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.

A100139 a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 3^k * 2^(n-6k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 67, 152, 376, 992, 2704, 7424, 20233, 54398, 144112, 376736, 974368, 2500544, 6385435, 16264220, 41396788, 105423776, 268818064, 686499008, 1755723793, 4495691834, 11521647916, 29543647160, 75774096832, 194353495424
Offset: 0

Views

Author

Paul Barry, Nov 06 2004

Keywords

Comments

Binomial transform of 1,1,1,1,1,1,4,4,13,13,31,... with g.f. (1-x)^2(1+x)^3/(1-3x^2+3x^4-4x^6)=(1+x)(1-x^2)^2/((1-x^2)^3-3x^6).

Links

Crossrefs

Cf. A097122, A100133, A100134, A100137, A100138.

Programs

  • Mathematica
    LinearRecurrence[{6,-12,8,0,0,3},{1,2,4,8,16,32},30] (* Harvey P. Dale, Sep 30 2015 *)

Formula

G.f.: (1-2x)^2/((1-2x)^3 - 3x^6).
a(n) = 6*a(n-1) -12*a(n-2) + 8*a(n-3) + 3*a(n-6).

A097123 Expansion of (1-x)^2/((1-x)^3 - 4*x^3).

Original entry on oeis.org

1, 1, 1, 5, 17, 41, 97, 253, 673, 1745, 4481, 11573, 30001, 77689, 200929, 519725, 1344833, 3479969, 9004033, 23296357, 60276817, 155961545, 403535969, 1044107357, 2701521889, 6989923441, 18085741441, 46795063445, 121077583217
Offset: 0

Views

Author

Paul Barry, Jul 25 2004

Keywords

Links

Crossrefs

Cf. A097122.

Programs

  • Mathematica
    LinearRecurrence[{3, -3, 5}, {1, 1, 1}, 30] (* Amiram Eldar, Oct 11 2021 *)
CoefficientList[Series[(1-x)^2/((1-x)^3-4x^3),{x,0,30}],x] (* Harvey P. Dale, May 06 2022 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n, 3*k) * 4^k); \\ Michel Marcus, Oct 11 2021

Formula

G.f.: (1-2*x+x^2)/(1-3*x+3*x^2-5*x^3).
a(n) = 3*a(n-1) - 3*a(n-2) + 5*a(n-3).
a(n) = Sum_{k=0..floor(n/3)} binomial(n, 3k) * 4^k.

A372076 The sequence T_{3,1}(n,3).

Original entry on oeis.org

0, 1, 2, 3, 7, 20, 51, 121, 290, 711, 1747, 4268, 10407, 25405, 62066, 151611, 370255, 904196, 2208267, 5393233, 13171682, 32168415, 78563131, 191870876, 468596895, 1144430581, 2794984562, 6826049523, 16670917207, 40714541300
Offset: 0

Views

Author

N. J. A. Sloane, Jun 17 2024

Keywords

References

  • Maribel Díaz Noguera [Maribel Del Carmen Díaz Noguera], Rigoberto Flores, Jose L. Ramirez, and Martha Romero Rojas, Catalan identities for generalized Fibonacci polynomials, Fib. Q., 62:2 (2024), 100-111. See Table 3.

Links

Crossrefs

Cf. A097122, A372077.

Programs

  • Mathematica
    a[n_] := Sum[Sum[Binomial[n - j - 1, 3*k]*3^k, {k,0,n-j-1}], {j,0,n-1}]; Table[a[n], {n,0,29}] (* Detlef Meya, Jun 22 2024 *)

Formula

a(n) = Sum_{j=0..n-1} Sum_{k=0..n-j-1} binomial(n - j - 1, 3*k)*3^k. a(n+1) = a(n) + A097122(n). - Detlef Meya, Jun 22 2024
G.f.: (x-x^2)/(1-3*x+3*x^2-4*x^3). - Georg Fischer, Apr 10 2025, from the reference, p. 108.

Extensions

a(11) and beyond from Detlef Meya, Jun 22 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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