A077939 -id:A077939 - cp's OEIS frontend

A089068 a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.

0, 0, 1, 3, 6, 12, 23, 43, 80, 148, 273, 503, 926, 1704, 3135, 5767, 10608, 19512, 35889, 66011, 121414, 223316, 410743, 755475, 1389536, 2555756, 4700769, 8646063, 15902590, 29249424, 53798079, 98950095, 181997600, 334745776, 615693473

Offset: 0

Roger L. Bagula, Dec 03 2003

The a(n+2) represent the Kn12 and Kn22 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums. - Johannes W. Meijer, Sep 21 2010

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

Cf. A000931, A000073, A077939, A113300, A001057, A006054, A033505.

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25). - Johannes W. Meijer, Sep 21 2010

Join[{a=0,b=0,c=1},Table[d=a+b+c+2;a=b;b=c;c=d,{n,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+2}, a[n],{n,40}] (* or *) LinearRecurrence[{2,0,0,-1},{0,0,1,3},40] (* Harvey P. Dale, Sep 19 2011 *)

a(n) = A008937(n-2)+A008937(n-1). - Johannes W. Meijer, Sep 21 2010
a(n) = A018921(n-5)+A018921(n-4), n>4. - Johannes W. Meijer, Sep 21 2010
a(n) = A000073(n+2)-1. - R. J. Mathar, Sep 22 2010
From Johannes W. Meijer, Sep 22 2010: (Start)
a(n) = a(n-1)+A001590(n+1).
a(n) = Sum_{m=0..n} A040000(m)*A000073(n-m).
a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+1,k+1).
G.f. = x^2*(1+x)/((1-x)*(1-x-x^2-x^3)). (End)
a(n) = 2*a(n-1)-a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - Bruno Berselli, Sep 23 2010

Corrected and information added by Johannes W. Meijer, Sep 22 2010, Oct 22 2010

Definition based on arbitrarily set floating-point precision removed by R. J. Mathar, Sep 30 2010

A103142 a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=5, a(4)=13.

Original entry on oeis.org

1, 2, 5, 13, 34, 88, 228, 591, 1532, 3971, 10293, 26680, 69156, 179256, 464641, 1204374, 3121801, 8091873, 20974562, 54367172, 140922580, 365278767, 946821848, 2454212215, 6361447625, 16489208080, 42740897848, 110786663616, 287164880785, 744346531114

Offset: 0

Paul Barry, Jan 24 2005

Row sums of generalized Pascal matrix A103141.
Generalized Pell numbers.
Row sums of the tetranacci convolution triangle A202193. - Philippe Deléham, Feb 16 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (2,1,1,1).

Cf. A000129, A077939.

Row sums of A103141 and of A202193.

a:=[1,2,5,13];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Feb 12 2020

I:=[1,2,5,13]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)+Self(n-3) +Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 05 2012

m:=40; S:=series(1/(1-2*x-x^2-x^3-x^4), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 12 2020

LinearRecurrence[{2,1,1,1}, {1,2,5,13}, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

Vec(1/(1-2*x-x^2-x^3-x^4)+O(x^40)) \\ Charles R Greathouse IV, Jun 20 2011

def A103142_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2-x^3-x^4) ).list()
A103142_list(40) # G. C. Greubel, Feb 12 2020

a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4).

G.f.: 1/(1 - 2*x - x^2 - x^3 - x^4).

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A077986 Expansion of 1/(1 + 2*x - x^2 + x^3).

