A137241 -id:A137241 - cp's OEIS frontend

A140359 a(n) = 2a(n-1) + a(n-2) - 2a(n-3).

1, 1, 6, 11, 26, 51, 106, 211, 426, 851, 1706, 3411, 6826, 13651, 27306, 54611, 109226, 218451, 436906, 873811, 1747626, 3495251, 6990506, 13981011, 27962026, 55924051, 111848106, 223696211, 447392426, 894784851, 1789569706, 3579139411

Offset: 0

Paul Curtz, Jun 24 2008

nonn

This is the sequence A(1,1;1,2;3) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A001045, A087288, A137241, A140360.

[(5*2^(n+1) -9 + 5*(-1)^n)/6: n in [0..50]]; // G. C. Greubel, Oct 10 2017

Table[(5*2^(n+1) -9 + 5*(-1)^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 10 2017 *)
LinearRecurrence[{2,1,-2},{1,1,6},40] (* Harvey P. Dale, Mar 24 2021 *)

for(n=0,50, print1((5*2^(n+1) -9 + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Oct 10 2017

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - a(n) = 5*A001045(n), Jacobsthal numbers.
a(n+1) - 2*a(n) = (-1)^(n+1)* A010685(n).
From R. J. Mathar, Jul 10 2008: (Start)
O.g.f.: (1-x+3*x^2)/((x-1)*(2*x-1)*(1+x)).
a(n) = (5*2^(n+1) - 9 + 5*(-1)^n)/6. (End)
a(n) = a(n-1) + 2*a(n-2) +3, n>1 - Gary Detlefs, Jun 20 2010

Extended by R. J. Mathar, Jul 10 2008

A140428 a(n) = A000045(n) + A113405(n).

Original entry on oeis.org

0, 1, 1, 3, 5, 9, 15, 27, 49, 91, 169, 317, 599, 1143, 2197, 4251, 8269, 16161, 31711, 62435, 123273, 243963, 483745, 960725, 1910503, 3803295, 7577933, 15109499, 30143973, 60166553, 120136687, 239955563, 479396897, 957961755, 1914577241

Offset: 0

Paul Curtz, Jun 19 2008

The inverse binomial transform yields the sequence (-1)^(n+1)*a(n). This property is inherited from the A000045 and A113405 sequences, which have the same property individually. The same sign flipping behavior under inverse binomial transform is found in A001045 and for the sequence with two zeros followed by A000975.

This is often, but not here, related to the recurrences a(n)=2a(n-1)+a(n-2)-2a(n-3) associated with denominators 1-2x-x^2+2x^3=(x-1)(2x-1)(x+1) in the o.g.f., which transform into the similar -(x-1)(2x+1)/(1+x)^4 under the inverse binomial transform, see A137241.

			a(n) and the repeated differences in the followup rows are:
    0,   1,   1,   3,   5,   9,  15, ...
    1,   0,   2,   2,   4,   6,  12, ...
   -1,   2,   0,   2,   2,   6,  10, ...
    3,  -2,   2,   0,   4,   4,  10, ...
   -5,   4,  -2,   4,   0,   6,   6, ...
    9,  -6,   6,  -4,   6,   0,  12, ...
  -15,  12, -10,  10,  -6, -12,   0, ...
The main diagonal consists of zeros.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,3,-1,-2).

I:=[0,1,1,3,5,9]; [n le 6 select I[n] else 3*Self(n-1)-Self(n-2) -3*Self(n-3)+3*Self(n-4)-Self(n-5)-2*Self(n-6): n in [1..30]]; // G. C. Greubel, Jan 15 2018

CoefficientList[Series[-x (1 - 2 x - 3 x^4 + x^2)/((1 - x - x^2) (2 x - 1) (1 + x) (x^2 - x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 11 2017 *)
LinearRecurrence[{3,-1,-3,3,-1,-2}, {0,1,1,3,5,9}, 30] (* G. C. Greubel, Jan 15 2018 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -2,-1,3,-3,-1,3]^n*[0;1;1;3;5;9])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

O.g.f.: -x*(1-2*x-3*x^4+x^2)/((1-x-x^2)*(2*x-1)*(1+x)*(x^2-x+1)). - R. J. Mathar, Jul 10 2008

a(n)= -A128834(n)/3 + 2^n/9 + A000045(n) - (-1)^n/9. - R. J. Mathar, Jul 10 2008

Edited and extended by R. J. Mathar, Jul 10 2008

A136161 a(n) = 2*a(n-3) - a(n-6), starting a(0..5) = 0, 5, 2, 1, 3, 1.

