A030191 -id:A030191 - cp's OEIS frontend

A094788 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 6.

1, 6, 27, 110, 428, 1624, 6069, 22458, 82555, 302082, 1101816, 4009616, 14567657, 52865230, 191684283, 694609494, 2515972324, 9110338728, 32981059485, 119377761602, 432046756571, 1563510554986, 5657752486512, 20472344560800

Offset: 2

Herbert Kociemba, Jun 15 2004

Diagonal of the square array A217593. - Philippe Deléham, Mar 28 2013

Michael De Vlieger, Table of n, a(n) for n = 2..1791
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Drop[CoefficientList[Series[-x^2*(-1 + 2 x)/((x^2 - 3 x + 1) (5 x^2 - 5 x + 1)), {x, 0, 25}], x], 2] (* Michael De Vlieger, Aug 04 2021 *)
LinearRecurrence[{8,-21,20,-5},{1,6,27,110},30] (* Harvey P. Dale, Aug 31 2021 *)

Vec(x^2*(1-2*x)/(1-8*x+21*x^2-20*x^3+5*x^4)+O(x^66)) /* Joerg Arndt, Mar 29 2013 */

a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)*sin(3*r*Pi/5)*(2*cos(r*Pi/10))^(2*n+1).
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).
G.f.: -x^2*(-1+2*x) / ( (x^2-3*x+1)*(5*x^2-5*x+1) ).
a(n+2) = A217593(n,n+5). - Philippe Deléham, Mar 28 2013
2*a(n) = A030191(n-1) - A001906(n). - R. J. Mathar, Nov 15 2019

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, -1, 0, 1, 4, 6, 0, -1, 0, 1, 5, 12, 9, -4, 0, 0, 1, 6, 20, 32, 9, -8, 1, 0, 1, 7, 30, 75, 80, 0, -8, 1, 0, 1, 8, 42, 144, 275, 192, -27, 0, 0, 0, 1, 9, 56, 245, 684, 1000, 448, -81, 16, -1, 0, 1, 10, 72, 384, 1421, 3240, 3625, 1024, -162, 32, -1, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1,  1,  1, 1,   1,    1, ...
  0,  1,  2, 3,   4,    5, ...
  0,  0,  2, 6,  12,   20, ...
  0, -1,  0, 9,  32,   75, ...
  0, -1, -4, 9,  80,  275, ...
  0,  0, -8, 0, 192, 1000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.
Rows 0..1 give A000012, A001477.
Main diagonal gives (-1) * A109519(n+1).
Cf. A342120, A342133, A342134.

T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021

T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));

T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).
T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

A112091 Number of idempotent order-preserving partial transformations (of an n-element chain).

Original entry on oeis.org

1, 2, 6, 21, 76, 276, 1001, 3626, 13126, 47501, 171876, 621876, 2250001, 8140626, 29453126, 106562501, 385546876, 1394921876, 5046875001, 18259765626, 66064453126, 239023437501, 864794921876, 3128857421876, 11320312500001

Offset: 0

Abdullahi Umar, Aug 25 2008

			a(2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 2-element chain), namely: the empty map, (1)->(1), (2)->(2), (1,2)->(1,1), (1,2)->(1,2), (1,2)->(2,2); the mappings are coordinate-wise.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Callan and T. Mansour, Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns, arXiv:1705.00933 [math.CO] (2017), Table 2 No 200 (offset 1 then).
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
Index entries for linear recurrences with constant coefficients, signature (6,-10,5).

