A091140 -id:A091140 - cp's OEIS frontend

A091141 a(n) = 2a(n-1) + 4a(n-2) - 2*a(n-3) with initial terms 1, 4, 13.

1, 4, 13, 40, 124, 382, 1180, 3640, 11236, 34672, 107008, 330232, 1019152, 3145216, 9706576, 29955712, 92447296, 285304288, 880486336, 2717295232, 8385927232, 25880062720, 79869243904, 246486884224, 760690618624, 2347590286336, 7244969278720, 22358918465536

Offset: 1

Gary W. Adamson, Dec 21 2003

One of 3 related sequences generated from finite difference operations. Let r(1)=s(1)=t(1)=1. Given r(n), s(n) and t(n), let f(x) = r(n) x^2 + s(n) x + t(n) and let r(n+1), s(n+1) and t(n+1) be the 0th, 1st and 2nd differences of f(x) at x=1. I.e., r(n+1) = f(1) = r(n)+s(n)+t(n), s(n+1) = f(2)-f(1) = 3r(n)+s(n) and t(n+1) = f(3)-2f(2)+f(1) = 2r(n). This sequence gives s(n).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,-2).

Cf. r(n) = A091140(n), t(n) = A091142(n).

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {2, 0, 0}}, n-1].{{1}, {1}, {1}})[[2, 1]]
LinearRecurrence[{2,4,-2},{1,4,13},30] (* Harvey P. Dale, Jun 19 2018 *)

Vec(x*(x+1)^2/(2*x^3-4*x^2-2*x+1) + O(x^100)) \\ Colin Barker, May 21 2015

Let v(n) be the column vector with elements r(n), s(n), t(n); then v(n) = [1 1 1 / 3 1 0 / 2 0 0] v(n-1).
The limit as n->infinity of a(n+1)/a(n) is the largest root of x^3 - 2x^2 - 4x + 2 = 0, which is about 3.086130197651494.
G.f.: x*(x+1)^2 / (2*x^3-4*x^2-2*x+1). - Colin Barker, May 21 2015

A091142 a(n) = 2a(n-1) + 4a(n-2) - 2*a(n-3) with initial terms 1, 2, 6.

Original entry on oeis.org

1, 2, 6, 18, 56, 172, 532, 1640, 5064, 15624, 48224, 148816, 459280, 1417376, 4374240, 13499424, 41661056, 128571328, 396788032, 1224539264, 3779088000, 11662756992, 35992787456, 111078426880, 342802489600, 1057933111808, 3264919328256, 10075966124544

Offset: 1

Gary W. Adamson, Dec 21 2003

One of 3 related sequences generated from finite difference operations. Let r(1)=s(1)=t(1)=1. Given r(n), s(n) and t(n), let f(x) = r(n) x^2 + s(n) x + t(n) and let r(n+1), s(n+1) and t(n+1) be the 0th, 1st and 2nd differences of f(x) at x=1. I.e. r(n+1) = f(1) = r(n)+s(n)+t(n), s(n+1) = f(2)-f(1) = 3r(n)+s(n) and t(n+1) = f(3)-2f(2)+f(1) = 2r(n). This sequence gives t(n).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,-2).

Cf. r(n) = A091140(n), s(n) = A091141(n).

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {2, 0, 0}}, n-1].{{1}, {1}, {1}})[[3, 1]]

Vec(-x*(2*x^2-1)/(2*x^3-4*x^2-2*x+1) + O(x^100)) \\ Colin Barker, May 21 2015

Let v(n) be the column vector with elements r(n), s(n), t(n); then v(n) = [1 1 1 / 3 1 0 / 2 0 0] v(n-1).
The limit as n->infinity of a(n+1)/a(n) is the largest root of x^3 - 2x^2 - 4x + 2 = 0, which is about 3.086130197651494.
G.f.: -x*(2*x^2-1) / (2*x^3-4*x^2-2*x+1). - Colin Barker, May 21 2015

A334293 First quadrisection of Padovan sequence.

Original entry on oeis.org

1, 0, 2, 5, 16, 49, 151, 465, 1432, 4410, 13581, 41824, 128801, 396655, 1221537, 3761840, 11584946, 35676949, 109870576, 338356945, 1042002567, 3208946545, 9882257736, 30433357674, 93722435101, 288627200960, 888855064897, 2737314167775, 8429820731201, 25960439030624

Offset: 0

Oboifeng Dira, Apr 21 2020

			For n=3, a(3) = 2*a(2) + 3*a(1) + a(0) = 2*2 + 3*0 + 1 = 5.

Colin Barker, Table of n, a(n) for n = 0..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (2,3,1).

Quadrisection of A000931.
Bisection (even part) of A099529.
Cf. A099098, A091140.

Vec((1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3) + O(x^30)) \\ Colin Barker, Apr 27 2020

a(n) = A000931(4n).
a(n) = A099529(2n).
a(n) = Sum_{k=0..n} binomial(2*n-k-1, 2*k-1).
a(n) = 2*a(n-1)+3*a(n-2)+a(n-3), a(0)=1, a(1)=0, a(2)=2 for n>=3.
G.f.: (1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3). - Colin Barker, Apr 27 2020

A095797 Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).

Original entry on oeis.org

1, 1, 1, 1, 4, 11, 14, 6, 35, 75, 70, 24, 204, 540, 570, 210, 1524, 3618, 3528, 1224, 9894, 25050, 25524, 9144, 69612, 169932, 168828, 59364, 467736, 1165908, 1175208, 417672, 3226524, 7947084, 7944648, 2806416, 21924672, 54371568, 54612456, 19359144, 150267840, 371199864

Offset: 0

Gary W. Adamson, Jun 06 2004

(n+1)-st set of 4 terms = leftmost finite differences of sequences generated from 3rd degree polynomials having n-th row coefficients, (given n = 1,2,3...) For example, first row is (1 1 1 1) with a corresponding polynomial x^3 + x^2 + x + 1. (f(x),x = 1,2,3...) = 4, 15, 40, 85, 156...Leftmost term of the sequence = 4, with finite difference rows: 11, 25, 45, 71...; 14, 20, 26, 32...; and 6, 6, 6, 6. Thus leftmost terms of the sequence 4, 15, 40...and the finite difference rows are (4 11 14 6) which is the second row.

The matrix generator is discussed in A028246, while 2nd degree polynomial examples are A091140, A091141 and A091140. The first degree case is A095795.

			3rd set of 4 terms = (35, 75, 70, 24) since M^2 * [1 1 1 1] = [35 75 70 24].
Array begins:
     1,    1,    1,   1;
     4,   11,   14,   6;
    35,   75,   70,  24;
   204,  540,  570, 210;
  1524, 3618, 3528,1224;
  9894,25050,25524,9144;

Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12).

Cf. A028246, A091140, A091141, A091142, A095795, A053698.

M := Matrix(4,4,[1,1,1,1,7,3,1,0,12,2,0,0,6,0,0,0]) ;
v := Vector(4,[1,1,1,1]) ;
for i from 0 to 20 do
        Mpr := (M ^ i).v ;
        for j from 1 to 4 do
                printf("%d,", Mpr[j]) ;
        end do;
end do; # R. J. Mathar, Jun 20 2011

LinearRecurrence[{0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12},{1,1,1,1,4,11,14,6,35,75,70,24,204,540,570,210},50] (* Harvey P. Dale, Feb 08 2013 *)

Vec((1+x+x^2+x^3+7*x^5+10*x^6+2*x^7-5*x^8+7*x^9-10*x^10-2*x^12 +6*x^13-16*x^14-24*x^11) / (1-4*x^4-24*x^8+30*x^12+12*x^16)+O(x^99)) \\ Charles R Greathouse IV, Jun 21 2011

G.f.: ( 1 +x +x^2 +x^3 +7*x^5 +10*x^6 +2*x^7 -5*x^8 +7*x^9 -10*x^10 -2*x^12 +6*x^13 -16*x^14 -24*x^11 ) / ( 1-4*x^4-24*x^8+30*x^12+12*x^16 ). - R. J. Mathar, Jun 20 2011

a(n) = +4*a(n-4) +24*a(n-8) -30*a(n-12) -12*a(n-16).

Name added by R. J. Mathar, several entries corrected by Charles R Greathouse IV, Jun 21 2011

cp's OEIS Frontend

A091141 a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 4, 13.

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A091142 a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 2, 6.

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A334293 First quadrisection of Padovan sequence.

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A095797 Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).

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A091141 a(n) = 2a(n-1) + 4a(n-2) - 2*a(n-3) with initial terms 1, 4, 13.

A091142 a(n) = 2a(n-1) + 4a(n-2) - 2*a(n-3) with initial terms 1, 2, 6.