Original entry on oeis.org

1, -2, 5, -13, 33, -84, 214, -545, 1388, -3535, 9003, -22929, 58396, -148724, 378773, -964666, 2456829, -6257097, 15935689, -40585304, 103363394, -263247781, 670444260, -1707499695, 4348691431, -11075326817, 28206844760, -71837707768, 182957587113, -465959726754

Offset: 0

N. J. A. Sloane, Nov 17 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,1,-1).

Cf. A077939.

a:=[1,-2,5];; for n in [4..30] do a[n]:=-2*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jun 25 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1+2*x-x^2+x^3) )); // G. C. Greubel, Jun 25 2019

m:=30; S:=series(1/(1+2*x-x^2+x^3), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 26 2020

CoefficientList[Series[1/(1+2x-x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {-2,1,-1},{1,-2,5},30] (* Harvey P. Dale, Feb 14 2014 *)
b[n_]:= b[n]= If[n<3, n, 2*b[n-1] +b[n-2] +b[n-3]]; Table[(-1)^n*b[n+1], {n, 0, 30}] (* Rigoberto Florez, Jan 23 2020 *)

Vec(1/(1+2*x-x^2+x^3)+O(x^30)) \\ Charles R Greathouse IV, Sep 26 2012

def A077986_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return ( 1/(1+2*x-x^2+x^3) ).list()
A077986_list(30) # G. C. Greubel, Jun 25 2019

a(n) = (-1)^n * A077939(n). - Joerg Arndt, Sep 30 2012

a(n) = -2*a(n-1) + a(n-2) - a(n-3), with a(0)=1, a(1)=-2, a(2)=5. - Harvey P. Dale, Feb 14 2014

A276225 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=3, a(1)=2, a(2)=6.

Original entry on oeis.org

3, 2, 6, 17, 42, 107, 273, 695, 1770, 4508, 11481, 29240, 74469, 189659, 483027, 1230182, 3133050, 7979309, 20321850, 51756059, 131813277, 335704463, 854978262, 2177474264, 5545631253, 14123715032, 35970535581, 91610417447, 233315085507, 594211124042, 1513347751038, 3854221711625, 9816002298330

Offset: 0

G. C. Greubel, Aug 24 2016

Also the number of maximal independent vertex sets (and minimal vertex covers) on the 2n-crossed prism graph. - Eric W. Weisstein, Jun 15 2017
Also the number of irredundant sets in the n-sun graph. - Eric W. Weisstein, Aug 07 2017
Let {x,y,z} be the simple roots of P(x) = x^3 + u*x^2 + v*x + w. For n>=0, let p(n) = x^n/((x-y)(x-z)) + y^n/((y-x)(y-z)) + z^n/((z-x)(z-y)), q(n) =  x^n + y^n + z^n. Then for n >= 0, q(n) = 3*p(n+2) + 2*u*p(n+1) + v*p(n). In this case, P(x) = x^3 - 2*x^2 - x - 1, q(n) = a(n), p(n) = A077939(n). - Kai Wang, Apr 15 2020
Also the number of tilings of a bracelet of length n with two colors of squares and one color of domino and tromino. - Greg Dresden and Arnim Kuchhal, Aug 05 2024

Robert Israel, Table of n, a(n) for n = 0..2450
Eric Weisstein's World of Mathematics, Crossed Prism Graph
Eric Weisstein's World of Mathematics, Irredundant Set
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Eric Weisstein's World of Mathematics, Sun Graph
Index entries for linear recurrences with constant coefficients, signature (2,1,1).

Cf. A077939, A276226.

I:=[3,2,6]; [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

f:= gfun:-rectoproc({a(n+3) = 2*a(n+2) + a(n+1) + a(n), a(0)=3, a(1)=2, a(2)=6},a(n),remember):
map(f, [$0..40]); # Robert Israel, Aug 29 2016

LinearRecurrence[{2, 1, 1}, {3, 2, 6}, 50]
CoefficientList[Series[(3 - 4 x - x^2)/(1 - 2 x - x^2 - x^3), {x, 0, 32}], x] (* Michael De Vlieger, Aug 25 2016 *)
Table[RootSum[-1 - #1 - 2 #1^2 + #1^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, Jun 15 2017 *)

Vec((3-4*x-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*a(n) + A276226(n)*(p + q) + A077939(n-1)*(p^2 + q^2)).
G.f.: (3 - 4*x - x^2)/(1 - 2*x - x^2 - x^3).
a(n) = b^n + c^n + d^n, where (b, c, d) are the three roots of the cubic equation x^3 = 2*x^2 + x + 1.
a(n) = 3*A077939(n+2) - 4*A077939(n+1) - A077939(n). - Kai Wang, Apr 15 2020

A077849 Expansion of (1-x)^(-1)/(1 - 2*x - x^2 - x^3).

