A069306 -id:A069306 - cp's OEIS frontend

A069299 Number of n X 8 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

985, 453423, 158755329, 48861110849, 14020698615685, 3860368291738151, 1036037626157286045, 273552889555043981667, 71469066339273464672861, 18543498700760853810422275, 4789484633305049073669332993, 1233335138386307928872371551119

Offset: 2

R. H. Hardin, Mar 14 2002

nonn

Cf. n X 2 A002450, n X 3 A069294, n X 4 A069295, n X 5 A069296, n X 6 A069297, n X 7 A069298, n X 9 A069300, n X 10 A069301, n X 11 A069302, n X 12 A069303, n X 13 A069304, n X 14 A069305, read by rows A069306-A069320.

a(7)-a(13) from Sean A. Irvine, Apr 25 2024

A069300 Number of n X 9 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

Original entry on oeis.org

2378, 2396868, 1771648672, 1130534931080, 665580139683936, 373423075547578456, 203239076083010593472, 108443838410200501279708, 57104283197019200756790188, 29803331617203085845704313992, 15460814292574112201614850454092, 7987403217621240109184116174886680

Offset: 2

R. H. Hardin, Mar 14 2002

nonn

Cf. n X 2 A002450, n X 3 A069294, n X 4 A069295, n X 5 A069296, n X 6 A069297, n X 7 A069298, n X 8 A069299, n X 10 A069301, n X 11 A069302, n X 12 A069303, n X 13 A069304, n X 14 A069305, read by rows A069306-A069320.

a(7)-a(13) from Sean A. Irvine, Apr 25 2024

A069301 Number of n X 10 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

Original entry on oeis.org

5741, 12670261, 19771329973, 26160665637805, 31605577678427531, 36146216722785667861, 39916149930281967457205, 43066986985849138740694549, 45738751568329997358178054329, 48049989770896486757102805620329, 50096394975413699108192666406325657

Offset: 2

R. H. Hardin, Mar 14 2002

nonn

Cf. n X 2 A002450, n X 3 A069294, n X 4 A069295, n X 5 A069296, n X 6 A069297, n X 7 A069298, n X 8 A069299, n X 9 A069300, n X 11 A069302, n X 12 A069303, n X 13 A069304, n X 14 A069305, read by rows A069306-A069320.

a(6)-a(12) from Sean A. Irvine, Apr 25 2024

A069302 Number of n X 11 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

Original entry on oeis.org

13860, 66977242, 220646655318, 605385196535596, 1501028805154667736, 3500044399151834410528, 7844661838902748923237822, 17121637307123834033940314568, 36691638903540665604692077389476, 77626289171582272584203485766947172

Offset: 2

R. H. Hardin, Mar 14 2002

nonn

Cf. n X 2 A002450, n X 3 A069294, n X 4 A069295, n X 5 A069296, n X 6 A069297, n X 7 A069298, n X 8 A069299, n X 9 A069300, n X 10 A069301, n X 12 A069303, n X 13 A069304, n X 14 A069305, read by rows A069306-A069320.

a(6)-a(11) from Sean A. Irvine, Apr 25 2024

A069303 Number of n X 12 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

Original entry on oeis.org

33461, 354053617, 2462407613981, 14009490587031167, 71292415421456529651, 338969996144009158522891, 1542263634548542336151173999, 6811155547203848959429866325465, 29462439537258441022480797977381647, 125575826853940088096659037959183104657

Offset: 2

R. H. Hardin, Mar 14 2002

nonn

Cf. n X 2 A002450, n X 3 A069294, n X 4 A069295, n X 5 A069296, n X 6 A069297, n X 7 A069298, n X 8 A069299, n X 9 A069300, n X 10 A069301, n X 11 A069302, n X 13 A069304, n X 14 A069305, read by rows A069306-A069320.