[ n eq 1 select 1 else n eq 2 select 2 else n eq 3 select 6 else 6*Self(n-1)-10*Self(n-2)+ 5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 21 2011

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==1+5(a[n-1]-a[n-2])},a[n], {n,30}] (* or *) LinearRecurrence[{6,-10,5},{1,2,6},31] (* Harvey P. Dale, Aug 20 2011 *)

Vec((2*x-1)^2/(1-x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = ((sqrt(5))^(n - 1))*(((sqrt(5) + 1)/2)^n - ((sqrt(5) - 1)/2)^n) + 1. [corrected by Jason Yuen, Sep 06 2024]
a(n) = 1 + 5*(a(n-1) - a(n-2)), a(0) = 1, a(1) = 2.
G.f.: (1 - 2*x)^2/((1 - x)*(1 - 5*x + 5*x^2)). Convolution of A081567 with the sequence 1, -1, -1, -1 (-1 continued). - R. J. Mathar, Sep 06 2008
a(n) = 1 + A030191(n-1). - R. J. Mathar, Jun 20 2011
a(n) = 6*a(n-1) - 10*a(n-2) + 5*a(n-3); a(0) = 1, a(1) = 2, a(2) = 6. - Harvey P. Dale, Aug 20 2011
E.g.f.: exp(x) + (exp((5 + sqrt(5))*x/2) - exp((5 - sqrt(5))*x/2))/sqrt(5). - Franck Maminirina Ramaharo, Nov 09 2018

A217593 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=1 or if k-n >= 9, T(0,k) = 1 for k = 0..8, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 1, 5, 9, 5, 0, 0, 0, 1, 6, 14, 14, 0, 0, 0, 0, 1, 7, 20, 28, 14, 0, 0, 0, 0, 0, 8, 27, 48, 42, 0, 0, 0, 0, 0, 0, 8, 35, 75, 90, 42, 0, 0, 0, 0, 0, 0, 0, 43, 110, 165, 132, 0, 0, 0, 0, 0, 0, 0, 0, 43, 153, 275, 297, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 196, 428, 572, 429, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 18 2013

			Square array begins :
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 0, ...
0, 0, 2, 5, 9, 14, 20, 27, 35, 43, 43, 0, 0, ...
0, 0, 0, 5, 14, 28, 75, 110, 153, 196, 196, 0, 0, ....
0, 0, 0, 0, 14, 42, 90, 165, 275, 428, 624, 820, 820, 0, 0, ...
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 7, 8, 8
2, 5, 9, 14, 20, 27, 35, 43, 43
5, 14, 28, 48, 75, 110, 153, 196, 196
14, 42, 90, 165, 275, 428, 624, 820, 820
42, 132, 297, 572, 1000, 1624, 2444, 3264, 3264
132, 429, 1001, 2001, 3625, 6069, 9333, 12597, 12597
429, 1430, 3431, 7056, 13125, 22458, 35055, 47652, 47652
...

A hexagon arithmetic of E. Lucas.

T(n,n)   = A033191(n).
T(n,n+1) = A033191(n+1).
T(n,n+2) = A033190(n+1).
T(n,n+3) = A094667(n+1).
T(n,n+4) = A093131(n+1) = A030191(n).
T(n,n+5) = A094788(n+2).
T(n,n+6) = A094825(n+3).
T(n,n+7) = T(n,n+8) = A094865(n+3).
Sum_{k, 0<=k<=n} T(n-k,k) = A178381(n).

A081576 Square array of binomial transforms of Fibonacci numbers, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 3, 0, 1, 7, 20, 21, 5, 0, 1, 9, 38, 75, 55, 8, 0, 1, 11, 62, 189, 275, 144, 13, 0, 1, 13, 92, 387, 905, 1000, 377, 21, 0, 1, 15, 128, 693, 2305, 4256, 3625, 987, 34, 0, 1, 17, 170, 1131, 4955, 13392, 19837, 13125, 2584, 55

Offset: 0

Paul Barry, Mar 22 2003

Array rows are solutions of the recurrence a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2) where a(0) = 0 and a(1) = 1.

