A116413 -id:A116413 - cp's OEIS frontend

A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

1, 3, 1, 6, 6, 1, 12, 21, 9, 1, 24, 60, 45, 12, 1, 48, 156, 171, 78, 15, 1, 96, 384, 558, 372, 120, 18, 1, 192, 912, 1656, 1473, 690, 171, 21, 1, 384, 2112, 4608, 5160, 3225, 1152, 231, 24, 1, 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1, 1536, 10752

Offset: 0

Paul Barry, Feb 13 2006

Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.

Subtriangle of triangle given by (0, 3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

			Triangle begins
1,
3, 1,
6, 6, 1,
12, 21, 9, 1,
24, 60, 45, 12, 1,
48, 156, 171, 78, 15, 1
Triangle T(n,k), 0<=k<=n, given by (0, 3, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins :
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 12, 21, 9, 1
0, 24, 60, 45, 12, 1
0, 48, 156, 171, 78, 15, 1
... - _Philippe Deléham_, Jan 18 2012

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A003688, A003945.

With[{n = 10}, DeleteCases[#, 0] & /@ CoefficientList[Series[(1 + x)/(1 - (y + 2) x - y x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Number triangle T(n,k)=sum{j=0..n, C(k+1,j)*C(n-j,k)2^(n-k-j)}
From Vladimir Kruchinin, Mar 17 2011: (Start)
T((m+1)*n+r-1, m*n+r-1) * r/(m*n+r) = sum(k=1..n, k/n * T((m+1)*n-k-1, m*n-1) * T(r+k-1,r-1)), n>=m>1.
T(n-1,m-1) = m/n * sum(k=1..n-m+1, k*A003945(k-1)*T(n-k-1,m-2)), n>=m>1. (End)
G.f.: (1+x)/(1-(y+2)*x -y*x^2). - Philippe Deléham, Jan 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A104537(n), A110523(n), (-2)^floor(n/2), A057079(n), A003945(n), A003688(n+1), A123347(n), A180035(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(2,0) = T(2,1) = 6, T(2,2) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 31 2013

A109545 a(n) = 2*a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 1, 2, 6, 15, 38, 97, 247, 629, 1602, 4080, 10391, 26464, 67399, 171653, 437169, 1113390, 2835602, 7221763, 18392518, 46842401, 119299083, 303833085, 773807654, 1970747476, 5019135691, 12782826512, 32555536191, 82913034585

Offset: 0

Roger L. Bagula, Jun 20 2005

David Garth and Kevin G. Hare, Comments on the spectra of Pisot numbers, J. Number Theory 121 (2006), 187-203.
Index entries for linear recurrences with constant coefficients, signature (2, 1, 1).

a = 2; b = -1; M = {{0, 1, 0, 0, 0}, { a - 2, a - 2, a - 2 - b, a - 2 - b, 0}, {1, 1, 1, 1, 0}, {0, 1, 1, 0, 0}, {0, 0, 0, 1, 1}} v[1] = {1, 1, 1, 1, 1} v[n_] := v[n] = M.v[n - 1] a0 = Table[Abs[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{2,1,1},{1,1,2},30] (* Harvey P. Dale, Aug 05 2015 *)
Lucas := 1 + x (1 + 2 x)/(1 - x - x^2); (* InvertTransform defined in A052987 *)
InvertTransform[Lucas, 28] (* Peter Luschny, Jan 10 2019 *)

lim_{n-> infinity} a(n)/a(n-1)= 2.54682...
G.f.: (1-x-x^2)/(1-2*x-x^2-x^3). [Sep 28 2009]
a(n) = A077939(n)-A116413(n-1).
G.f.: (-1+x+x^2)/(-1+2*x+x^2+x^3). a(n) = A077997(n)-A077939(n-2). [From R. J. Mathar, Sep 27 2009]

Definition replaced by recurrence by the Associate Editors of the OEIS, Sep 28 2009

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

Original entry on oeis.org

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]

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A116412 Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).

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A109545 a(n) = 2*a(n-1) + a(n-2) + a(n-3).

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

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