a(6)-a(11) from Sean A. Irvine, Apr 25 2024

A069304 Number of n X 13 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

Original entry on oeis.org

80782, 1871590506, 27480393415984, 324202249915245972, 3386189554819795682250, 32831359005580196869725148, 303271759994928192091591236664, 2710567644648890574598792001139520, 23671928574368475557299894398216017188, 203322469544117825662881107321485212321552

Offset: 2

R. H. Hardin, Mar 14 2002

nonn

Cf. n X 2 A002450, n X 3 A069294, n X 4 A069295, n X 5 A069296, n X 6 A069297, n X 7 A069298, n X 8 A069299, n X 9 A069300, n X 10 A069301, n X 11 A069302, n X 12 A069303, n X 14 A069305, read by rows A069306-A069320.

a(6)-a(11) from Sean A. Irvine, Apr 26 2024

A215928 a(n) = 2*a(n-1) + a(n-2) for n > 2, a(0) = a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149, 107578520350, 259717522849

Offset: 0

Michael Somos, Aug 27 2012

Number of 132-avoiding two-stack sortable permutations. See Theorem 2.2 of Egge and Mansour which gives a generating function equation P(x) = 1 + x + 2*x^2 + x*(P(x) - 1 - x) + x^2*(P(x) - 1) + x*(P(x) - 1 - x).
Row sums of triangle A155161. - Philippe Deléham, Aug 31 2012
a(n) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 1, 1, 1; 0, 1, 0] or [1, 1, 0; 1, 1, 1; 1, 1, 0] or [1, 1, 1; 0, 0, 1; 1, 1, 1] or [1, 0, 1; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
For n > 0, A001333(n)/a(n) = A001333(n)/A000129(n), which converges to sqrt(2). - Karl V. Keller, Jr., May 17 2015

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 70*x^6 + 169*x^7 + 408*x^8 + 985*x^9 + ...

Karl V. Keller, Jr., Table of n, a(n) for n = 0..500
Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.
E. S. Egge and T. Mansour, 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, arXiv:math/0205206 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A000045, A024537, A069306, A078057, A097075, A142474.

[1] cat [ n le 2 select (n) else 2*Self(n-1)+Self(n-2): n in [1..35] ]; // Vincenzo Librandi, May 14 2015

f:= gfun:-rectoproc({a(n)=2*a(n-1)+a(n-2), a(0)=1, a(1)=1, a(2)=2}, a(n), remember):
map(f, [$0..100]); # Robert Israel, May 29 2015

CoefficientList[Series[(1 - x - x^2)/(1 - 2 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 14 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / (1 - x / (1 - x / (1 - x / (1 + x)))) + x * O(x^n), n))};

a(n) = 2*a(n-1) + a(n-2) for n > 2, a(0) = a(1) = 1, a(2) = 2.
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 + x)))) = (1 - x - x^2) / (1 - 2*x - x^2).
a(n) = A000129(n) unless n = 0.
a(n+1) - a(n) = A078057(n-1).
PSUM transform is A024537.
PSUMSIGN transform is A097075.
INVERT transform of A000045(n). [Corrected by Wolfdieter Lang, Dec 07 2020]
G.f.: 1/( 1 - (Sum_{k>=0} x*(x + x^2)^k) ) = 1/( 1 - (Sum_{k>=1} (x/(1 - x^2))^k) ). - Joerg Arndt, Sep 30 2012
G.f.: 1 + Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 + x)/( x*(4*k + 4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013
a(n) = A069306(n-1) if n > 1. - Michael Somos, Oct 23 2018
E.g.f.: 1 + exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Franck Maminirina Ramaharo, Nov 29 2018

A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 4, 1, 1, 2, 4, 6, 3, 2, 1, 2, 5, 8, 6, 6, 1, 1, 2, 6, 10, 10, 12, 4, 2, 1, 2, 7, 12, 15, 20, 10, 8, 1, 1, 2, 8, 14, 21, 30, 20, 20, 5, 2, 1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1, 1, 2, 10, 18, 36, 56, 56, 70, 35, 30, 6, 2

