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.

Previous Showing 31-37 of 37 results.

A124237 Riordan array (1/(1-2x), x/((1-x)(1-2x))).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 17, 8, 1, 16, 49, 39, 11, 1, 32, 129, 150, 70, 14, 1, 64, 321, 501, 338, 110, 17, 1, 128, 769, 1524, 1375, 640, 159, 20, 1, 256, 1793, 4339, 4973, 3075, 1083, 217, 23, 1, 512, 4097, 11762, 16508
Offset: 0

Views

Author

Paul Barry, Oct 22 2006

Keywords

Comments

Row sums are A007052. Diagonal sums are F(2n+1)=A001519(n+1)=A122367(n). Product of A007318 and Delannoy triangle A008288.

Examples

			Triangle begins
1,
2, 1,
4, 5, 1,
8, 17, 8, 1,
16, 49, 39, 11, 1,
32, 129, 150, 70, 14, 1

Formula

Number triangle T(n,k)=sum{j=0..n-k, C(n-j,k)C(n+k,j)}
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014

A173731 a(n) = a(n-1) * (11*a(n-1) - a(n-2)) / (a(n-1) + 4*a(n-2)), a(0) = a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 0, 1, 11, 88, 638, 4466, 30856, 212135, 1455685, 9981840, 68428140, 469043796, 3214953456, 22035826813, 151036348463, 1035219958696, 7095506886986, 48633337477670, 333337879614520, 2284731883069955, 15659785467455305
Offset: 0

Views

Author

Michael Somos, Feb 23 2010

Keywords

Examples

			x^2 + 11*x^3 + 88*x^4 + 638*x^5 + 4466*x^6 + 30856*x^7 + 212135*x^8 + ...

Links

Crossrefs

Cf. A172511

Programs

  • Magma
    [(4+Fibonacci(4*n+1)/3+Fibonacci(4*n + 3)/3-5* Fibonacci(2*n+1)) / 20: n in [0..25]]; // Vincenzo Librandi, Nov 30 2016
  • Mathematica
    Table[(4 + Fibonacci[4*n + 1]/3 + Fibonacci[4*n + 3]/3 - 5*Fibonacci[2*n + 1])/20, {n, 0, 25}] (* or *) LinearRecurrence[{11, -33, 33, -11, 1}, {0, 0, 1, 11, 88}, 25] (* G. C. Greubel, Nov 29 2016 *)
  • PARI
    {a(n) = (4 + fibonacci(4*n + 1)/3 + fibonacci(4*n + 3)/3 - 5 * fibonacci(2*n + 1)) / 20}

Formula

a(n) = (4 + A049685(n) - 5 * A122367(n)) / 20 = a(1 - n).
G.f.: x^2 / ((1 - x) * (1 - 3*x + x^2) * (1 - 7*x + x^2)) = ( 4 / (1 - x) - 5 * (1 - x) / (1 - 3*x + x^2) + (1 - x) / (1 - 7*x + x^2) ) / 20.
From G. C. Greubel, Nov 29 2016: (Start)
a(n) = 11*a(n-1) - 33*a(n-2) + 33*a(n-3) - 11*a(n-4) + a(n-5).
a(n) = (12 + Fibonacci(4*n + 1) + Fibonacci(4*n + 3) - 15*Fibonacci[2*n + 1] ) / 60. (End)

A202209 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 13, 5, 0, 0, 34, 19, 1, 0, 0, 89, 65, 8, 0, 0, 0, 233, 210, 42, 1, 0, 0, 0, 610, 654, 183, 11, 0, 0, 0, 0, 1597, 1985, 717, 74, 1, 0, 0, 0, 0, 4181, 5911, 2622, 394, 14, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 14 2011

Keywords

Comments

Riordan array ((1-x)/(1-3x+x^2), x^2/(1-3x+x^2)) .

Examples

			Triangle begins :
1
2, 0
5, 1, 0
13, 5, 0, 0
34, 19, 1, 0, 0
89, 65, 8, 0, 0, 0
233, 210, 42, 1, 0, 0, 0

Crossrefs

Cf. A000045, A000079, A001519, A001870, A001906, A126124, A202207 (antidiagonal sums)

Formula

T(n,k) = 3*T(n-1,k) - T(n-2,k) + T(n-2,k-1).
G.f.: (1-x)/(1-3x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively.
T(n,0) = A122367(n) = A000045(2n+1).

A373644 a(n) = Sum_{k=0..floor(n/4)} binomial(2*n-7*k,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 8, 11, 18, 29, 44, 64, 96, 148, 228, 345, 519, 786, 1198, 1824, 2766, 4190, 6358, 9661, 14674, 22268, 33786, 51284, 77866, 118212, 179426, 272333, 413391, 627547, 952613, 1445995, 2194911, 3331793, 5057593, 7677250, 11653681, 17689720
Offset: 0

Views

Author

Seiichi Manyama, Jun 12 2024

Keywords

Links

Crossrefs

Cf. A052535, A122367, A373639.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(2*n-7*k, k));

Formula

G.f.: 1 / (1 - x^4 - x/(1 - x^4)).
a(n) = a(n-1) + 2*a(n-4) - a(n-8) for n > 7.

A156561 Floor(Fibonacci(2n+1)/9).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 25, 67, 177, 464, 1216, 3184, 8336, 21824, 57136, 149585, 391619, 1025273, 2684201, 7027331, 18397793, 48166048, 126100352, 330135008, 864304672, 2262779008, 5924032352, 15509318049, 40603921795, 106302447337, 278303420217
Offset: 0

Views

Author

Paul Curtz, Feb 10 2009

Keywords

Links

Crossrefs

Cf. A069403.

Programs

  • Mathematica
    Floor[Fibonacci[2*Range[0,30]+1]/9] (* or *) LinearRecurrence[{4,-4,1,0,0,-1,4,-4,1},{0,0,0,1,3,9,25,67,177},31] (* Harvey P. Dale, Jun 06 2016 *)

Formula

a(n) = ( A000045(2n+1)-A154811(n) )/9 = floor(A122367(n)/9) = floor(A001519(n+1)/9) = floor( |A099496(n)|/9).
a(n)=3a(n-1)-a(n-2)+|A112690(n+10)|, i.e., a(n)-3a(n-1)+a(n-2) is a sequence of period 12 containing 0's and 1's. - R. J. Mathar, Feb 23 2009
G.f.: (1-x+x^2)/((1-x)(1+x^2)(1-3x+x^2)(1-x^2+x^4)). - R. J. Mathar, Feb 23 2009

Extensions

Edited and extended by R. J. Mathar, Jan 23 2009, Feb 23 2009

A183189 Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 6, 1, 0, 18, 5, 0, 0, 54, 21, 1, 0, 0, 162, 81, 8, 0, 0, 0, 486, 297, 45, 1, 0, 0, 0, 1458, 1053, 216, 11, 0, 0, 0, 0, 4374, 3645, 945, 78, 1, 0, 0, 0, 0, 13122, 12393, 3888, 450, 14, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 14 2011

Keywords

Comments

Riordan array ((1-x)/(1-3x), x^2/(1-3x)).
A skewed version of triangular array in A193723.
A202209*A007318 as infinite lower triangular matrices.

Examples

			Triangle begins:
  1
  2, 0
  6, 1, 0
  18, 5, 0, 0
  54, 21, 1, 0, 0
  162, 81, 8, 0, 0, 0
  486, 297, 45, 1, 0, 0, 0

Crossrefs

Cf. A000244, A025192, A081038, A183188 (antidiagonal sums).

Formula

G.f.: (1-x)/(1-3*x-y*x^2).
T(n,k) = Sum_{j, j>=0} T(n-2-j,k-1)*3^j.
T(n,k) = 3*T(n-1,k) + T(n-2,k-1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively.

A238156 Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 7, 4, 0, 2, 11, 16, 5, 0, 2, 15, 36, 30, 6, 0, 2, 19, 64, 91, 50, 7, 0, 2, 23, 100, 204, 196, 77, 8, 0, 2, 27, 144, 385, 540, 378, 112, 9, 0, 2, 31, 196, 650, 1210, 1254, 672, 156, 10, 0, 2, 35, 256, 1015, 2366, 3289, 2640, 1122, 210, 11
Offset: 0

Views

Author

Philippe Deléham, Feb 18 2014

Keywords

Comments

Row sums are A001519(n+1) = A122367(n).
Diagonal sums are A052969(n).

Examples

			Triangle begins:
1;
0, 2;
0, 2, 3;
0, 2, 7, 4;
0, 2, 11, 16, 5;
0, 2, 15, 36, 30, 6;
0, 2, 19, 64, 91, 50, 7;
0, 2, 23, 100, 204, 196, 77, 8;
0, 2, 27, 144, 385, 540, 278, 112, 9;
0, 2, 31, 196, 650, 1210, 1254, 672, 156, 10;
0, 2, 35, 256, 1015, 2366, 3289, 2640, 1122, 210, 11;
...

Crossrefs

Cf. A016742, A004767, A005581, A005583

Formula

G.f.: (1-x)/(1-x-2*x*y+x^2*y^2).
Sum_{k=0..n} T(n,k)*2^k = 4^n = A000302(n).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
Previous Showing 31-37 of 37 results.

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