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.

A215455 a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3), with a(0)=3, a(1)=6 and a(2)=18.

Original entry on oeis.org

3, 6, 18, 57, 186, 621, 2109, 7251, 25146, 87726, 307293, 1079370, 3798309, 13382817, 47191491, 166501902, 587670810, 2074699233, 7325660010, 25869337773, 91359785781, 322660334739, 1139593274178, 4024976418198, 14216179376325, 50211881768346, 177350652641349
Offset: 0

Views

Author

Roman Witula, Aug 11 2012

Keywords

Comments

The Berndt-type sequence number 1 for the argument 2*Pi/9 (see also A215007, A215008) is connected with the following trigonometric identities: f(n;x)=g(n;x)=const for n=1,2 (and are equal to 6 and 18 respectively), f(n;x)+g(n;x)=const for n=3,4,5 (and are equal to 120, 420 and 1512 respectively). Moreover each of the functions f(3;x), g(3;x) and f(6;x)+g(6;x) is not the constant function. Here f(n;x) := (2*cos(x))^(2n) + (2*cos(x-Pi/3))^(2n) + (2*cos(x+Pi/3))^(2n), and g(n;x) := (2*sin(x))^(2n) + (2*cos(x-Pi/6))^(2n) + (2*cos(x+Pi/6))^(2n), for every n=1,2,..., and x in R (see Witula-Slota paper for details).

Examples

			From the identity c(j)^2 = 2 + c(2*j) we deduce that a(1)=6 is equivalent with c(2) + c(4) + c(8) = 0, where c(j) := 2*cos(Pi*j/9).

Links

Crossrefs

Cf. A094831, A215007, A215008.

Programs

  • Mathematica
    LinearRecurrence[{6,-9,1}, {3,6,18}, 50]
  • PARI
    Vec((3-12*x+9*x^2)/(1-6*x+9*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

a(n) = c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n), where c(j) = 2*cos(Pi*j/9).
G.f.: 3*(1 - x)*(1 - 3*x)/(1 - 6*x + 9*x^2 - x^3).
a(n) = 3*A094831(n). - Andrew Howroyd, Apr 28 2020

Extensions

Terms a(22) and beyond from Andrew Howroyd, Apr 28 2020

A188048 Expansion of (1 - x^2)/(1 - 3*x^2 - x^3).

Original entry on oeis.org

1, 0, 2, 1, 6, 5, 19, 21, 62, 82, 207, 308, 703, 1131, 2417, 4096, 8382, 14705, 29242, 52497, 102431, 186733, 359790, 662630, 1266103, 2347680, 4460939, 8309143, 15730497, 29388368, 55500634, 103895601, 195890270, 367187437, 691566411, 1297452581
Offset: 0

Views

Author

L. Edson Jeffery, Mar 19 2011

Keywords

Comments

Sequence is related to rhombus substitution tilings.

Links

Crossrefs

Cf. A052931.

Programs

  • Magma
    I:=[1,0,2,1]; [n le 4 select I[n] else Self(n-1)+3*Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2015
  • Maple
    F:= gfun:-rectoproc({a(n)=3*a(n-2)+a(n-3),a(0)=1,a(1)=0,a(2)=2},a(n),remember):
map(F, [$0..100]); # Robert Israel, Jun 21 2015
  • Mathematica
    CoefficientList[Series[(1-x^2)/(1-3x^2-x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 31 2011 *)
LinearRecurrence[{0,3,1}, {1,0,2}, 50] (* Roman Witula, Aug 20 2012 *)
  • PARI
    abs(polsym(1-3*x+x^3,66)/3) /* Joerg Arndt, Aug 19 2012 */

Formula

G.f.: (1 - x^2)/(1 - 3*x^2 - x^3).
a(n) = 3*a(n-2)+a(n-3), for n>=3, with a(0)=1, a(1)=0, a(2)=2.
a(n) = a(n-1)+3*a(n-2)-2*a(n-3)-a(n-4), for n>=4, with {a(k)}={1,0,2,1}, k=0,1,2,3.
a(n) = A187497(3*n+1).
a(n) = m_(3,3), where (m_(i,j)) = (U_1)^n, i,j=1,2,3,4 and U_1 is the tridiagonal unit-primitive matrix [0, 1, 0, 0; 1, 0, 1, 0; 0, 1, 0, 1; 0, 0, 1, 1].
3*(-1)^n*a(n) = A215664(n). - Roman Witula, Aug 20 2012
a(2n) = A094831(n); a(2n+1) = A094834(n). - John Blythe Dobson, Jun 20 2015
a(n) = A052931(n)-A052931(n-2). - R. J. Mathar, Nov 03 2020
a(n) = (2^n/3)*(cos^n(Pi/9) + cos^n(5*Pi/9) + cos^n(7*Pi/9)). - Greg Dresden, Sep 24 2022

A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 3, 0, 1, 5, 10, 9, 0, 0, 0, 6, 15, 19, 9, 0, 0, 0, 6, 21, 34, 28, 0, 0, 0, 0, 0, 27, 55, 62, 28, 0, 0, 0, 0, 0, 27, 82, 117, 90, 0, 0, 0, 0, 0, 0, 0, 109, 199, 207, 90, 0, 0, 0, 0, 0, 0, 0, 109, 308, 406, 297, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Comments

A hexagon arithmetic of E. Lucas.

Examples

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, ... row n=0
1, 2, 3, 4, 5, 6, 6, 0, 0, ... row n=1
1, 3, 6, 10, 15, 21, 27, 27, 0, 0, ... row n=2
0, 3, 9, 19, 34, 55, 82, 109, 109, 0, 0, ... row n=3
0, 0, 9, 28, 62, 117, 199, 308, 417, 417, 0, 0, ... row n=4
0, 0, 0, 28, 90, 207, 406, 714, 1131, 1548, 1548, 0, 0, ... row n=5
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 6
1, 3, 6, 10, 15, 21, 27, 27
3, 9, 19, 34, 55, 82, 109, 109
9, 28, 62, 117, 199, 308, 417, 417
28, 90, 207, 406, 714, 1131, 1548, 1548
90, 297, 703, 1417, 2548, 4096, 5644, 5644
297, 1000, 2417, 4965, 9061, 14705, 20349, 20349
1000, 3417, 8382, 17443, 32148, 52497, 72846, 72846
3417, 11799, 29242, 61390, 113887, 186733, 259579, 259579
11799, 41041, 102431, 216318, 403051, 662630, 922209, 922209
...

Crossrefs

Cf. Similar sequences: A216201, A216210, A216216, A216218, ...

Formula

T(n,n+4) = T(n,n+5) = A094829(n+2).
T(n,n+3) = A094834(n+1).
T(n,n+2) = A094833(n+1).
T(n,n+1) = A094832(n).
T(n,n) = A094831(n).
T(n+1,n) = T(n+2,n) = A094826(n).
sum(T(n-k,k), 0<=k<=n) = A065455(n).
Showing 1-3 of 3 results.

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