Offset: 0

Peter Luschny, Jul 22 2024

There are several versions of Lucas and Fibonacci polynomials in this database. Our naming follows the convention of calling polynomials after the values of the polynomials at x = 1. This assumes a regular sequence of polynomials, that is, a sequence of polynomials where degree(p(n)) = n. This view makes the coefficients of the polynomials (the terms of a row) a refinement of the values at the unity.
A remarkable property of the polynomials under consideration is that they are dual in this respect. This means they give the Lucas numbers at x = 1 and the Fibonacci numbers at x = -1 (except for the sign). See the example section.
The Pell numbers and the dual Pell numbers are also values of the polynomials, at the points x = -1/2 and x = 1/2 (up to the normalization factor 2^n). This suggests a harmonized terminology: To call 2^n*P(n, -1/2) = 1, 0, 1, 2, 5, ... the Pell numbers (A000129) and 2^n*P(n, 1/2) = 1, 4, 9, 22, ... the dual Pell numbers (A048654).
Based on our naming convention one could call A162515 (without the prepended 0) the Fibonacci polynomials. In the definition above only the initial values would change to: T(n, k) = k + 1 for k < 1. To extend this line of thought we introduce A374438 as the third triangle of this family.
The triangle is closely related to the qStirling2 numbers at q = -1. For the definition of these numbers see A333143. This relates the triangle to A065941 and A103631.

			Triangle starts:
  [ 0] [1]
  [ 1] [1, 2]
  [ 2] [1, 2, 1]
  [ 3] [1, 2, 2,  2]
  [ 4] [1, 2, 3,  4,  1]
  [ 5] [1, 2, 4,  6,  3,  2]
  [ 6] [1, 2, 5,  8,  6,  6,  1]
  [ 7] [1, 2, 6, 10, 10, 12,  4,  2]
  [ 8] [1, 2, 7, 12, 15, 20, 10,  8,  1]
  [ 9] [1, 2, 8, 14, 21, 30, 20, 20,  5,  2]
  [10] [1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1]
.
Table of interpolated sequences:
  |  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |
  |  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|
  |    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |
  |  0 |        -1         |     1   |       1     |       1    |
  |  1 |         1         |     3   |       0     |       4    |
  |  2 |         0         |     4   |       1     |       9    |
  |  3 |         1         |     7   |       2     |      22    |
  |  4 |         1         |    11   |       5     |      53    |
  |  5 |         2         |    18   |      12     |     128    |
  |  6 |         3         |    29   |      29     |     309    |
  |  7 |         5         |    47   |      70     |     746    |
  |  8 |         8         |    76   |     169     |    1801    |
  |  9 |        13         |   123   |     408     |    4348    |

Paolo Xausa, Rows n = 0..150 of the triangle, flattened
Peter Luschny, Illustrating the polynomials.

Triangles related to Lucas polynomials: A034807, A114525, A122075, A061896, A352362.
Triangles related to Fibonacci polynomials: A162515, A053119, A168561, A049310, A374441.
Sums include: A000204 (Lucas numbers, row), A000045 & A212804 (even sums, Fibonacci numbers), A006355 (odd sums), A039834 (alternating sign row).
Type m^n*P(n, 1/m): A000129 & A048654 (Pell, m=2), A108300 & A003688 (m=3), A001077 & A048875 (m=4).
Adding and subtracting the values in a row of the table (plus halving the values obtained in this way): A022087, A055389, A118658, A052542, A163271, A371596, A324969, A212804, A077985, A069306, A215928.
Columns include: A040000 (k=1), A000027 (k=2), A005843 (k=3), A000217 (k=4), A002378 (k=5).
Diagonals include: A000034 (k=n), A029578 (k=n-1), abs(A131259) (k=n-2).
Cf. A029578 (subdiagonal), A124038 (row reversed triangle, signed).
Cf. A039961, A065941 (qStirling2), A103631, A108299, A131259. A133607, A194005, A333143, A374438 (m=3).

function T(n,k) // T = A374439
  if k lt 0 or k gt n then return 0;
  elif k le 1 then return k+1;
  else return T(n-1,k) + T(n-2,k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2025

