A001891 -id:A001891 - cp's OEIS frontend

A175105 Triangle T(n,k) read by rows. T(n,1)=1; T(n,k) = Sum_{i=1..k-1} ( T(n-i,k-1) + T(n-i,k) ), k>1.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 6, 1, 1, 5, 21, 22, 8, 1, 1, 6, 40, 64, 38, 10, 1, 1, 7, 72, 163, 140, 58, 12, 1, 1, 8, 125, 382, 442, 256, 82, 14, 1, 1, 9, 212, 846, 1259, 954, 420, 110, 16, 1, 1, 10, 354, 1800, 3334, 3166, 1794, 640, 142, 18, 1, 1, 11, 585, 3719, 8366, 9657, 6754, 3074, 924, 178, 20, 1

Offset: 1

Mats Granvik, Feb 10 2010

Are there closed forms for diagonals and columns?
With the definition of the array, I note that the sequence (phi(k)) (phi(k)= g.f. of the column number k) is given by the recurrence relation: phi(k+1)=phi(k)*(1-z^k)/(1-2*z+z^(k+1)). The consequence is: the sequence number k+1 column is the convolution of the k-one and a "-acci like" sequence whose g.f. is given by (1-z^k)/(1-2*z+z^(k+1)). E.g., the 2-column is the convolution of the 1-column and the sequence 1, 2, 3, 5, ... classical Fibonacci sequence without the first 1. The 3-column is the convolution of the 2-column and 1, 2, 4, 7, 13, ... tribonacci like-sequence (exactly: A000073 without beginning 0, 0, 1). - Richard Choulet, Feb 19 2010
Relation to metallic means:
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 0*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the golden ratio, A001622.
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 1*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the silver ratio, A014176.
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 2*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the bronze ratio, A098316.
A similar point can be made about variations of the Pascal triangle.

			Table begins:
  n/k| 1    2    3    4    5    6    7    8    9   10   11
  ---+-----------------------------------------------------
   1 | 1
   2 | 1    1
   3 | 1    2    1
   4 | 1    3    4    1
   5 | 1    4   10    6    1
   6 | 1    5   21   22    8    1
   7 | 1    6   40   64   38   10    1
   8 | 1    7   72  163  140   58   12    1
   9 | 1    8  125  382  442  256   82   14    1
  10 | 1    9  212  846 1259  954  420  110   16    1
  11 | 1   10  354 1800 3334 3166 1794  640  142   18    1
Example: T(8,4) = 163 because it is the sum of the numbers:
  10    6
  21   22
  40   64
For k=1, we obtain phi(k)(z)=1/(1-z) which is clear; for k=2, we obtain phi(k)(z)=1/(1-z)^2. For k=3, we obtain phi(3)(z)=(1+z)/((1-2*z+z^3)*(1-z)); this is A001891 without the beginning zero. - _Richard Choulet_, Feb 19 2010

Wikipedia, Metallic mean

Cf. A172119, A051731, A001891 (column k=3), A176084 (row sums).
(1-((-1)^T(n, k)))/2 = T(n, k) mod 2 = A051731.
Cf. A179807=antidiagonal sums. A179748 has simpler recurrence.

=if(column()=1;1;if(row()>=column();sum(indirect(address(row()-column()+1;column()-1;4)&":"&address(row()-column()+column()-1;column()-1;4);4))+sum(indirect(address(row()-column()+1;column();4)&":"&address(row()-column()+column()-1;column();4);4));0)) ' Mats Granvik, Mar 28 2010

A175105 := proc(n,k) if k =1 then 1; elif k > n or k< 1 then 0 ; else    add(procname(n-i,k-1)+procname(n-i,k),i=1..k-1) ; end if; end proc; # R. J. Mathar, Feb 16 2011

T[_, 1] = 1;
T[n_, k_] /; 1, ] = 0;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 19 2019 *)

The g.f of the number k column is phi(k)(z) = (1/(1-z))*Product_{i=1..k-1}(1-z^i)/(1-2*z+z^(i+1)). - Richard Choulet, Feb 19 2010

Corrected and edited by Mats Granvik, Jul 28 2010, Dec 09 2010

Choulet formulas indices shifted (to adapt to the new column index) by R. J. Mathar, Dec 13 2010

A251128 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with no 2X2 subblock having the sum of its diagonal elements greater than the maximum of its antidiagonal elements.

Original entry on oeis.org

10, 21, 21, 40, 40, 40, 72, 69, 69, 72, 125, 117, 108, 117, 125, 212, 193, 173, 173, 193, 212, 354, 315, 272, 266, 272, 315, 354, 585, 510, 430, 401, 401, 430, 510, 585, 960, 823, 680, 612, 580, 612, 680, 823, 960, 1568, 1326, 1080, 938, 854, 854, 938, 1080, 1326

Offset: 1

R. H. Hardin, Nov 30 2014

Table starts
...10...21...40...72..125..212..354..585...960..1568..2553..4148..6730.10909
...21...40...69..117..193..315..510..823..1326..2136..3442..5550..8955.14458
...40...69..108..173..272..430..680.1080..1721..2752..4413..7093.11421.18415
...72..117..173..266..401..612..938.1452..2266..3565..5646..8991.14379.23071
..125..193..272..401..580..854.1268.1912..2921..4520..7069.11153.17717.28291
..212..315..430..612..854.1214.1743.2550..3795..5747..8835.13757.21640.34309
..354..510..680..938.1268.1743.2420.3429..4957..7321.11025.16890.26241.41224
..585..823.1080.1452.1912.2550.3429.4702..6585..9447.13873.20817.31818.49369
..960.1326.1721.2266.2921.3795.4957.6585..8926.12405.17694.25890.38762.59176
.1568.2136.2752.3565.4520.5747.7321.9447.12405.16680.23037.32726.47762.71410

			Some solutions for n=4 k=4
..0..1..1..1..1....0..0..0..1..1....0..0..1..0..1....1..0..0..0..1
..0..0..0..0..0....0..0..0..1..0....1..0..1..0..1....1..0..0..0..1
..1..1..1..1..1....0..0..0..1..0....1..0..1..0..1....1..0..0..0..0
..0..0..0..0..0....0..0..0..1..0....1..0..1..0..1....1..0..0..0..0
..0..0..0..0..0....1..1..0..1..0....1..0..1..0..0....1..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1104

Column 1 is A001891(n+2)

Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4)
k=2: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5)
k=3: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
k=4: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
k=5: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
k=6: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
k=7: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6

A213584 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 21, 16, 10, 4, 40, 32, 22, 13, 5, 72, 59, 43, 28, 16, 6, 125, 104, 78, 54, 34, 19, 7, 212, 178, 136, 97, 65, 40, 22, 8, 354, 299, 231, 168, 116, 76, 46, 25, 9, 585, 496, 386, 284, 200, 135, 87, 52, 28, 10, 960, 816, 638, 473, 337, 232, 154, 98, 58, 31, 11

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213585.
Antidiagonal sums: A213586.
Row 1, (1,2,3,5,...)**(1,2,3,4,...): A001891.
Row 2, (1,2,3,5,...)**(2,3,4,5,...): A023550.
Row 3, (1,2,3,5,...)**(3,4,5,6,...): A023554.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....10...21...40....72
2...7....16...32...59....104
3...10...22...43...78....136
4...13...28...54...97....168
5...16...34...65...116...200
6...19...40...76...135...232

Clark Kimberling, Antidiagonals n = 1..60

Cf. A000045, A213500, A213579.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5)))) # G. C. Greubel, Jul 08 2019

[[Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n-k+5] + k*Fibonacci[n-k+4] -2*n-5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

t(n,k) = fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n, k) = Fibonacci(k+4) + n*Fibonacci(k+3) - 2*(n+k) - 3. - G. C. Greubel, Jul 08 2019

A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2n-3+2h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 4, 3, 10, 8, 5, 21, 18, 12, 7, 40, 35, 26, 16, 9, 72, 64, 49, 34, 20, 11, 125, 112, 88, 63, 42, 24, 13, 212, 191, 152, 112, 77, 50, 28, 15, 354, 320, 257, 192, 136, 91, 58, 32, 17, 585, 530, 428, 323, 232, 160, 105, 66, 36, 19

Offset: 1

Clark Kimberling, Jun 21 2012

Principal diagonal:  A213769.
Antidiagonal sums:  A213770.
Row 1,  (1,1,2,3,5,...)**(1,3,5,7,9,...): A001891.
Row 2,  (1,1,2,3,5,...)**(3,5,7,9,11,...).
Row 3,  (1,1,2,3,5,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....4....10...21...40....72....125
3....8....18...35...64....112...191
5....12...26...49...88....152...257
7....16...34...63...112...192...323
9....20...42...77...136...232...389
11...24...50...91...160...272...455

Clark Kimberling, Antidiagonals n=1..80, flattened

Cf. A001891, A213500, A213769, A213770.

b[n_] := Fibonacci[n]; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213768 *)
Table[t[n, n], {n, 1, 40}] (* A213769 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213770 *)

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - x - x^2)(1 - x )^2.
T(n,k) = 2*n*Fibonacci(k+2) + Lucas(k+2) - 2*(k+n) - 3. - Ehren Metcalfe, Jul 08 2019

A033937 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= 3.

Original entry on oeis.org

2, 7, 17, 35, 66, 118, 204, 345, 575, 949, 1556, 2540, 4134, 6715, 10893, 17655, 28598, 46306, 74960, 121325, 196347, 317737, 514152, 831960, 1346186, 2178223, 3524489, 5702795, 9227370, 14930254, 24157716

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001924, A001891.

List([0..40], n-> Fibonacci(n+7) -3*n-11) # G. C. Greubel, Jul 05 2019

[Fibonacci(n+7) -3*n-11: n in [0..40]]; // G. C. Greubel, Jul 05 2019

Table[Fibonacci[n+7] -3*n-11, {n,0,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, n--; fibonacci(n+7) -3*n-11) \\ G. C. Greubel, Jul 05 2019

[fibonacci(n+7) -3*n-11 for n in (0..40)] # G. C. Greubel, Jul 05 2019

a(n) = Fibonacci(n+7) - (11+3*n).

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

A213576 Rectangular array: (row n) = bc, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 3, 1, 7, 4, 2, 14, 10, 7, 3, 26, 21, 17, 11, 5, 46, 40, 35, 27, 18, 8, 79, 72, 66, 56, 44, 29, 13, 133, 125, 118, 106, 91, 71, 47, 21, 221, 212, 204, 190, 172, 147, 115, 76, 34, 364, 354, 345, 329, 308, 278, 238, 186, 123, 55, 596, 585, 575, 557, 533, 498, 450, 385, 301, 199, 89

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213577.
Antidiagonal sums:  A213578.
Row 1,  (1,2,3,...)**(1,1,2,3,5,...): A001924;
Row 2,  (1,2,3,...)**(1,2,3,5,8,...): A001891;
Row 3,  (1,2,3,...)**(2,3,5,8,13,...): A033937;
Row 4,  (1,2,3,...)**(3,5,8,13,21,...): A033960;
Row 5,  (1,2,3,...)**(5,8,13,21,...): A037140;
Row 6,  (1,2,3,...)**(8,13,21,34,...): A037157.
For a guide to related arrays, see A213500.
The falling antidiagonal rows can be computed by the sum Sum_{j=0..n-k} (n-k-j+1)*Fibonacci(k+j) which can also be seen as Fibonacci(n+4) - Lucas(k+2) - (n-k)*Fibonacci(k+1). - G. C. Greubel, Jul 05 2019

