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.

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A164414 Number of binary strings of length n with no substrings equal to 0000, 0001, or 1010.

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 37, 63, 107, 181, 307, 521, 883, 1497, 2539, 4305, 7299, 12377, 20987, 35585, 60339, 102313, 173483, 294161, 498787, 845753, 1434075, 2431649, 4123155, 6991305, 11854603, 20100913, 34083523, 57792729, 97994555, 166161601, 281747059
Offset: 0

Views

Author

R. H. Hardin, Aug 14 2009

Keywords

Links

Crossrefs

Cf. A077949.

Formula

From R. J. Mathar, Nov 30 2011: (Start)
a(n) = 13*A077949(n-4) + 9*A077949(n-5) + 15*A077949(n-6).
G.f.: (x^6-x^5-x^4-2*x^3-2*x^2-x-1)/(2*x^3+x-1).
(End)

Extensions

a(0)-a(3) from Alois P. Heinz, Jun 10 2021

A180675 The Ca3 sums of the Pell-Jacobsthal triangle A013609.

Original entry on oeis.org

1, 1, 1, 9, 97, 1041, 11169, 119833, 1285697, 13794337, 148000449, 1587907625, 17036776865, 182788823089, 1961154631009, 21041371248697, 225754408665729, 2422135536207937, 25987269043538817, 278819307278968905
Offset: 0

Views

Author

Johannes W. Meijer, Sep 21 2010

Keywords

Comments

The a(n+3) represent the Ca3 sums of the Pell-Jacobsthal triangle A013609. See A180662 for information about these camel and other chess sums.

Links

Crossrefs

Cf. A077949 (Ca1), A008998 (Ca2), A180675 (Ca3), A092467 (Ca4).

Programs

  • Maple
    nmax:=20: a(0):=1: a(1):=1: a(2):=1: for n from 3 to nmax do a(n) := 11*a(n-1)-3*a(n-2)+a(n-3) od: seq(a(n),n=0..nmax);
  • Mathematica
    LinearRecurrence[{11,-3,1},{1,1,1},20] (* Harvey P. Dale, Jun 23 2013 *)

Formula

a(n) = 11*a(n-1)-3*a(n-2)+a(n-3) with a(0)=1, a(1)=1 and a(2)=1.
a(n+2) = add(A013609(n+2*k,3*k),k=0..floor(n)).
GF(x) = (1-10*x-7*x^2)/(1-11*x+3*x^2-x^3).

A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 9, 9, 12, 12, 15, 16, 17, 21, 21, 27, 28, 32, 37, 38, 48, 49, 59, 65, 70, 85, 87, 107, 114, 129, 150, 157, 192, 201, 236, 264, 286, 342, 358, 428, 465, 522, 606, 644, 770, 823, 950, 1071, 1166, 1376
Offset: 1

Views

Author

Vicente Jesús Maniega Granado, Oct 03 2016

Keywords

Comments

Generalized Fibonacci growth sequence using i = 2 as maturity period, j = 5 as conception period, and k = 2 as growth factor.
Maturity period is the number of periods that a Fibonacci tree node needs for being able to start developing branches. Conception period is the number of periods in a Fibonacci tree node needed to develop new branches since its maturity. Growth factor is the number of additional branches developed by a Fibonacci tree node, plus 1, and equals the base of the exponential series related to the given tree if maturity factor would be zero. Standard Fibonacci would use 1 as maturity period, 1 as conception period, and 2 as growth factor as the series becomes equal to 2^n with a maturity period of 0. Related to Lucas sequences.

Examples

			For n = 13 the a(13) = a(8) + a(6) = 2 + 1 = 3.

Links

Crossrefs

Cf. A000079 (i = 0, j = 1, k = 2), A000244 (i = 0, j = 1, k = 3), A000302 (i = 0, j = 1, k = 4), A000351 (i = 0, j = 1, k = 5), A000400 (i = 0, j = 1, k = 6), A000420 (i = 0, j = 1, k = 7), A001018 (i = 0, j = 1, k = 8), A001019 (i = 0, j = 1, k = 9), A011557 (i = 0, j = 1, k = 10), A001020 (i = 0, j = 1, k = 11), A001021 (i = 0, j = 1, k = 12), A016116 (i = 0, j = 2, k = 2), A108411 (i = 0, j = 2, k = 3), A213173 (i = 0, j = 2, k = 4), A074872 (i = 0, j = 2, k = 5), A173862 (i = 0, j = 3, k = 2), A127975 (i = 0, j = 3, k = 3), A200675 (i = 0, j = 4, k = 2), A111575 (i = 0, j = 4, k = 3), A000045 (i = 1, j = 1, k = 2), A001045 (i = 1, j = 1, k = 3), A006130 (i = 1, j = 1, k = 4), A006131 (i = 1, j = 1, k = 5), A015440 (i = 1, j = 1, k = 6), A015441 (i = 1, j = 1, k = 7), A015442 (i = 1, j = 1, k = 8), A015443 (i = 1, j = 1, k = 9), A015445 (i = 1, j = 1, k = 10), A015446 (i = 1, j = 1, k = 11), A015447 (i = 1, j = 1, k = 12), A000931 (i = 1, j = 2, k = 2), A159284 (i = 1, j = 2, k = 3), A238389 (i = 1, j = 2, k = 4), A097041 (i = 1, j = 2, k = 10), A079398 (i = 1, j = 3, k = 2), A103372 (i = 1, j = 4, k = 2), A103373 (i = 1, j = 5, k = 2), A103374 (i = 1, j = 6, k = 2), A000930 (i = 2, j = 1, k = 2), A077949 (i = 2, j = 1, k = 3), A084386 (i = 2, j = 1, k = 4), A089977 (i = 2, j = 1, k = 5), A178205 (i = 2, j = 1, k = 11), A103609 (i = 2, j = 2, k = 2), A077953 (i = 2, j = 2, k = 3), A226503 (i = 2, j = 3, k = 2), A122521 (i = 2, j = 6, k = 2), A003269 (i = 3, j = 1, k = 2), A052942 (i = 3, j = 1, k = 3), A005686 (i = 3, j = 2, k = 2), A237714 (i = 3, j = 2, k = 3), A238391 (i = 3, j = 2, k = 4), A247049 (i = 3, j = 3, k = 2), A077886 (i = 3, j = 3, k = 3), A003520 (i = 4, j = 1, k = 2), A108104 (i = 4, j = 2, k = 2), A005708 (i = 5, j = 1, k = 2), A237716 (i = 5, j = 2, k = 3), A005709 (i = 6, j = 1, k = 2), A122522 (i = 6, j = 2, k = 2), A005710 (i = 7, j = 1, k = 2), A237718 (i = 7, j = 2, k = 3), A017903 (i = 8, j = 1, k = 2).

Programs

  • Magma
    [n le 7 select 1 else Self(n-5)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Nov 30 2016
  • Mathematica
    LinearRecurrence[{0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 70] (*  or *)
CoefficientList[ Series[(1+x+x^2+x^3+x^4)/(1-x^5-x^7), {x, 0, 70}], x] (* Robert G. Wilson v, Nov 25 2016 *)
nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,a+c}; NestList[nxt,{1,1,1,1,1,1,1},70][[;;,1]] (* Harvey P. Dale, Oct 22 2024 *)
  • PARI
    Vec(x*(1+x+x^2+x^3+x^4)/((1-x+x^2)*(1+x-x^3-x^4-x^5)) + O(x^100)) \\ Colin Barker, Oct 27 2016
  • SageMath
    @CachedFunction # a = A242763
def a(n): return 1 if n<8 else a(n-5) +a(n-7)
[a(n) for n in range(1,76)] # G. C. Greubel, Oct 23 2024

Formula

Generic a(n) = 1 for n <= i+j; a(n) = a(n-j) + (k-1)*a(n-(i+j)) for n>i+j where i = maturity period, j = conception period, k = growth factor.
G.f.: x*(1+x+x^2+x^3+x^4) / ((1-x+x^2)*(1+x-x^3-x^4-x^5)). - Colin Barker, Oct 09 2016
Generic g.f.: x*(Sum_{l=0..j-1} x^l) / (1-x^j-(k-1)*x^(i+j)), with i > 0, j > 0 and k > 1.

A052610 E.g.f. 1/(1-x-2x^3).

Original entry on oeis.org

1, 1, 2, 18, 120, 840, 9360, 115920, 1491840, 22861440, 395539200, 7304774400, 148011494400, 3281639961600, 77850214041600, 1975895970048000, 53666956062720000, 1547595999645696000, 47204701332332544000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Programs

  • Maple
    spec := [S,{S=Sequence(Union(Z,Prod(Z,Z,Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Formula

E.g.f.: -1/(-1+x+2*x^3)
Recurrence: {a(1)=1, a(0)=1, a(2)=2, (-12*n^2-22*n-12-2*n^3)*a(n) +(-n-3)*a(n+2) +a(n+3)=0}
Sum(1/29*(2+6*_alpha^2+9*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+2*_Z^3))*n!
a(n)=n!*A077949(n). - R. J. Mathar, Jun 03 2022
Previous Showing 11-14 of 14 results.

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