A374439 := (n, k) -> ifelse(k::odd, 2, 1)*binomial(n - irem(k, 2) - iquo(k, 2), iquo(k, 2)):
# Alternative, using the function qStirling2 from A333143:
T := (n, k) -> 2^irem(k, 2)*qStirling2(n, k, -1):
seq(seq(T(n, k), k = 0..n), n = 0..10);

A374439[n_, k_] := (# + 1)*Binomial[n - (k + #)/2, (k - #)/2] & [Mod[k, 2]];
Table[A374439[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, Jul 24 2024 *)

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k > n: return 0
    if k < 2: return k + 1
    return T(n - 1, k) + T(n - 2, k - 2)

from math import comb as binomial
def T(n: int, k: int) -> int:
    o = k & 1
    return binomial(n - o - (k - o) // 2, (k - o) // 2) << o

def P(n, x):
    if n < 0: return P(n, x)
    return sum(T(n, k)*x**k for k in range(n + 1))
def sgn(x: int) -> int: return (x > 0) - (x < 0)
# Table of interpolated sequences
print("|  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |")
print("|  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|")
print("|    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |")
f = "| {0:2d} | {1:9d}         |  {2:4d}   |   {3:5d}     |    {4:4d}    |"
for n in range(10): print(f.format(n, -P(n, -1), P(n, 1), int(2**n*P(n, -1/2)), int(2**n*P(n, 1/2))))

from sage.combinat.q_analogues import q_stirling_number2
def A374439(n,k): return (-1)^((k+1)//2)*2^(k%2)*q_stirling_number2(n+1, k+1, -1)
print(flatten([[A374439(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 23 2025

T(n, k) = 2^k' * binomial(n - k' - (k - k') / 2, (k - k') / 2) where k' = 1 if k is odd and otherwise 0.
T(n, k) = (1 + (k mod 2))*qStirling2(n, k, -1), see A333143.
2^n*P(n, -1/2) = A000129(n - 1), Pell numbers, P(-1) = 1.
2^n*P(n, 1/2) = A048654(n), dual Pell numbers.
T(2*n, n) = (1/2)*(-1)^n*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)*A045721((n-1)/2) ). - G. C. Greubel, Jan 23 2025

A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, 1, -2, 5, -12, 29, -70, 169, -408, 985, -2378, 5741, -13860, 33461, -80782, 195025, -470832, 1136689, -2744210, 6625109, -15994428, 38613965, -93222358, 225058681, -543339720, 1311738121, -3166815962, 7645370045, -18457556052, 44560482149

Offset: 0

Michael Somos, Aug 28 2012

BINOMIAL transform is A052955.

Essentially the same as A000129, A069306, A048624, A215928, A077985, and A176981. - R. J. Mathar, Sep 08 2013

			G.f. = 1 + x + x^3 - 2*x^4 + 5*x^5 - 12*x^6 + 29*x^7 - 70*x^8 + 169*x^9 - 408*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
Index entries for linear recurrences with constant coefficients, signature (-2,1).

Cf. A001542, A001653, A052955, A077985.

[1,1] cat [n le 2 select (n-1) else -2*Self(n-1)+Self(n-2): n in [1..35] ]; // Vincenzo Librandi, Sep 09 2013

CoefficientList[Series[(1 + 3 x + x^2)/(1 + 2 x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 09 2013 *)
a[ n_] := With[ {m = If[ n < 1, 1 - n, n], s = If[ n < 1, (-1)^n, 1]}, s SeriesCoefficient[ x (1 + 2 x) / (1 + 2 x - x^2), {x, 0, m}]]; (* Michael Somos, Mar 19 2019 *)

{a(n) = my(m=n, s=1); if(n<1, m=1-n; s=(-1)^n); s * polcoeff( x * (1 + 2*x) / (1 + 2*x - x^2) + x * O(x^m), m)}; /* Michael Somos, Mar 19 2019 */

G.f.: 1 / (1 - x / (1 + x / (1 + x / (1 + x)))) = (1 + 3*x + x^2) / (1 + 2*x - x^2).
a(n + 3) = A077985(n). a(n) * a(n+2) - a(n+1)^2 = -(-1)^n.
a(2*n + 1) = A001653(n). a(2*n + 2) = -A001542(n).
a(n) = Sum_{k=0..n} A147746(n,k)*(-1)^(n-k). - Philippe Deléham, Aug 30 2012
G.f.: 1 + x + x^2/(1-x)  - G(0)*x^2 /(2-2*x), where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 10 2013
a(n) = (-1)^n a(1-n) = A000129(-1-n) if n < 0. a(n-2) = 2*a(n-1) + a(n) if n<1 or n>2. - Michael Somos, Mar 19 2019
E.g.f.: exp(-x)*(4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/2 - 1. - Stefano Spezia, Oct 31 2024

A176981 Expansion of 2+(1-2x)/(-1+2x+x^2).

Original entry on oeis.org

1, 0, -1, -2, -5, -12, -29, -70, -169, -408, -985, -2378, -5741, -13860, -33461, -80782, -195025, -470832, -1136689, -2744210, -6625109, -15994428, -38613965, -93222358, -225058681, -543339720, -1311738121, -3166815962, -7645370045, -18457556052, -44560482149

Offset: 0

Roger L. Bagula, Apr 30 2010

It is essentially A000129, A077985 and A069306 except for signs and offsets.

			1 - x^2 - 2*x^3 - 5*x^4 - 12*x^5 - 29*x^6 - 70*x^7 - 169*x^8 - 408*x^9 - 985*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129.

a[0] = 1; a[n_] := a[n] = a[n - 1] - Sqrt[2*a[n - 1]^2 + (-1)^n]; Table[a[n], {n, 0, 30}]
Join[{1}, LinearRecurrence[{2, 1}, {0, -1}, 30]] (* or *) Join[{1}, Rest[ CoefficientList[Series[1 + (1 - 2 x)/(-1 + 2 x + x^2), {x, 0, 30}], x]]] (* Harvey P. Dale, Dec 24 2011 *)
FullSimplify[Join[{1}, Table[((1 - Sqrt[2])^(n-1) - (1 + Sqrt[2])^(n-1)) / 2^(3/2), {n, 1, 30}]]] (* Vaclav Kotesovec, Sep 01 2025 *)

a(n) = a(n-1) - sqrt(2*a(n-1)^2 + (-1)^n) = a(n-1)*(1-sqrt(2+(-1)^n/a(n-1)^2)) for n>0.
So lim_{n->infinity} a(n+1)/a(n) = 1+sqrt(2).
Matches the A000045 formula: Fibonacci(n) = Fibonacci(n-1)*(1 + sqrt(5+4*(-1)^(n-1)/Fibonacci(n-1)^2))/2.
a(0)=1, a(1)=0, a(2)=-1, a(n) = 2*a(n-1)+a(n-2). - Harvey P. Dale, Dec 24 2011
G.f.: 1 / (1 + x^2 / (1 - 2*x / (1 - x / (1 + x)))). - Michael Somos, Jan 03 2013
G.f.: 1 - Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
For n>0, a(n) = ((1 - sqrt(2))^(n-1) - (1 + sqrt(2))^(n-1)) / 2^(3/2). - Vaclav Kotesovec, Sep 01 2025

Name corrected by Jason Yuen, Sep 01 2025

cp's OEIS Frontend

A069299 Number of n X 8 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

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A069300 Number of n X 9 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

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A069301 Number of n X 10 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

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A069302 Number of n X 11 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

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A069303 Number of n X 12 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

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A069304 Number of n X 13 binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.

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A215928 a(n) = 2*a(n-1) + a(n-2) for n > 2, a(0) = a(1) = 1, a(2) = 2.

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A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

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A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0.

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A176981 Expansion of 2+(1-2x)/(-1+2x+x^2).