A077939 -id:A077939 - cp's OEIS frontend

A141448 Generalized Pell numbers P(n,5,5).

0, 1, 2, 5, 13, 34, 89, 232, 605, 1578, 4116, 10736, 28003, 73041, 190515, 496926, 1296147, 3380779, 8818187, 23000741, 59993521, 156482896, 408159020, 1064613385, 2776862948, 7242974718, 18892067685, 49276745441, 128530009618

Offset: 0

R. J. Mathar, Aug 07 2008

P(n,2,2) and P(n,2,1) are in A000129.
P(n,3,2) is A116413. P(n,3,1) and P(n,3,3) are A077939.
P(n,4,1) and P(n,4,4) are A103142.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
E. Kilic and D. Tasci, The generalized Binet formula, representation and sums of the generalizedorder-k Pell numbers, Taiwanese J of Math vol 10 no 6 (2006), 1661-1670.
Index entries for linear recurrences with constant coefficients, signature (2,1,1,1,1).

I:=[0,1,2,5,13]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Dec 13 2012

P := proc(n,k,i) option remember ; if n = 1-i then 1; elif n <= 0 then 0; else 2*P(n-1,k,i)+add(P(n-j,k,i),j=2..k) ; fi ; end: for n from 0 to 40 do printf("%d,",P(n,5,5)) ; od:

CoefficientList[Series[x/(1 - 2*x - x^2 - x^3 - x^4 - x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
LinearRecurrence[{2,1,1,1,1},{0,1,2,5,13},40] (* Harvey P. Dale, Jan 08 2016 *)

a(n):=b(n+1);
b(n):=sum(sum(binomial(k,r)*2^(k-r)*sum((sum(binomial(j,-r+n-m-k-j)*binomial(m,j),j,0,m))*binomial(r,m),m,0,r),r,0,k),k,1,n); /* Vladimir Kruchinin, May 05 2011 */

From R. J. Mathar, Nov 28 2008: (Start)
a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5).
G.f.: x/(1-2*x-x^2-x^3-x^4-x^5). (End)
a(n+1) = Sum_(k=1..n, Sum_(r=0..k, binomial(k,r)*2^(k-r)*Sum_(m=0..r,(Sum_(j=0..m, binomial(j,-r+n-m-k-j)*binomial(m,j)))*binomial(r,m)))), a(0)=0, a(1)=1. [Vladimir Kruchinin, May 05 2011]

A276226 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=0, a(1)=6, a(2)=8.

Original entry on oeis.org

0, 6, 8, 22, 58, 146, 372, 948, 2414, 6148, 15658, 39878, 101562, 258660, 658760, 1677742, 4272904, 10882310, 27715266, 70585746, 179769068, 457839148, 1166033110, 2969674436, 7563221130, 19262149806, 49057195178, 124939761292, 318198867568, 810394691606, 2063928012072, 5256449583318, 13387221870314, 34094821336018

Offset: 0

G. C. Greubel, Aug 24 2016

Index entries for linear recurrences with constant coefficients, signature (2,1,1).

Cf. A276225, A077939.

I:=[0,6,8]; [n le 3 select I[n] else 2*Self(n-1)+ Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016

LinearRecurrence[{2, 1, 1}, {0, 6, 8}, 50]
CoefficientList[Series[2 (3 x - 2 x^2)/(1 - 2 x - x^2 - x^3), {x, 0, 33}], x] (* Michael De Vlieger, Aug 25 2016 *)

concat(0, Vec(2*(3*x-2*x^2)/(1-2*x-x^2-x^3) + O(x^99))) \\ Altug Alkan, Aug 25 2016

Let p = (4*(61 + 9*sqrt(29)))^(1/3), q = (4*(61 - 9*sqrt(29)))^(1/3), and x = (1/6)*(4 + p + q) then x^n = (1/6)*(2*A276225(n) + a(n)*(p + q) + A077939(n-1)*(p^2 + q^2)).G.f.: 2*(3*x - 2*x^2)/(1 - 2*x - x^2 - x^3).

A276229 a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=0, a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, -1, -1, 4, -3, -6, 16, -7, -31, 61, -6, -147, 220, 68, -655, 739, 639, -2772, 2233, 3950, -11188, 5521, 20805, -43035, 6946, 99929, -156856, -36056, 449697, -534441, -401009, 1919588, -1652011, -2588174, 7811784, -4287447, -13924295, 30310973

Offset: 0

G. C. Greubel, Aug 24 2016

Essentially the same as A077978. - Georg Fischer, Oct 02 2018

Index entries for linear recurrences with constant coefficients, signature (-1,-2,1).

Cf. A077939, A276228.