			Northwest corner (the array is read by falling antidiagonals):
  1,   3,   7,  14,  26,  46,  79
  1,   4,  10,  21,  40,  72, 125
  2,   7,  17,  35,  66, 118, 204
  3,  11,  27,  56, 106, 190, 329
  5,  18,  44,  91, 172, 308, 533
  8,  29,  71, 147, 278, 498, 862

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

Flat(List([1..10], n-> List([1..n], k-> Fibonacci(n+4) - (n-k+1) *Fibonacci(k+1) - Fibonacci(k+3)))); # G. C. Greubel, Jul 05 2019

[[Fibonacci(n+4) -(n-k)*Fibonacci(k+1) -Lucas(k+2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Jul 05 2019

(* First Program *)
b[n_]:= n; c[n_]:= Fibonacci[n];
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213576 *)
r[n_]:= Table[t[n, k], {k,1,40}]  (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n, 1, 40}] (* A213577 *)
s[n_]:= Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* Second Program *)
T[n_, k_]:= Fibonacci[n+4] - (n-k)*Fibonacci[k+1] - LucasL[k+2];
Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

T(n, k)= fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3);
for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

[[fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Jul 05 2019

Rows: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
Columns: T(n,k) = T(n-1,k) + T(n-2,k).
G.f. for row n: f(x)/g(x), where f(x) = F(n) - F(n-1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n,k) = F(n+k+3) - k*F(n+1) - F(n+3). - Ehren Metcalfe, Jul 04 2019

A213765 Rectangular array: (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 4, 1, 10, 5, 2, 21, 14, 9, 3, 40, 31, 24, 14, 5, 72, 61, 52, 38, 23, 8, 125, 112, 101, 83, 62, 37, 13, 212, 197, 184, 162, 135, 100, 60, 21, 354, 337, 322, 296, 263, 218, 162, 97, 34, 585, 566, 549, 519, 480, 425, 353, 262, 157, 55, 960, 939, 920, 886

Offset: 1

Clark Kimberling, Jun 21 2012

Principal diagonal: A213766.
Antidiagonal sums: A213767.
Row 1,  (1,3,5,7,9,...)**(1,1,2,3,5,...): A001891.
Row 2,  (1,3,5,7,9,...)**(1,2,3,5,8,...): A023652.
Row 3,  (1,3,5,7,9,...)**(2,3,5,8,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....4....10....21....40....72
1....5....14....31....61....112
2....9....24....52....101...184
3....14...38....83....162...296
5....23...62....135...263...480
8....37...100...218...425...776
13...60...162...353...688...1256

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

b[n_] := 2 n - 1; c[n_] := Fibonacci[n];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213765 *)
Table[t[n, n], {n, 1, 40}] (* A213766 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213767 *)

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = x*(F(n) + F(n+1)*x - F(n-1)*x^2) and g(x) = (1 - x - x^2)(1 - x )^2.
T(n,k) = F(n+k+4) - 2*k*F(n+1) - F(n+4), F = A000045. - Ehren Metcalfe, Jul 10 2019

A001887 Number of permutations p of {1,2,...,n} such that p(i) - i < 0 or p(i) - i > 2 for all i.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 33, 236, 1918, 17440, 175649, 1942171, 23396353, 305055960, 4280721564, 64330087888, 1030831875953, 17545848553729, 316150872317105, 6012076099604308, 120330082937778554

Offset: 0

N. J. A. Sloane

nonn

Previous name was: Hit polynomials.

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..300
E. R. Canfield, N. C. Wormald, Menage numbers, bijections and P-recursiveness, Discr. Math. 63 (1987) 117, table Section 7.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 224.
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A000255, A000271, A001883-A001891, A075851, A075852.

