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 1-6 of 6 results.

A223968 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 5 or if k-n >= 6, T(4,0) = T(3,0) = 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, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 0, 0, 6, 15, 20, 15, 5, 0, 0, 6, 21, 35, 35, 20, 0, 0, 0, 0, 27, 56, 70, 55, 20, 0, 0, 0, 0, 27, 83, 126, 125, 75, 0, 0, 0, 0, 0, 0, 110, 209, 251, 200, 75, 0, 0, 0, 0, 0, 0, 110, 319, 460, 451, 275, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Mar 30 2013

Keywords

Examples

			Square array begins:
1....1....1....1....1....1....0....0....0....0....0....0
1....2....3....4....5....6....6....0....0....0....0....0
1....3....6...10...15...21...27...27....0....0....0....0
1....4...10...20...35...56...83..110..110....0....0....0
1....5...15...35...70..126..209..319..429..429....0....0
0....5...20...55..125..251..460..779.1208.1637.1637....0
0....0...20...75..200..451..911.1690.2898.4535.6172.6172
...
Square array, read by diagonals, with 0 omitted:
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 4, 15, 55, 200, 726, 2638, 9604, 35037, ...
1, 3, 10, 35, 125, 451, 1637, 5965, 21794, ...
1, 2, 6, 20, 70, 251, 911, 3327, 12190, 44744, ...
1, 3, 10, 35, 126, 460, 1690, 6225, 22950, ...
1, 4, 15, 56, 209, 779, 2898, 10760, 39882, ...
1, 5, 21, 83, 319, 1208, 4535, 16932, 62986, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...

Crossrefs

Cf. Similar sequences: A214846, A216054, A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236, A216238, A217257, A217315, A217593, A217765, A217770

Formula

sum(T(n-k,k), 0<=k<=n) = A223940(n).
T(n,n+5) = T(n,n+4) = A221863(n).
T(n,n+3) = A221862(n).
T(n,n+2) = A221859(n).
T(n,n+1) = A216710(n).
T(n,n) = A224514(n).
T(n+1,n) = A224509(n).
T(n+2,n) = A220948(n).
T(n+3,n) = T(n+4,n) = A224422(n). - Philippe Deléham, Apr 13 2013

A217777 Expansion of (1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4).

Original entry on oeis.org

1, 2, 4, 8, 15, 30, 55, 110, 200, 400, 725, 1450, 2625, 5250, 9500, 19000, 34375, 68750, 124375, 248750, 450000, 900000, 1628125, 3256250, 5890625, 11781250, 21312500, 42625000, 77109375, 154218750, 278984375, 557968750, 1009375000, 2018750000, 3651953125
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Links

Crossrefs

Cf. A217770

Programs

  • PARI
    Vec((1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4)+O(x^66)) /* Joerg Arndt, Mar 29 2013 */

Formula

a(n) = A216212(n+1)/2.
a(2n) = A039717(n+1), a(2n+1) = 2*a(2n) = 2*A039717(n+1).
a(n) = sum(A217770(n-k,k), 0<=k<=n).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=4, a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8.
G.f.: (1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4).

Extensions

Corrected name (g.f.), Joerg Arndt, Mar 29 2013

A217778 Expansion of (1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)).

Original entry on oeis.org

1, 3, 10, 34, 117, 407, 1429, 5055, 17986, 64278, 230473, 828391, 2982825, 10754459, 38811802, 140165322, 506449789, 1830590295, 6618524221, 23933966743, 86562282258, 313102489406, 1132598701585, 4097213146599, 14822370816337, 53623952036787
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Comments

A diagonal of the square array A217770.

Links

Crossrefs

Cf. A217770.

Programs

  • Magma
    m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)))); // Bruno Berselli, Mar 28 2013
  • Mathematica
    LinearRecurrence[{8, -21, 20, -5}, {1, 3, 10, 34}, 26] (* Bruno Berselli, Mar 28 2013 *)
CoefficientList[Series[(1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)),{x,0,30}],x] (* Harvey P. Dale, Sep 26 2023 *)
  • Maxima
    makelist(expand(((3+sqrt(5))*(5+sqrt(5))^n-(3-sqrt(5))*(5-sqrt(5))^n+(1+sqrt(5))*(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(4*2^n*sqrt(5))), n, 0, 25); /* Bruno Berselli, Mar 28 2013 */

Formula

G.f.: (1-5*x+7*x^2-3*x^3)/(1-8*x+21*x^2-20*x^3+5*x^4).
a(n) = A081567(n) - A094865(n).
a(n) = A217770(n+1,n).
a(n) = 8*a(n-1) -21*a(n-2) +20*a(n-3) -5*a(n-4) for n>3, a(0)=1, a(1)=3, a(2)=10, a(3)=34.
a(n) = ((3+r)*(5+r)^n-(3-r)*(5-r)^n+(1+r)*(3+r)^n-(1-r)*(3-r)^n)/(4*r*2^n), where r=sqrt(5). [Bruno Berselli, Mar 28 2013]

A217779 Expansion of (1-4x+4*x^2)/((1-5x+5*x^2)*(1-3x+x^2)).

Original entry on oeis.org

1, 4, 15, 56, 208, 768, 2821, 10320, 37639, 136972, 497652, 1805984, 6548425, 23729916, 85953823, 311240928, 1126753336, 4078394080, 14760382029, 53415642632, 193291233367, 699417041844, 2530731376540, 9156839587776, 33131242464913, 119873850697588
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Comments

A diagonal of the square array A217770.

Links

Crossrefs

Cf. A217770, A217777, A217778.

Formula

a(n) = A217770(n,n+2).
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4) for n>=4, a(0) = 1, a(1) = 4, a(2) = 15, a(3) = 56.

A217782 Expansion of (1-x)*(1-2x)*(1-3x)/((1-5x+5*x^2)*(1-3x+x^2)).

Original entry on oeis.org

1, 2, 6, 20, 69, 242, 857, 3054, 10930, 39236, 141153, 508598, 1834641, 6623450, 23926334, 86468052, 312587197, 1130277914, 4087621545, 14784539846, 53478888618, 193456813508, 699850536281, 2531866279710, 9159810802849, 33139021206962, 119894215708662
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Comments

A diagonal of the square array A217770.

Links

Crossrefs

Cf. A217770, A217777, A217778, A217779.

Formula

a(n) = A217770(n,n).
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) -5*a(n-4) for n>=4, a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 20.

A217783 Expansion of (1-x)*(1-2x)/((1-5x+5*x^2)*(1-3x+x^2)).

Original entry on oeis.org

1, 5, 21, 83, 318, 1196, 4445, 16389, 60097, 219527, 799734, 2907800, 10558041, 38297573, 138819053, 502925211, 1821362830, 6594366404, 23870720757, 86396702117, 312668994969, 1131463798415, 4094241931526, 14814592074288, 53603587025713, 193949782284101
Offset: 0

Views

Author

Philippe Deléham, Mar 24 2013

Keywords

Comments

A diagonal of the square array A217770.

Links

Crossrefs

Cf. A217770, A217777, A217778, A217779, A217782.

Formula

a(n) = A217770(n,n+3).
a(n) = 8*a(n-1) -21*a(n-2) + 20*a(n-3) -5*a(n-4) for n>=4, a(0) = 1, a(1) = 5, a(2) = 21, a(3) = 83.
Showing 1-6 of 6 results.

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