Original entry on oeis.org

0, 5, 2, 1, 3, 1, 2, 1, 0, 3, -1, -1, 4, -3, -2, 5, -5, -3, 6, -7, -4, 7, -9, -5, 8, -11, -6, 9, -13, -7, 10, -15, -8, 11, -17, -9, 12, -19, -10, 13, -21, -11, 14, -23, -12, 15, -25, -13, 16, -27, -14

Offset: 0

Paul Curtz, Mar 16 2008

Consider the general recurrence a(n) = k*a(n-1) + (5-2*k)*a(n-2) + (2-k)*a(n-3). The coefficients, in k, can be used to form the triple (k, 5-2*k, 2-k). Each triple is associated with a sequence, for example (0, 5, 2) leads to A111108, A112685, ..., (1, 3, 1) leads to A051927, A097075, ..., and so on. This sequence is formed from the triples {(0, 5, 2), (1, 3, 1), (2, 1, 0), (3, -1, -1), (4, -3, -2), ...}, for k >= 0. (Comment modified by G. C. Greubel, Dec 31 2023).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A097076, A099254, A112685, A135138, A135997, A137241.

Cf. A051927, A097075, A111108, A112685.

I:=[0,5,2,1,3,1]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..60]]; // G. C. Greubel, Dec 26 2023

LinearRecurrence[{0,0,2,0,0,-1},{0,5,2,1,3,1},60] (* Harvey P. Dale, Aug 16 2012 *)
Table[PadRight[{n, 5-2*n, 2-n}], {n,0,20}]//Flatten (* _G. C. Greubel, Dec 26 2023 *)

Vec(x*(5+2*x+x^2-7*x^3-3*x^4)/((1-x)^2*(1+x+x^2)^2+O(x^99))) \\ Charles R Greathouse IV, Jul 06 2011

def a(n): # a = A136161
    if n<6: return (0,5,2,1,3,1)[n]
    else: return 2*a(n-3) - a(n-6)
[a(n) for n in range(61)] # G. C. Greubel, Dec 26 2023

G.f.: x*(5+2*x+x^2-7*x^3-3*x^4) / ( (1-x)^2*(1+x+x^2)^2 ). - R. J. Mathar, Jul 06 2011
a(3n) = n.
a(3n+1) = 5 - 2*n.
a(3n+3) = 2 - n.
a(n) = (1/9)*( 27 - 2*(n+1) - 34*ChebyshevU(n, -1/2) + (-1)^n*(9*A099254(n) - 6*A099254(n-1)) ). - G. C. Greubel, Dec 26 2023

A136249 a(n)=-a(n-1)+4a(n-2)+4a(n-3).

Original entry on oeis.org

4, -2, 1, 7, -11, 43, -59, 187, -251, 763, -1019, 3067, -4091, 12283, -16379, 49147, -65531, 196603, -262139, 786427, -1048571, 3145723, -4194299, 12582907, -16777211, 50331643, -67108859, 201326587, -268435451, 805306363, -1073741819, 3221225467

Offset: 0

Paul Curtz, Mar 17 2008

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 4, 4).

Cf. A137241, A135529.

[2^(n-2)+5*(-1)^n*(1-2^(n-2)): n in [0..40]]; // Vincenzo Librandi, Aug 09 2011

LinearRecurrence[{-1,4,4},{4,-2,1},50] (* or *) Table[(5(-2)^n- 40(-1)^n+2^n)/8,{n,50}] (* Harvey P. Dale, Jun 10 2011 *)

a(2*n)=5-2^(2*n), a(2*n+1)=10-3*a(2n).
a(n)+a(n+1)=A135520(n).
a(n) = 1/6*2^n*a(0) + 1/4*2^n*a(1) - 1/2*a(0)*(-2)^n - 1/3*(-1)^n*a(2) - 1/4*a(1)*(-2)^n + 4/3*(-1)^n*a(0) + 1/4*(-2)^n*a(2) + 1/12*2^n*a(2). - Alexander R. Povolotsky, Mar 31 2008
G.f.: (4+2*x-17*x^2)/((1+2*x)*(1-2*x)*(1+x)). a(n)=2^(n-2)+5*(-1)^n*(1-2^(n-2)). - R. J. Mathar, Jun 15 2009
a(n)=(5*(-2)^n-40*(-1)^n+2^n)/8. - Harvey P. Dale, Jun 10 2011

Edited by N. J. A. Sloane, Apr 18 2008

More terms from Harvey P. Dale, Jun 10 2011

cp's OEIS Frontend

A140359 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).

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A140428 a(n) = A000045(n) + A113405(n).

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A136161 a(n) = 2*a(n-3) - a(n-6), starting a(0..5) = 0, 5, 2, 1, 3, 1.

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A136249 a(n)=-a(n-1)+4*a(n-2)+4*a(n-3).

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A140359 a(n) = 2a(n-1) + a(n-2) - 2a(n-3).

A136249 a(n)=-a(n-1)+4a(n-2)+4a(n-3).