First column of A080061.

nmax = 21;
gf = 1/(x^2-1)(x-Sum[n! (x(x-1)/(x^3-2x-1))^n + O[x]^nmax, {n, 0, nmax}]);
CoefficientList[gf, x] (* Jean-François Alcover, Aug 19 2018 *)

G.f.: (1/(x^2-1))*(x-Sum_{n>=0} n!*(x*(x-1)/(x^3-2*x-1))^n). - Vladeta Jovovic, Jun 30 2007
D-finite with recurrence (P. Flajolet, 1997): a(n) = (n-1)*a(n-1) + (n+2)*a(n-2) - (3*n-13)*a(n-3) - (2*n-8)*a(n-4) + (3*n-15)*a(n-5) + (n-4)*a(n-6) - (n-7)*a(n-7) - a(n-8), n>8.
a(n) ~ exp(-3) * n!. - Vaclav Kotesovec, Sep 10 2014

More terms from Vladimir Baltic and Vladeta Jovovic, Jan 05 2003

New name from Vaclav Kotesovec using a former comment by Vladimir Baltic and Vladeta Jovovic, Sep 16 2014

A033960 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= 4.

Original entry on oeis.org

3, 11, 27, 56, 106, 190, 329, 557, 929, 1534, 2516, 4108, 6687, 10863, 17623, 28564, 46270, 74922, 121285, 196305, 317693, 514106, 831912, 1346136, 2178171, 3524435, 5702739, 9227312, 14930194, 24157654, 39088001

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001924, A001891, A033937.

List([0..40], n-> Fibonacci(n+8) -5*n-18) # G. C. Greubel, Jul 05 2019

[Fibonacci(n+8) -5*n-18: n in [0..40]]; // G. C. Greubel, Jul 05 2019

Table[Fibonacci[n+8] -5*n-18, {n,0,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, n--; fibonacci(n+8) -5*n-18) \\ G. C. Greubel, Jul 05 2019

[fibonacci(n+8) -5*n-18 for n in (0..40)] # G. C. Greubel, Jul 05 2019

a(n) = Fibonacci(n+8) - (18+5*n).

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

A037140 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), for k >= 5.

Original entry on oeis.org

5, 18, 44, 91, 172, 308, 533, 902, 1504, 2483, 4072, 6648, 10821, 17578, 28516, 46219, 74868, 121228, 196245, 317630, 514040, 831843, 1346064, 2178096, 3524357, 5702658, 9227228, 14930107, 24157564, 39087908, 63245717, 102333878, 165579856, 267914003

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001924, A001891, A033937, A033960.

List([0..40], n-> Fibonacci(n+9) -8*n-29) # G. C. Greubel, Jul 05 2019

[Fibonacci(n+9) -8*n-29: n in [0..40]]; // G. C. Greubel, Jul 05 2019

Table[Fibonacci[n+9] -8*n-29, {n,0,40}] (* G. C. Greubel, Jul 05 2019 *)

vector(40, n, n--; fibonacci(n+9) -8*n-29) \\ G. C. Greubel, Jul 05 2019

[fibonacci(n+9) -8*n-29 for n in (0..40)] # G. C. Greubel, Jul 05 2019

a(n) = Fibonacci(n+9) - (29+8*n).

G.f.: (5+3*x)/((1-x-x^2)*(1-x)^2).

Corrected by Franklin T. Adams-Watters, Oct 25 2006

cp's OEIS Frontend

A175105 Triangle T(n,k) read by rows. T(n,1)=1; T(n,k) = Sum_{i=1..k-1} ( T(n-i,k-1) + T(n-i,k) ), k>1.

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A251128 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with no 2X2 subblock having the sum of its diagonal elements greater than the maximum of its antidiagonal elements.

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A213584 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A033937 Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= 3.

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A213576 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and ** = convolution.

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A213765 Rectangular array: (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A213584 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2n-3+2h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

A213576 Rectangular array: (row n) = bc, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and = convolution.