence[{2,-2,2}, {1,1,0}, 50] (* or *) CoefficientList[
Series[(1-x)/(1-2*x+2*x^2-2*x^3), {x,0,50}], x] (* G. C. Greubel, Jun 27 2019 *)

Vec((1-x)/(1-2*x+2*x^2-2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012

((1-x)/(1-2*x+2*x^2-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(n) = A077943(n) - A077943(n-1). - R. J. Mathar, Aug 04 2008

A104767 a(n)=n for n <= 3, a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) for n >= 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 10, 16, 24, 36, 56, 88, 136, 208, 320, 496, 768, 1184, 1824, 2816, 4352, 6720, 10368, 16000, 24704, 38144, 58880, 90880, 140288, 216576, 334336, 516096, 796672, 1229824, 1898496, 2930688, 4524032, 6983680, 10780672, 16642048, 25690112, 39657472

Offset: 0

Don N. Page, Oct 13 2005

nonn

Also a(n) for n > 0 is the number of terms in the expansion of (x - y) * (x - y) * (x^2 - y^2) * (x^3 - y^3) * ... * (x^F_n-1 - y^F_n-1), where F_n is the n-th Fibonacci number. In this definition one can take y=1. In other words the sequence gives the number of nonzero terms in the polynomial Product {k=1..n-1}, (1 - x^F_k). - Robert G. Wilson v, May 12 2013

Also a(n) for n > 0 is the number of terms in the expansion of Product_{k=2..n+1} (x^F_k - y^F_k) with coefficient +1 (same with -1). We can take y=1 and the Product_{k=2..n+1} (x^F_k - 1) has a(n) terms with coefficient +1 and same with -1. Note that no coefficient is greater than 1 in absolute value. - Michael Somos, May 17 2018

			From _Michael Somos_, May 17 2018: (Start)
For n=3, (x - y) * (x - y) = x^2 - 2*x*y + y^2 has a(3) = 3 terms.
For n=4, (x - y) * (x - y) * (x^2 - y^2) = x^4 - 2*x^3*y + 2*x*y^3 - y^4 has a(4) = 4 terms.
for n=2, (x - y) * (x^2 - y^2) = x^3 - x^2*y - x*y^2 + y^3 has a(2) = 2 terms with + sign and also with - sign.
For n=3, (x - y) * (x^2 - y^2) * (x^3 - y^3) = x^6 - x^5*y - x^4*y^2 + x^2*y^4 + x*y^5 - y^6 has a(3) = 3 terms with + sign and also with - sign. (End)

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Richard P. Stanley, Theorems and Conjectures on Some Rational Generating Functions, arXiv:2101.02131 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2).

Cf. A093996.

a:=[0,1,2,3,4];; for n in [5..50] do a[n]:=2*a[n-1]-2*a[n-2]+2*a[n-3]; od; a; # Muniru A Asiru, May 17 2018

f:=proc(n) option remember; if n <= 4 then RETURN(n); fi; 2*f(n-4)+f(n-1); end;

a[n_] := a[n] = If[n < 4, n, 2a[n - 1] - 2a[n - 2] + 2a[n - 3]]; Table[ a[n], {n, 0, 39}] (* Robert G. Wilson v *)
Join[{0}, LinearRecurrence[{2, -2, 2}, {1, 2, 3}, 41]] (* Robert G. Wilson v, May 12 2013 *)
Join[{0}, LinearRecurrence[{1, 0, 0, 2}, {1, 2, 3, 4}, 41]] (* Robert G. Wilson v, May 12 2013 *)
a[n_] := Length@ ExpandAll@ Product[1 - x^Fibonacci[k], {k, n-1}]; a[1] = 1; (* Robert G. Wilson v, May 12 2013 *)
nxt[{a_,b_,c_}]:={b,c,2c-2b+2a}; Join[{0},NestList[nxt,{1,2,3},40][[All,1]]] (* Harvey P. Dale, Nov 30 2021 *)

a=vector(100); a[1]=1;a[2]=2;a[3]=3; for(n=4, #a, a[n] = 2*a[n-1]-2*a[n-2]+2*a[n-3]); concat(0,a) \\ Altug Alkan, May 18 2018

a(n) = n for n <= 4; for n >= 5, a(n) = 2a(n-4) + a(n-1).

G.f.: (x + x^3)/(-2*x^3 + 2*x^2 - 2*x + 1). a(n) = A077943(n-3) + A077943(n-1).

More terms from Robert G. Wilson v, Oct 14 2005

A077858 Expansion of (1-x)^(-1)/(1-2x+2x^2-2*x^3).

Original entry on oeis.org

1, 3, 5, 7, 11, 19, 31, 47, 71, 111, 175, 271, 415, 639, 991, 1535, 2367, 3647, 5631, 8703, 13439, 20735, 31999, 49407, 76287, 117759, 181759, 280575, 433151, 668671, 1032191, 1593343, 2459647, 3796991, 5861375, 9048063, 13967359, 21561343, 33284095, 51380223

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-2).

Cf. A077943.

CoefficientList[Series[(1-x)^(-1)/(1-2x+2x^2-2x^3), {x,0,40}], x]  (* Harvey P. Dale, Mar 31 2011 *)

a(n)-a(n-1) = A077943(n). - R. J. Mathar, Mar 13 2021

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A075115 Binomial transform of A073145: a(n)=Sum(binomial(n,k)*A073145(k),(k=0,..,n)).

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A077993 Expansion of 1/(1+2*x+2*x^2+2*x^3).

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A078003 Expansion of (1-x)/(1-2*x+2*x^2-2*x^3).

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A104767 a(n)=n for n <= 3, a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) for n >= 4.

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A077858 Expansion of (1-x)^(-1)/(1-2*x+2*x^2-2*x^3).

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A077993 Expansion of 1/(1+2x+2x^2+2*x^3).

A078003 Expansion of (1-x)/(1-2x+2x^2-2*x^3).

A077858 Expansion of (1-x)^(-1)/(1-2x+2x^2-2*x^3).