A111575 -id:A111575 - cp's OEIS frontend

A133463 Partial sums of the sequence that starts with 2 and is followed by A111575.

2, 3, 4, 5, 6, 9, 12, 15, 18, 27, 36, 45, 54, 81, 108, 135, 162, 243, 324, 405, 486, 729, 972, 1215, 1458, 2187, 2916, 3645, 4374, 6561, 8748, 10935, 13122, 19683, 26244, 32805, 39366, 59049, 78732, 98415, 118098, 177147, 236196, 295245, 354294

Offset: 0

Paul Curtz, Nov 28 2007

nonn

A111575 := proc(n) 3^(floor(n/4)) ; end: A133463 := proc(n) 2+add( A111575(i),i=0..n-1) ; end: seq(A133463(n),n=0..80) ; # R. J. Mathar, Jan 12 2008

Accumulate[Join[{2},With[{pt=3^Range[0,15]},Sort[Join[pt,pt,pt,pt]]]]] (* Harvey P. Dale, Jul 21 2021 *)

a(4n) = 2*A111575(4n). a(4n+1)= 3*A111575(4n+1). a(4n+2)= 4*A111575(4n+2). a(4n+3)= 5*A111575(4n+3). - R. J. Mathar, Jan 12 2008

Edited and extended by R. J. Mathar, Jan 12 2008

A111573 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 0,1,3,3.

Original entry on oeis.org

0, 1, 3, 3, 4, 8, 14, 21, 33, 55, 90, 144, 232, 377, 611, 987, 1596, 2584, 4182, 6765, 10945, 17711, 28658, 46368, 75024, 121393, 196419, 317811, 514228, 832040, 1346270, 2178309, 3524577, 5702887, 9227466, 14930352, 24157816, 39088169

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. A001638, A111569, A111570, A111571, A111572, A111574, A111575, A111576.

Table[Fibonacci[n + 1] - Cos[n*Pi/2], {n, 0, 40}] (* Greg Dresden, Oct 16 2021 *)

G.f.: -x*(1+2*x)/((x^2+x-1)*(x^2+1)).
a(n) = A056594(n+3) + A000045(n+1). - R. J. Mathar, Nov 10 2009
From Greg Dresden, Jan 15 2024: (Start)
a(2*n) = Fibonacci(n)*Lucas(n+1);
a(2*n+1) = Fibonacci(2*n+1). (End)

Name clarified by Robert C. Lyons, Feb 06 2025

A127975 Repeat 3^n three times.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 9, 9, 27, 27, 27, 81, 81, 81, 243, 243, 243, 729, 729, 729, 2187, 2187, 2187, 6561, 6561, 6561, 19683, 19683, 19683, 59049, 59049, 59049, 177147, 177147, 177147, 531441, 531441, 531441, 1594323, 1594323, 1594323, 4782969, 4782969, 4782969

Offset: 0

Paul Barry, Feb 09 2007

a(n) is the number of functions f:[n+1]->[3] with f(1)=1 and with f(x)=f(y) whenever y=ceiling(x/3). - Dennis P. Walsh, Sep 06 2018

			a(6)=9 since there are exactly 9 functions f:[7]->[3], denoted by <f(1),f(2),...,f(7)>, with f(1)=1 and with f(x)=f(y) whenever y=ceiling(x/3). The nine functions are <1,1,1,1,1,1,1>, <1,1,1,1,1,1,2>, <1,1,1,1,1,1,3>, <1,1,1,2,2,2,1>, <1,1,1,2,2,2,2>, <1,1,1,2,2,2,3>, <1,1,1,3,3,3,1>, <1,1,1,3,3,3,2>, and <1,1,1,3,3,3,3>. - _Dennis P. Walsh_, Sep 06 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..6000
Index entries for linear recurrences with constant coefficients, signature (0,0,3).

Cf. A108411, A111575.

[3^(Floor(n/3)):n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

seq(3^floor(n/3),n=0..45); # Dennis P. Walsh, Sep 06 2018

CoefficientList[Series[(1+x+x^2)/(1-3*x^3), {x,0,50}], x] (* or *) Table[3^(Floor[n/3]), {n,0,50}] (* G. C. Greubel, Apr 30 2017 *)

a(n)=3^(n\3) \\ Charles R Greathouse IV, Oct 03 2016

G.f.: (1+x+x^2)/(1-3*x^3).

Edited and corrected by R. J. Mathar, Jun 14 2008

A111572 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms -1,3,2,1.

Original entry on oeis.org

-1, 3, 2, 1, 3, 8, 11, 15, 26, 45, 71, 112, 183, 299, 482, 777, 1259, 2040, 3299, 5335, 8634, 13973, 22607, 36576, 59183, 95763, 154946, 250705, 405651, 656360, 1062011, 1718367, 2780378, 4498749, 7279127, 11777872, 19056999, 30834875, 49891874, 80726745

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.
Floretion Algebra Multiplication Program, FAMP Code: 4ibaseseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e
From Greg Dresden and Jiaqi Wang, Jun 24 2023: (Start)
For n >= 5, a(n) is also the number of ways to tile this "central staircase" figure of length n-2 with squares and dominoes. This is the picture for length 9; there are a(11)=112 ways to tile it:
             _
   _______|_|___
  |||_|||_|||_|
        |_|            (End)

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1).

Cf. A001638, A111569, A111570, A111571, A111573, A111574, A111575, A111576.

Cf. A000032, A000045.

G.f.: (1-4*x+x^2)/((1+x^2)*(x^2+x-1))
From Greg Dresden and Jiaqi Wang, Jun 24 2023: (Start)
a(2*n) = F(n+1)*L(n-1) + F(n)*F(n-1),
a(2*n+1) = F(n+1)*(F(n+1) + 2*F(n-1)), for F(n) and L(n) the Fibonacci and Lucas numbers.
(End)

Name clarified by Robert C. Lyons, Feb 06 2025

A111574 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,-1,2,3.

Original entry on oeis.org

1, -1, 2, 3, 3, 4, 9, 15, 22, 35, 59, 96, 153, 247, 402, 651, 1051, 1700, 2753, 4455, 7206, 11659, 18867, 30528, 49393, 79919, 129314, 209235, 338547, 547780, 886329, 1434111, 2320438, 3754547, 6074987, 9829536, 15904521, 25734055, 41638578, 67372635

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

Floretion Algebra Multiplication Program, FAMP Code: -4baseiseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1).

Cf. A001638, A111569, A111570, A111571, A111572, A111573, A111575, A111576.

LinearRecurrence[{1,0,1,1},{1,-1,2,3},40] (* Harvey P. Dale, Jan 24 2017 *)

G.f.: (-1+2*x-3*x^2)/((x^2+x-1)*(x^2+1)).

Name clarified by Robert C. Lyons, Feb 06 2025

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

Vicente Jesús Maniega Granado, Oct 03 2016

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.

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

Colin Barker, Table of n, a(n) for n = 1..1000
Julia Collins, Fibonacci Tree
Fractal Foundation, Fibonacci Fractals
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,1).

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).

[n le 7 select 1 else Self(n-5)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Nov 30 2016

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 *)

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

@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

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.

cp's OEIS Frontend

A133463 Partial sums of the sequence that starts with 2 and is followed by A111575.

Views

Author

Keywords

Programs

Maple

Mathematica

Formula

Extensions

A111573 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 0,1,3,3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A127975 Repeat 3^n three times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A111572 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms -1,3,2,1.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A111574 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,-1,2,3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula