A008288 -id:A008288 - cp's OEIS frontend

A344576 a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

0, 2, 10, 52, 278, 1510, 8288, 45834, 254922, 1424252, 7986550, 44921582, 253320352, 1431678194, 8106897418, 45982821860, 261206625526, 1485765938390, 8461264982176, 48237937154554, 275275548126890, 1572297656021292, 8987888015996790, 51417128080562142

Offset: 0

José María Grau Ribas, May 24 2021

nonn

a(n+1)/a(n) tends to A156035.

Cf. A000045, A001850, A008288, A156035.

F[0, 0] = 0; F[m_, 0] := Fibonacci[m]; F[0, n_] := Fibonacci[n];
F[m_, n_] := F[m, n] =   F[m - 1 , n ] + F[m , n - 1] +  F[m - 1, n - 1];
Table[F[n, n], {n, 0, 100}]

\\ here D(n,k) is A008288(n,k).
D(n, k) = {sum(d = 0, min(n,k), binomial(k, d)*binomial(n+k-d, k))}
a(n) = {2*sum(k=1, n, fibonacci(k)*(D(n-1,n-k) + D(n-1,n-k-1)))} \\ Andrew Howroyd, May 29 2021

a(n) = 2*Sum_{k=1..n} Fibonacci(k)*(A008288(n-1,n-k) + A008288(n-1,n-k-1)). - Andrew Howroyd, May 29 2021
G.f.: x*(3*x^2-18*x+3-(x+1)*sqrt(x^2-6*x+1))/((x^2-7*x+1)*(x^2-6*x+1)). - Alois P. Heinz, May 29 2021
a(n) = ((79-97*n+26*n^2)*a(n-1) + (-9+9*n-2*n^2)*a(n-4) + (107-111*n+26*n^2)*a(n-3) + (-322+352*n-88*n^2)*a(n-2)) / (5-7*n+2*n^2) for n >= 4. - José María Grau Ribas, Jun 19 2021

A364553 Number of edges in the n-Pell graph.

Original entry on oeis.org

0, 1, 5, 18, 58, 175, 507, 1428, 3940, 10701, 28705, 76230, 200766, 525083, 1365175, 3531240, 9093512, 23325785, 59625981, 151947066, 386139650, 978834759, 2475645491, 6248406780, 15740857452, 39585199525, 99389810585, 249177006702, 623846750086, 1559888545075

Offset: 0

Eric W. Weisstein, Jul 28 2023

For n > 0, also the number of maximum and maximal cliques in the n-Pell graph.

E. Munarini, Pell Graphs, Disc. Math., 342 (2019), 2415-2428.
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Pell Graph
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000129, A001333, A008288, A026937, A093967, A364636, row sums of A364361.

A364553 := n -> (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n): seq(simplify(A364553(n)), n=0..29); # Peter Luschny, Jul 30 2023

Table[n Fibonacci[n + 1, 2]/2, {n, 0, 20}]
Table[n (Fibonacci[n, 2] + (-I)^n ChebyshevT[n, I])/2, {n, 0, 20}]
Table[With[{s = Sqrt[2]}, n ((s + 2) (1 + s)^n - (s - 2) (1 - s)^n)/8], {n, 0, 20}] // Expand
LinearRecurrence[{4, -2, -4, -1}, {0, 1, 5, 18}, 20]
CoefficientList[Series[x (1 + x)/(-1 + 2 x + x^2)^2, {x, 0, 20}], x]

# Using function 'delannoy_row' from A008288.
def A364553(n:int) -> int:
    return sum(k * delannoy_row(n)[k] for k in range(n + 1))
print([A364553(n) for n in range(30)])  # Peter Luschny, Jul 30 2023

a(n) = n*(A000129(n) + A001333(n))/2.
a(n) = n*A000129(n+1)/2.
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4).
G.f.: x*(1+x)/(-1+2*x+x^2)^2.
From Peter Luschny, Jul 31 2023:  (Start)
a(n) = (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n).
With this formula, the sequence can be continued to the left half of the number line: a(-n) = -(-1)^n*A026937(n-2) for n >= 0.
a(n) = (A093967(n) + A364636(n)) / 2.
a(n) = Sum_{k=0..n} k * A008288(n, k).  (End)

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 9, 17, 9, 1, 1, 12, 36, 36, 12, 1, 1, 15, 64, 101, 64, 15, 1, 1, 18, 101, 227, 227, 101, 18, 1, 1, 21, 147, 440, 627, 440, 147, 21, 1, 1, 24, 202, 767, 1459, 1459, 767, 202, 24, 1, 1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1

Offset: 0

F. Chapoton, Mar 25 2025

The original definition was "Decomposition of A077938".

Every row is symmetric.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   6,    1;
  1,  9,  17,    9,    1;
  1, 12,  36,   36,   12,    1;
  1, 15,  64,  101,   64,   15,    1;
  1, 18, 101,  227,  227,  101,   18,    1;
  1, 21, 147,  440,  627,  440,  147,   21,   1;
  1, 24, 202,  767, 1459, 1459,  767,  202,  24,  1;
  1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1;
  ...

Similar to A008288, A103450, and A382444.
Row sums are A077938.
T(2n, n) gives A339565.
Cf. A056594.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
inverse = 1 + (-y - 1)*x - y*x^2 + (-y^2 - y)*x^3
gf = 1 / inverse
[list(u) for u in list(gf.O(11))]

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

Sum_{k=0..n} (-1)^k * T(n,k) = A056594(n). - Alois P. Heinz, Mar 25 2025

A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

Original entry on oeis.org

1, 2, 2, 1, 5, 4, 4, 16, 20, 8, 1, 14, 41, 44, 16, 6, 50, 146, 198, 128, 32, 1, 27, 155, 377, 456, 272, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584, 512, 1, 65

Offset: 0

Paul D. Hanna, Oct 25 2004

Row sums form A099606, where A099606(n) = Pell(n+1)*2^[(n+1)/2]. Central coefficients of even-indexed rows form A026000, where A026000(n) = T(2n,n), where T = Delannoy triangle (A008288). Antidiagonal sums form A099607.

			Rows begin:
[1],
[2,2],
[1,5,4],
[4,16,20,8],
[1,14,41,44,16],
[6,50,146,198,128,32],
[1,27,155,377,456,272,64],
[8,112,560,1408,1992,1616,704,128],
[1,44,406,1652,3649,4712,3568,1472,256],
[10,210,1572,6084,14002,20330,18880,10912,3584,512],
[1,65,870,5202,17469,36365,48940,42800,23552,7424,1024],...
The binomial transform of row 2 equals column 2 of A034870:
BINOMIAL[1,5,4] = [1,6,15,28,45,66,91,120,153,...].
The binomial transform of row 3 equals column 3 of A034870:
BINOMIAL[4,16,20,8] = [4,20,56,120,220,364,560,...].
The binomial transform of row 4 equals column 4 of A034870:
BINOMIAL[1,14,41,44,16] = [1,15,70,210,495,1001,...].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A099602, A099605, A034870, A000129.

CoefficientList[CoefficientList[Series[(1 + 2*(y + 1)*x - (y + 1)*x^2)/(1 - (2*y + 1)*(2*y + 2)*x^2 + (y + 1)^2*x^4), {x, 0, 49}, {y, 0, 49}], x],
  y] // Flatten (* G. C. Greubel, Apr 14 2017 *)

{T(n,k)=polcoeff(polcoeff((1+2*(y+1)*x-(y+1)*x^2)/(1-(2*y+1)*(2*y+2)*x^2+(y+1)^2*x^4)+x*O(x^n),n,x)+y*O(y^k),k,y)}

G.f.: (1+2*(y+1)*x-(y+1)*x^2)/(1-(2*y+1)*(2*y+2)*x^2+(y+1)^2*x^4). T(n, n) = 2^n.

A102662 Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 1, 7, 11, 3, 1, 9, 23, 17, 3, 1, 11, 39, 51, 23, 3, 1, 13, 59, 113, 91, 29, 3, 1, 15, 83, 211, 255, 143, 35, 3, 1, 17, 111, 353, 579, 489, 207, 41, 3, 1, 19, 143, 547, 1143, 1323, 839, 283, 47, 3, 1, 21, 179, 801, 2043, 3045, 2651, 1329, 371, 53, 3, 1, 23, 219

Offset: 1

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 03 2005

Generalization of A008288 (use initial terms 1,1,3). Triangle seen as lower triangular matrix: The absolute values of the coefficients of the characteristic polynomials of the n X n matrix are the (n+1)th row of A038763. Row sums give A048654.

			Triangle begins:
  1
  1 3
  1 5 3
  1 7 11 3
  1 9 23 17 3

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Reinhard Zumkeller, Rows n=0..149 of triangle, flattened
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.

Cf. A038763, A048654, A008288.

a102662 n k = a102662_tabl !! n !! k
a102662_row n = a102662_tabl !! n
a102662_tabl = [1] : [1,3] : f [1] [1,3] where
   f xs ys = zs : f ys zs where
     zs = zipWith (+) ([0] ++ xs ++ [0]) $
                      zipWith (+) ([0] ++ ys) (ys ++ [0])
-- Reinhard Zumkeller, Feb 23 2012

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207624 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A102662 *)
(* Clark Kimberling, Feb 20 2012 *)

T(k,r)=if(r>k,0,if(k==1,1,if(k==2,if(r==1,1,3),if(r==1,1,if(r==k,3,T(k-1,r-1)+T(k-1,r)+T(k-2,r-1))))))
BM(n) = M=matrix(n,n);for(i=1,n, for(j=1,n,M[i,j]=T(i,j)));M
M=BM(10)
for(i=1,10,s=0;for(j=1,i,s+=M[i,j]);print1(s,","))

From Clark Kimberling, Feb 20 2012: (Start)
A102662=v and A207624=u, defined together as follows:
u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=2*x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1; see the Mathematica section. (End)

A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)binomial(n-j,k)((j+1) mod 2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 32, 21, 6, 1, 1, 7, 31, 65, 65, 31, 7, 1, 1, 8, 43, 116, 161, 116, 43, 8, 1, 1, 9, 57, 189, 341, 341, 189, 57, 9, 1, 1, 10, 73, 288, 645, 842, 645, 288, 73, 10, 1, 1, 11, 91, 417, 1121, 1827, 1827, 1121, 417, 91

Offset: 0

Paul Barry, May 31 2005

Or as a square array read by antidiagonals, T(n,k) = Sum_{j=0..n} binomial(k,j)*binomial(n+k-j,k)*((j+1) mod 2).
A symmetric number triangle based on 1/(1-x^2).
The construction of a symmetric triangle in this example is general. Let f(n) be a sequence, preferably with f(0)=1. Then T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*f(j) yields a symmetric triangle. When f(n)=1^n, we get Pascal's triangle. When f(n)=2^n, we get the Delannoy triangle (see A008288). In general, f(n)=k^n yields a (1,k,1)-Pascal triangle (see A081577, A081578). Row sums of triangle are A100131. Diagonal sums of the triangle are A108351. Triangle mod 2 is A106465.

			Triangle rows begin
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 21, 32, 21,  6,  1;
As a square array read by antidiagonals, rows begin
  1, 1,  1,   1,   1,    1,    1, ...
  1, 2,  3,   4,   5,    6,    7, ...
  1, 3,  7,  13,  21,   31,   43, ...
  1, 4, 13,  32,  65,  116,  189, ...
  1, 5, 21,  65, 161,  341,  645, ...
  1, 6, 31, 116, 341,  842, 1827, ...
  1, 7, 43, 189, 645, 1827, 4495, ...

trgn(nn) = {for (n= 0, nn, for (k = 0, n, print1(sum(j=0, n-k, binomial(k,j)*binomial(n-j,k)*((j+1) % 2)), ", ");); print(););} \\ Michel Marcus, Sep 11 2013

Row k (and column k) has g.f. (1+C(k,2)x^2)/(1-x)^(k+1).

A121832 Expansion of 1/(1-x-x^5-x^6).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 6, 8, 10, 13, 19, 29, 43, 61, 84, 116, 164, 236, 340, 485, 685, 965, 1365, 1941, 2766, 3936, 5586, 7916, 11222, 15929, 22631, 32153, 45655, 64793, 91944, 130504, 185288, 263096, 373544, 530281, 752729, 1068521, 1516905

Offset: 0

Jon E. Schoenfield, Aug 27 2006

Number of compositions of n into parts 1, 5, and 6. [Joerg Arndt, Sep 03 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 18.
Said Amrouche and Hacène Belbachir, Unimodality and linear recurrences associated with rays in the Delannoy triangle, Turkish Journal of Mathematics (2019) Vol. 44, 118-130.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,1).

Cf. A008288.

I:=[1,1,1,1,1,2]; [n le 6 select I[n] else Self(n-1)+Self(n-5)+Self(n-6): n in [1..50]]; // Vincenzo Librandi, Sep 03 2013

CoefficientList[Series[1 / (1 - x - x^5 - x^6), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 03 2013 *)
LinearRecurrence[{1,0,0,0,1,1},{1,1,1,1,1,2},50] (* Harvey P. Dale, Mar 30 2018 *)

a(n) = a(n-1) + a(n-5) + a(n-6).

a(n) = Sum_{k=0..floor(n/5)} A008288(n-4*k, k). - Johannes W. Meijer, Jul 19 2013

A152548 Sum of squared terms in rows of triangle A152547: a(n) = Sum_{k=0..C(n,[n/2])-1} A152547(n,k)^2.

Original entry on oeis.org

1, 4, 10, 24, 54, 120, 260, 560, 1190, 2520, 5292, 11088, 23100, 48048, 99528, 205920, 424710, 875160, 1798940, 3695120, 7574996, 15519504, 31744440, 64899744, 132503644, 270415600, 551231800, 1123264800, 2286646200, 4653525600

Offset: 0

Paul D. Hanna, Dec 14 2008

nonn

Cf. A008288, A152547, A107233, A014477.

seq(simplify((-2)^n*hypergeom([-n,3/2], [1], 2)),n=0..29); # Peter Luschny, Apr 26 2016

CoefficientList[Series[Sqrt[(1+2x)/(1-2x)^3],{x,0,30}],x] (* Harvey P. Dale, Jan 04 2016 *)

a(n)=sum(k=0,floor((n+1)/2),binomial(n+1, k)*(n+1-2*k)^3)/(n+1)

G.f.: A(x) = sqrt( (1+2x)/(1-2x)^3 ).
a(n) = Sum_{k=0..[(n+1)/2]} C(n+1, k)*(n+1-2k)^3/(n+1).
a(n) = A107233(n)/(n+1).
Self-convolution equals A014477.
E.g.f.: ((1 + 4*x)*BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). - Peter Luschny, Aug 26 2012
a(n) = (-2)^n*hypergeom([-n,3/2], [1], 2). - Peter Luschny, Apr 26 2016
D-finite with recurrence: (n+1)*a(n+1) = 4*a(n) + 4*n*a(n-1). - Vladimir Reshetnikov, Oct 10 2016
a(n) ~ 2^(n + 3/2) * sqrt(n/Pi). - Vaclav Kotesovec, Oct 11 2016
From Peter Bala, Mar 31 2024: (Start)
a(n) = (2^n) * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(-3/2, n-k).
a(n) = (2^n) * Sum_{k = 0..n} (2^k)*binomial(n, k)*binomial(1/2, k).
a(n) = (2^n)* Sum_{k = 0..n} binomial(n, k)*binomial(k+1/2, n). See A008288.
a(n) = (2*n + 1)!/(2^n * n!^2) * hypergeom([-n, -1/2], [-n-1/2], -1).
a(n) = 2^n * hypergeom([-n, -1/2], [1], 2).
a(n) = (-1/2)^n * binomial(2*n, n)/(1 - 2*n) * hypergeom([-n, 3/2], [-n+3/2], -1).(End)

A209695 Triangle of coefficients of polynomials u(n,x) jointly generated with A209696; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 8, 18, 12, 1, 11, 40, 58, 29, 1, 14, 71, 164, 175, 70, 1, 17, 111, 357, 601, 507, 169, 1, 20, 160, 664, 1550, 2048, 1428, 408, 1, 23, 218, 1112, 3346, 6106, 6632, 3940, 985, 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378

Offset: 1

Clark Kimberling, Mar 13 2012

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012

			First five rows:
  1;
  1,  2;
  1,  5,  5;
  1,  8, 18, 12;
  1, 11, 40, 58, 29;
First three polynomials u(n,x):
  1
  1 + 2x
  1 + 5x + 5x^2.
From _Philippe Deléham_, Mar 24 2012: (Start)
(1, 0, 1/2, -1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  5,  0;
  1,  8, 18, 12,  0;
  1, 11, 40, 58, 29, 0; (End)

Cf. A209696, A208510, A000129, A008288.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209695 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209696 *)

u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 24 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-2*y*x-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

A211314 Square array of Delannoy numbers D(i,j) mod 7 (i >= 0, j >= 0) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 0, 6, 0, 1, 1, 2, 4, 4, 2, 1, 1, 4, 6, 0, 6, 4, 1, 1, 6, 5, 3, 3, 5, 6, 1, 1, 1, 1, 0, 6, 0, 1, 1, 1, 1, 3, 1, 6, 2, 2, 6, 1, 3, 1, 1, 5, 5, 1, 1, 3, 1, 1, 5, 5, 1, 1, 0, 6, 0, 1, 6, 6, 1, 0, 6, 0, 1, 1, 2, 4, 4, 2, 1, 1, 1, 2, 4, 4, 2, 1

Offset: 0

N. J. A. Sloane, Apr 15 2012

			Written as a triangle:
1,
1, 1,
1, 3, 1,
1, 5, 5, 1,
1, 0, 6, 0, 1,
1, 2, 4, 4, 2, 1,
1, 4, 6, 0, 6, 4, 1,
1, 6, 5, 3, 3, 5, 6, 1,
1, 1, 1, 0, 6, 0, 1, 1, 1,
...

Marko Razpet, A self-similarity structure generated by king's walk, Algebraic and topological methods in graph theory (Lake Bled, 1999). Discrete Math. 244 (2002), no. 1-3, 423--433. MR1844050 (2002k:05022).

Cf. A008288, A211312, A211313, A211315.

cp's OEIS Frontend

A344576 a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A364553 Number of edges in the n-Pell graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

Formula

A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A102662 Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).

Views

Author

Keywords

Comments

Examples

Programs

PARI

Formula

A121832 Expansion of 1/(1-x-x^5-x^6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A152548 Sum of squared terms in rows of triangle A152547: a(n) = Sum_{k=0..C(n,[n/2])-1} A152547(n,k)^2.

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).

A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)binomial(n-j,k)((j+1) mod 2).