			Square array begins as:
  0, 1,  1,   2,    3,    5,     8, ... A000045;
  0, 1,  3,   8,   21,   55,   144, ... A001906;
  0, 1,  5,  20,   75,  275,  1000, ... A030191;
  0, 1,  7,  38,  189,  905,  4256, ... A099453;
  0, 1,  9,  62,  387, 2305, 13392, ... A081574;
  0, 1, 11,  92,  693, 4955, 34408, ... A081575;
  0, 1, 13, 128, 1131, 9455, 76544, ...
The antidiagonal triangle begins as:
  0;
  0, 1;
  0, 1,  1;
  0, 1,  3,  2;
  0, 1,  5,  8,   3;
  0, 1,  7, 20,  21,   5;
  0, 1,  9, 38,  75,  55,   8;
  0, 1, 11, 62, 189, 275, 144, 13;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Array row n: A000045 (n=0), A001906 (n=1), A030191 (n=2), A099453 (n=3), A081574 (n=4), A081575 (n=5).
Array columns k: A005408 (k=3), A077588 (k=4).
Cf. A028387, A081573, A081574, A081575.

A081576:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j)*(n-k)^(k-j): j in [0..k]]) >;
[A081576(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

T[n_, k_]:= If[n==0, Fibonacci[k], Sum[Binomial[k, j]*Fibonacci[j]*n^(k-j), {j, 0, k}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, May 26 2021 *)

def A081576(n,k): return sum( binomial(k,j)*fibonacci(j)*(n-k)^(k-j) for j in (0..k) )
flatten([[A081576(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Rows are successive binomial transforms of F(n).
T(n, k) = ( ( (2*n + 1 + sqrt(5))/2 )^k - ( (2*n + 1 - sqrt(5))/2 )^k )/sqrt(5).
From G. C. Greubel, May 26 2021: (Start)
T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*n^(k-j) with T(0, k) = Fibonacci(k) (square array).
T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*(n-k)^(k-j) (antidiagonal triangle). (End)

A084330 a(0)=0, a(1)=1, a(n) = 31a(n-1) - 29a(n-2).

Original entry on oeis.org

0, 1, 31, 932, 27993, 840755, 25251608, 758417953, 22778659911, 684144336604, 20547893297305, 617144506454939, 18535590794481264, 556706123941725953, 16720357709153547887, 502186611389449931860, 15082894579507494998937, 453006320234438296943107

Offset: 0

Benoit Cloitre, Jun 21 2003

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (31,-29).

Cf. A030191.

I:=[0,1]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 02 2015

f:=proc(n) option remember; if n <=1 then n else 31*f(n-1)-29*f(n-2); fi; end;

LinearRecurrence[{31,-29},{0,1},30] (* Harvey P. Dale, Jul 11 2014 *)

a(n)=(1/13)*sum(k=0,n,binomial(n,k)*fibonacci(7*k))

a(n) = (1/13)*sum(k=0, n, binomial(n, k)*F(7*k)) where F(k) denotes the k-th Fibonacci number.

G.f.: x / (29*x^2-31*x+1). - Colin Barker, Jun 26 2013

Corrected by N. J. A. Sloane, Sep 16 2005

A087453 a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)Fibonacci(mk).

Original entry on oeis.org

0, 5, 65, 790, 9555, 115525, 1396720, 16886585, 204161685, 2468349470, 29842764575, 360804095305, 4362182828640, 52739531723965, 637629901296505, 7709053867890950, 93203771368320795, 1126849435241369885, 13623801173086279760, 164714071462466568145

Offset: 0

Benoit Cloitre, Oct 23 2003

Colin Barker, Table of n, a(n) for n = 0..923
Index entries for linear recurrences with constant coefficients, signature (13,-11).

Cf. A001906 (S(n,1)), A030191 (S(n,2)).

I:=[0,5]; [n le 2 select I[n] else 13*Self(n-1)-11*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Apr 27 2015

Table[Sum[Binomial[n,k]Fibonacci[5k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 03 2014 *)
LinearRecurrence[{13, -11}, {0, 5}, 20] (* Vincenzo Librandi, Apr 27 2015 *)

concat(0, Vec(5*x/(11*x^2-13*x+1) + O(x^100))) \\ Colin Barker, Apr 27 2015

a(n) = 13*a(n-1)-11*a(n-2).
a(n) = (1/sqrt(5))*(((13+5*sqrt(5))/2)^n-((13-5*sqrt(5))/2)^n).
G.f.: 5*x / (11*x^2-13*x+1). - Colin Barker, Apr 27 2015

A087635 a(n) = S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)Fibonacci(mk).

Original entry on oeis.org

0, 2, 12, 64, 336, 1760, 9216, 48256, 252672, 1323008, 6927360, 36272128, 189923328, 994451456, 5207015424, 27264286720, 142757658624, 747488804864, 3913902194688, 20493457948672, 107305138913280, 561857001684992

Offset: 0

Benoit Cloitre, Oct 23 2003

Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A000045, A001906 (S(n, 1)), A030191 (S(n, 2)).

Cf. A084326.

LinearRecurrence[{6,-4}, {0, 2}, 22] (* Amiram Eldar, Apr 29 2025 *)

a(n) = 6*a(n-1)-4*a(n-2) = 2*A084326(n).
a(n) = Sum_{0<=j<=i<=n} C(i,j)*C(n,i)*Fibonacci(i+j). - Benoit Cloitre, May 21 2005
a(n) = 2^n*Fibonacci(2*n). - Benoit Cloitre, Sep 13 2005
a(n) = Sum_{k=0..n} C(n,k)*Fibonacci(k)*Lucas(n-k). - Ross La Haye, Aug 14 2006
G.f.: 2*x/(1-6*x+4*x^2). - Colin Barker, Jun 19 2012

A099449 An Alexander sequence for the knot 7_6.

Original entry on oeis.org

1, 5, 18, 60, 197, 650, 2153, 7140, 23682, 78545, 260498, 863945, 2865282, 9502740, 31515953, 104523050, 346651997, 1149675660, 3812913618, 12645575405, 41939208002, 139091904605, 461300030418, 1529907284460, 5073956524397

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 7_6. The g.f. is the image of the g.f. of A030191 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2))

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (5,-7,5,-1).

Cf. A030191, A099448.

I:=[1,5,18,60,197,650,2153,7140]; [n le 8 select I[n] else 5*Self(n-1)-7*Self(n-2)+5*Self(n-3)-Self(n-4) : n in [1..30]]; // Vincenzo Librandi, Feb 12 2014

CoefficientList[Series[(1 - x) (x + 1) (x^2 + 1)/(x^4 -5 x^3 + 7 x^2 - 5 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{5,-7,5,-1},{1,5,18,60,197},30] (* Harvey P. Dale, Oct 06 2015 *)

Vec(-(x-1)*(x+1)*(x^2+1)/(x^4-5*x^3+7*x^2-5*x+1) + O(x^100)) \\ Colin Barker, Feb 10 2014

G.f.: -(x-1)*(x+1)*(x^2+1) / (x^4-5*x^3+7*x^2-5*x+1). - Colin Barker, Feb 10 2014
a(n) = A099448(n) - A099448(n-2).
a(n) = 5*a(n-1)-7*a(n-2)+5*a(n-3)-a(n-4) for n>4. - Colin Barker, Feb 10 2014

G.f. corrected by Colin Barker, Feb 10 2014

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

cp's OEIS Frontend

A094788 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 6.

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x + k*x^2).

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A112091 Number of idempotent order-preserving partial transformations (of an n-element chain).

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A217593 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=1 or if k-n >= 9, T(0,k) = 1 for k = 0..8, T(n,k) = T(n-1,k) + T(n,k-1).

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Formula

A081576 Square array of binomial transforms of Fibonacci numbers, read by antidiagonals.

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Magma

Mathematica

Sage

Formula

A084330 a(0)=0, a(1)=1, a(n) = 31*a(n-1) - 29*a(n-2).

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A087453 a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

A084330 a(0)=0, a(1)=1, a(n) = 31a(n-1) - 29a(n-2).

A087453 a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)Fibonacci(mk).

A087635 a(n) = S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)Fibonacci(mk).