Original entry on oeis.org

1, 3, 8, 21, 54, 138, 352, 897, 2285, 5820, 14823, 37752, 96148, 244872, 623645, 1588311, 4045140, 10302237, 26237926, 66823230, 170186624, 433434405, 1103878665, 2811378360, 7160069791, 18235396608, 46442241368, 118279949136, 301237536249, 767197263003

Offset: 0

N. J. A. Sloane, Nov 17 2002

I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
Index entries for linear recurrences with constant coefficients, signature (3,-1,0,-1)

Partial sums of A077939.

A077939 := proc(n) if n< 0 then 0; else coeftayl( 1/(1-2*x-x^2-x^3) ,x=0,n) ; end if; end proc:
A077849 := proc(n) (-1+4*A077939(n)+2*A077939(n-1)+A077939(n-2))/3 ; end proc:
seq(A077849(n),n=0..20) ; # R. J. Mathar, Mar 22 2011

CoefficientList[Series[(1-x)^(-1)/(1-2x-x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,-1,0,-1},{1,3,8,21},40] (* Harvey P. Dale, Nov 01 2016 *)

Vec((1-x)^(-1)/(1-2*x-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

A104580 Tribonacci convolution triangle.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 7, 12, 9, 4, 1, 13, 26, 25, 14, 5, 1, 24, 56, 63, 44, 20, 6, 1, 44, 118, 153, 125, 70, 27, 7, 1, 81, 244, 359, 336, 220, 104, 35, 8, 1, 149, 499, 819, 864, 646, 357, 147, 44, 9, 1, 274, 1010, 1830, 2144, 1800, 1134, 546, 200, 54, 10, 1

Offset: 0

Paul Barry, Mar 16 2005

			Rows begin
  {1},
  {1,1},
  {2,2,1},
  {4,5,3,1},
  {7,12,9,4,1},
   ...

First column is A000073(n+2). Row sums are A077939. Diagonal sums are A002478.

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> A000073(n+1)); # Peter Luschny, Oct 19 2022

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(sum(binomial(i+k,k)*trinomial(i,n-k-i),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 15 2011 */

Riordan array (1/(1-x-x^2-x^3), x/(1-x-x^2-x^3)).
From Paul Barry, Jun 02 2009: (Start)
T(n,m) = T'(n-1,m-1) + T'(n-1,m) + T'(n-2,m) + T'(n-3,m), where T'(n,m) = T(n,m) for n >= 0 and 0 <= m <= n and T'(n,m) = 0 otherwise. (End)
T(n,k) = Sum_{i=0..n-k} binomial(i+k,k)*A027907(i,n-k-i). - Emanuele Munarini, Mar 15 2011

A186575 Expansion of (1 + 2x + 6x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

Original entry on oeis.org

1, 3, 10, 15, 31, 66, 127, 255, 514, 1023, 2047, 4098, 8191, 16383, 32770, 65535, 131071, 262146, 524287, 1048575, 2097154, 4194303, 8388607, 16777218, 33554431, 67108863, 134217730, 268435455, 536870911, 1073741826, 2147483647, 4294967295

Offset: 0

Vladimir Kruchinin, Feb 23 2011

From Kai Wang, May 23 2020: (Start)
Let f(t) = t^3 + u*t^2 + v*t + w and {x,y,z} be the simple roots of f(t).
For n >= 0, let p(n) = x^n/((x-y)*(x-z)) + y^n/((y-x)*(y-z)) + z^n/((z-x)*(z-y)) and q(n) = x^n + y^n + z^n.
Then for n >= 0, q(n) = 3*p(n+2) + 2*u*p(n+1) + v*p(n).
In this case, f(t) = t^3 - t^2 - t - 2. q(n) = 3*p(n+2) - 2*p(n+1) - p(n).
p(n) = {0, 0, 1, 1, 2, 5, 9,...}, q(n) = {3, 1, 3, 10, 15, 31,...}.
a(n) = q(n+1), A077939(n) = p(n+2). (End)

			G.f. = 1 + 3*x + 10*x^2 + 15*x^3 + 31*x^4 + 66*x^5 + 127*x^6 + 255*x^7 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Gamaliel Cerda-Morales, A note on Modified Third-order Jacobsthal numbers, arXiv:1905.00725 [math.CO], 2019. See pp. 3-4.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Kai Wang, Closed Forms and Generating Functions For Power Sums, 2020.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A099837.

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3))); // Marius A. Burtea, Jan 31 2020

CoefficientList[Series[(1+2x+6x^2)/(1-x-x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 14 2011 *)

Vec((1 + 2*x + 6*x^2) / ((1 - 2*x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, May 03 2019

polsym(polrecip(1 - x - x^2 - 2*x^3),44)[^1] \\ Joerg Arndt, Jun 23 2020

a(n+1) = n*Sum_{k=1..n} Sum_{j=n-3*k..k} 2^(k-j)*binomial(j,n-3*k+2*j)*binomial(k,j)/k.
G.f.: [log(1/(1 - x - x^2 - 2*x^3))]', (x + x^2 + 2*x^3)^k = Sum_{n>=k} Sum_{j=n-3*k..k} 2^(k-j)*binomial(j,n-3*k+2*j)*binomial(k,j)*x^n (see link).
a(n) = 2^(n+1) + A099837(n+1). - R. J. Mathar, Mar 18 2011
a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n>2. - Colin Barker, May 03 2019
From Kai Wang, May 23 2020: (Start)
a(n) = 3*A077947(n+1) - 2*A077947(n) - A077947(n-1).
A077947(n) = (-8*a(n+3) + 27*a(n+2) - a(n+1))/147. (End)

More terms from Harvey P. Dale, Mar 14 2011

A102036 Triangle, read by rows, where the terms are generated by the rule: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-3,k-1), with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 9, 15, 7, 1, 1, 12, 33, 28, 9, 1, 1, 15, 60, 81, 45, 11, 1, 1, 18, 96, 189, 161, 66, 13, 1, 1, 21, 141, 378, 459, 281, 91, 15, 1, 1, 24, 195, 675, 1107, 946, 449, 120, 17, 1, 1, 27, 258, 1107, 2349, 2673, 1742, 673, 153, 19, 1

Offset: 0

Paul D. Hanna, Dec 30 2004

Row sums form A077939. This sequence was inspired by Luke Hanna.
Diagonal sums are A000078(n+3). - Philippe Deléham, Feb 16 2014
Riordan array (1/(1-x), x*(1+x+x^2)/(1-x)). - Philippe Deléham, Feb 16 2014

			Generated by adding preceding terms in the triangle at positions that form the letter 'L':
T(n,k) =
T(n-3,k-1) +
T(n-2,k-1) +
T(n-1,k-1) + T(n-1,k).
Rows begin:
  [1],
  [1,  1],
  [1,  3,   1],
  [1,  6,   5,   1],
  [1,  9,  15,   7,   1],
  [1, 12,  33,  28,   9,   1],
  [1, 15,  60,  81,  45,  11,  1],
  [1, 18,  96, 189, 161,  66, 13,  1],
  [1, 21, 141, 378, 459, 281, 91, 15, 1], ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee The k-nacci triangle and applications. Cogent Math. 4, Article ID 1333293, 13 p. (2017).
J. L. Ramírez, V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.

Cf. A077939, A103141.