I:=[0,0,1]; [n le 3 select I[n] else -Self(n-1)- 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016

LinearRecurrence[{-1, -2, 1}, {0, 0, 1}, 50]
CoefficientList[Series[x^2/(1 + x + 2 x^2 - x^3), {x, 0, 39}], x] (* Michael De Vlieger, Aug 25 2016 *)

concat([0, 0], Vec(x^2/(1+x+2*x^2-x^3) + O(x^99))) \\ Altug Alkan, Aug 25 2016

G.f.: x^2/(1 + x + 2*x^2 - x^3).
Let P = (b-c)*(b-d), Q = (c-b)*(b-d), R = (d-b)*(d-c), (b, c, d) be the three roots of x^3 = 2*x^2 + x + 1, then a(n) = P^(-1)*b^(1-n) + Q^(-1)*c^(1-n) + R^(-1)*d^(1-n).
a(2*n) = -3*a(2*n-2) - 6*a(2*n-4) + a(2*n-6).

A375821 Number of ways to tile a 3-row parallelogram of length n with triangular and rectangular tiles, each of size 3.

Original entry on oeis.org

1, 1, 2, 7, 17, 41, 107, 274, 693, 1766, 4504, 11465, 29194, 74364, 189391, 482327, 1228412, 3128559, 7967841, 20292639, 51681711, 131623900, 335222103, 853749852, 2174345752, 5537663377, 14103422348, 35918853952, 91478793557, 232979863277, 593357374262

Offset: 0

Greg Dresden and Mingjun Oliver Ouyang, Aug 30 2024

Here is the 3-row parallelogram of length 6 (with 18 cells):
     _ ___ _ ___ _ ___
    |   |   |   |   |   |   |
   |__|___|_|___| |___|
  |   |   |   |   |   |   |
 |__|___|_|___| |___|
|   |   |   |   |   |   |
|_|___|_|___|_|___|,
and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:
   _
  |   |
 |__|_     _________
|   |   |   |   |   |   |
|_|___|,  |_|___|_|.
As an example, here is one of the a(6) = 107 ways to tile the 3 x 6 parallelogram:
     _ _______ _________
    |   |       |           |
   |  |_     |__________|
  |   |   |   |           |
 |  |   |_|___________|
|   |       |           |
|_|_______|_________|.

Index entries for linear recurrences with constant coefficients, signature (2,0,4,-1,0,-1).

Cf. A077939, A077961, A375823.

LinearRecurrence[{2, 0, 4, -1, 0, -1}, {1, 1, 2, 7, 17, 41}, 40]

a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).
G.f.: (1 - x - x^3)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).
a(n) = (A077939(n) + A077961(n))/2.

A375823 Number of ways to tile a 3-row trapezoid of average length n with triangular and rectangular tiles, each of size 3.

Original entry on oeis.org

0, 1, 3, 6, 16, 43, 107, 271, 695, 1769, 4499, 11464, 29202, 74360, 189382, 482339, 1228417, 3128538, 7967848, 20292665, 51681683, 131623881, 335222157, 853749843, 2174345679, 5537663440, 14103422412, 35918853816, 91478793556, 232979863477, 593357374127

Offset: 0

Greg Dresden and Mingjun Oliver Ouyang, Aug 30 2024

Here is the 3-row trapezoid of average length 6 (with 18 cells):
     _ ___ _ ___ _
    |   |   |   |   |   |
   |__|___|_|___| |_
  |   |   |   |   |   |   |
 |__|___|_|___| |___|_
|   |   |   |   |   |   |   |
|_|___|_|___|_|___|_|,
and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:
   _
  |   |
 |__|_     _________
|   |   |   |   |   |   |
|_|___|,  |_|___|_|.
As an example, here is one of the a(6) = 107 ways to tile the 3-row trapezoid
     _ ___ _________
    |   |   |           |
   |  |_  |_________|_
  |   |   |   |       |   |
 |  |   |  |     |   |
|   |       |   |   |       |
|_|_______|_|___|_____|.

Index entries for linear recurrences with constant coefficients, signature (2,0,4,-1,0,-1).

Cf. A077939, A077961, A375821.

LinearRecurrence[{2, 0, 4, -1, 0, -1}, {0, 1, 3, 6, 16, 43}, 40]

a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).
G.f.: x*(1 + x)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).
a(n) = (A077939(n) - A077961(n))/2.

cp's OEIS Frontend

A141448 Generalized Pell numbers P(n,5,5).

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A276226 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=0, a(1)=6, a(2)=8.

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A276229 a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=0, a(1)=0, a(2)=1.

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A375821 Number of ways to tile a 3-row parallelogram of length n with triangular and rectangular tiles, each of size 3.

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Mathematica

Formula

A375823 Number of ways to tile a 3-row trapezoid of average length n with triangular and rectangular tiles, each of size 3.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula