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.

Showing 1-3 of 3 results.

A091140 a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 3, 9.

Original entry on oeis.org

1, 3, 9, 28, 86, 266, 820, 2532, 7812, 24112, 74408, 229640, 708688, 2187120, 6749712, 20830528, 64285664, 198394016, 612269632, 1889544000, 5831378496, 17996393728, 55539213440, 171401244800, 528966555904, 1632459664128, 5037983062272, 15547871669248
Offset: 1

Views

Author

Gary W. Adamson, Dec 21 2003

Keywords

Comments

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 r(n).

Links

Crossrefs

Cf. s(n) = A091141(n), t(n) = A091142(n).

Programs

  • Mathematica
    a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {2, 0, 0}}, n-1].{{1}, {1}, {1}})[[1, 1]]
LinearRecurrence[{2,4,-2},{1,3,9},30] (* Harvey P. Dale, May 18 2021 *)
  • PARI
    Vec(-x*(x^2-x-1)/(2*x^3-4*x^2-2*x+1) + O(x^100)) \\ Colin Barker, May 21 2015

Formula

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^2-x-1) / (2*x^3-4*x^2-2*x+1). - Colin Barker, May 21 2015

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

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Dec 21 2003

Keywords

Comments

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).

Links

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Gary W. Adamson, Jun 06 2004

Keywords

Comments

(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.

Examples

			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;

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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).

Extensions

Name added by R. J. Mathar, several entries corrected by Charles R Greathouse IV, Jun 21 2011
Showing 1-3 of 3 results.

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