[[(&+[Binomial(n-m,k)*(&+[Binomial(j,m-j)*Binomial(k,j):j in [0..k]]): m in [0..n-k]]): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Dec 11 2018

T:=(n,k)->add(add((binomial(j,m-j)*binomial(k,j))*binomial(n-m,k),j=0..k),m=0..n-k): seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Dec 11 2018

T[n_, k_] := If[n < k || k < 0, 0, If[n == 0, 1, T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k - 1] + T[n - 3, k - 1]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 07 2018 *)
Table[Sum[Binomial[n-m, k]*Sum[Binomial[j, m-j]*Binomial[k, j], {j, 0, k}], {m, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 11 2018 *)

T(n,k):=sum((sum(binomial(j,m-j)*binomial(k,j),j,0,k))*binomial(n-m,k),m,0,n-k); /* Vladimir Kruchinin, Apr 21 2015 */

PARI
```
{T(n,k)=if(n
				
```

[[sum(binomial(n-m,k)*sum(binomial(j,m-j)*binomial(k,j) for j in (0..k)) for m in (0..n-k)) for k in (0..n)] for n in range(15)] # G. C. Greubel, Dec 11 2018

G.f.: 1/(1-y-x*(1+y+y^2)). - Vladimir Kruchinin, Apr 21 2015
T(n,k) = Sum_{m=0..(n-k)} (Sum_{j=0..k} C(j,m-j)*C(k,j))*C(n-m,k). - Vladimir Kruchinin, Apr 21 2015
From Werner Schulte, Dec 07 2018: (Start)
G.f. of column k: Sum_{n>=0} T(n+k,k) * x^n = (1+x+x^2)^k / (1-x)^(k+1) = (1-x^3)^k / (1-x)^(2*k+1).
Let k >= 0 be some fixed integer and a_k(n) be multiplicative with a_k(p^e) = T(e+k,k) for prime p and e >= 0. Then we have the Dirichlet g.f.: Sum{n>0} a_k(n) / n^s = (zeta(s))^(2*k+1) / (zeta(3*s))^k. (End)

A121574 Riordan array (1/(1-2x), x(1+x)/(1-2*x)).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 16, 8, 1, 16, 44, 37, 11, 1, 32, 112, 134, 67, 14, 1, 64, 272, 424, 305, 106, 17, 1, 128, 640, 1232, 1168, 584, 154, 20, 1, 256, 1472, 3376, 3992, 2641, 998, 211, 23, 1, 512, 3328, 8864, 12592, 10442, 5221, 1574, 277, 26, 1

Offset: 0

Paul Barry, Aug 08 2006

Row sums are A006190(n+1); diagonal sums are A077939.
Inverse is A121575.
A generalized Delannoy number triangle.
Antidiagonal sums are A002478. - Philippe Deléham, Nov 10 2011.
From Peter Bala, Feb 07 2024: (Start)
The following remarks assume the row indexing starts at n = 1.
The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = 2 + x, R(3,x) = 4 + 5*x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)

			Triangle begins
   1;
   2,   1;
   4,   5,   1;
   8,  16,   8,   1;
  16,  44,  37,  11,   1;
  32, 112, 134,  67,  14,  1;
  64, 272, 424, 305, 106, 17, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.

Cf. Diagonals: A000012, A016789, A080855, A000079, A053220.

T:=Flat(List([0..9],n->List([0..n],k->Sum([0..n-k],j->Binomial(k,j)*Binomial(n-j,k)*2^(n-k-j))))); # Muniru A Asiru, Nov 02 2018

[[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018

T:=(n,k)->add(binomial(k,j)*binomial(n-j,k)*2^(n-k-j),j=0..n-k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Nov 02 2018

Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 02 2018 *)

for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ G. C. Greubel, Nov 02 2018

Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1). - Philippe Deléham, Nov 10 2011
Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = (x + 2)*R(n-1,x) + x*R(n-2,x) with R(1,x) = 1 and R(2,x) = x + 2. - Peter Bala, Feb 07 2024

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

cp's OEIS Frontend

A089068 a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.

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A103142 a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=5, a(4)=13.

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A077986 Expansion of 1/(1 + 2*x - x^2 + x^3).

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A276225 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=3, a(1)=2, a(2)=6.

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A077849 Expansion of (1-x)^(-1)/(1 - 2*x - x^2 - x^3).

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A104580 Tribonacci convolution triangle.

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A186575 Expansion of (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

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A186575 Expansion of (1 + 2x + 6x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

A121574 Riordan array (1/(1-2x), x(1+x)/(1-2*x)).