A107839 -id:A107839 - cp's OEIS frontend

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

Original entry on oeis.org

1, 2, 3, 5, 10, 8, 12, 34, 38, 21, 29, 104, 161, 130, 55, 70, 305, 592, 654, 420, 144, 169, 866, 2023, 2788, 2436, 1308, 377, 408, 2404, 6556, 10810, 11756, 8574, 3970, 987, 985, 6560, 20446, 39164, 50779, 46064, 28987, 11822, 2584, 2378, 17663, 61912, 134960, 202630, 218717, 171232, 95078, 34690, 6765

Offset: 1

Clark Kimberling, Nov 26 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
    1
    2    3
    5   10    8
   12   34   38    21
   29  104  161   130    55
   70  305  592   654   420  144
  169  866 2023  2788  2436 1308  377
  408 2404 6556 10810 11756 8574 3970 987
Row 4 represents the polynomial p(4,x) = 12 + 34*x + 38*x^2 + 21*x^3, so (T(4,k)) = (12,34,38,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000129 (column 1), A001906 (p(n,n-1)), A107839 (row sums, p(n,1)), A077925 (alternating row sums, p(n,-1)), A023000 (p(n,2)), A001076 (p(n,-2)), A186446 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367298, A367299, A367300, A367301.

p[1, x_] := 1; p[2, x_] := 2 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 2 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 4*x + 5*x^2)), b = (1/2)*(3*x + 2 + 1/k), c = (1/2)*(3*x + 2 - 1/k).

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

Original entry on oeis.org

1, 3, 2, 10, 10, 3, 33, 46, 22, 4, 109, 194, 131, 40, 5, 360, 780, 678, 296, 65, 6, 1189, 3036, 3228, 1828, 581, 98, 7, 3927, 11546, 14514, 10100, 4194, 1036, 140, 8, 12970, 43150, 62601, 51664, 26479, 8604, 1722, 192, 9, 42837, 159082, 261598, 249720, 152245, 61318, 16248, 2712, 255, 10

Offset: 1

Clark Kimberling, Dec 23 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
     1
     3      2
    10     10      3
    33     46     22      4
   109    194    131     40     5
   360    780    678    296    65     6
  1189   3036   3228   1828   581    98    7
  3927  11546  14514  10100  4194  1036  140  8
Row 4 represents the polynomial p(4,x) = 33 + 46*x + 22*x^2 + 4*x^3, so (T(4,k)) = (33,46,22,4), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A006190 (column 1); A000027 (p(n,n-1)); A107839 (row sums, p(n,1)); A001045 (alternating row sums, p(n,-1)); A030240 (p(n,2)); A039834 (signed Fibonacci numbers, p(n,-2)); A016130 (p(n,3)); A225883 (p(n,-3)); A099450 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299.

p[1, x_] := 1; p[2, x_] := 3 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 3 + 2*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 4*x)), b = (1/2) (2*x + 3 + 1/k), c = (1/2) (2*x + 3 - 1/k).

A052984 a(n) = 5a(n-1) - 2a(n-2) for n>1, with a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 13, 59, 269, 1227, 5597, 25531, 116461, 531243, 2423293, 11053979, 50423309, 230008587, 1049196317, 4785964411, 21831429421, 99585218283, 454263232573, 2072145726299, 9452202166349, 43116719379147, 196679192563037, 897162524056891, 4092454235158381

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) = A020698(n) - 4*A020698(n-1) + 4*A020698(n-2) (n>=2). Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1058
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. See p. 3.
Zeying Xu, Graphical zonotopes with the same face vector, arXiv:1809.08764 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-2).

Cf. A005824, A020698, A107839, A147703.

a:=[1,3];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 23 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-5*x+2*x^2) )); // G. C. Greubel, Feb 10 2019

a:=[1,3]; [n le 2 select a[n] else 5*Self(n-1)-2*Self(n-2):n in [1..25]]; // Marius A. Burtea, Oct 23 2019

spec:= [S,{S=Sequence(Union(Prod(Sequence(Union(Z,Z)),Union(Z,Z)),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
a[0]:=1: a[1]:=3: for n from 2 to 25 do a[n]:=5*a[n-1]-2*a[n-2] od: seq(a[n],n=0..25); # Emeric Deutsch

a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}]
LinearRecurrence[{5,-2},{1,3},30] (* Harvey P. Dale, Apr 08 2014 *)
CoefficientList[Series[(1-2x)/(1-5x+2x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 09 2014 *)

Vec((1-2*x)/(1-5*x+2*x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011

def A052984_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x)/(1-5*x+2*x^2) ).list()
A052984_list(30) # G. C. Greubel, Feb 10 2019

a(n) = A005824(2n).
G.f.: (1-2*x)/(1-5*x+2*x^2).
a(n) = Sum_{alpha=RootOf(1-5*z+2*z^2)} (1 + 6*alpha)*alpha^(-1-n)/17.
a(n) = [M^(n+1)]2,2, where M is the 3 X 3 matrix defined as follows: M = [2,1,2; 1,1,1; 2,1,2]. - _Simone Severini, Jun 12 2006
a(n-1) = Sum_{k=0..n} A147703(n,k)*(-1)^k*2^(n-k), n>1. - Philippe Deléham, Nov 29 2008
a(n) = (a(n-1)^2 + 2^n)/a(n-2). - Irene Sermon, Oct 29 2013
a(n) = A107839(n) - 2*A107839(n-1). - R. J. Mathar, Feb 27 2019
E.g.f.: exp(5*x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - Stefano Spezia, Jun 17 2025

Edited by Robert G. Wilson v, Dec 29 2002

A005824 a(n) = 5a(n-2) - 2a(n-4), with initial terms 0,1,1,3.

Original entry on oeis.org

0, 1, 1, 3, 5, 13, 23, 59, 105, 269, 479, 1227, 2185, 5597, 9967, 25531, 45465, 116461, 207391, 531243, 946025, 2423293, 4315343, 11053979, 19684665, 50423309, 89792639, 230008587, 409593865, 1049196317, 1868384047, 4785964411, 8522732505, 21831429421, 38876894431

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 0..3036
Milica Anđelić and Carlos M. da Fonseca, On the constant coefficients of a certain recurrence relation: A simple proof, Heliyon (2021) Vol. 7, No. 8, e07764.
Sean A. Irvine, Walks on Graphs.
D. Panario, M. Sahin and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Jeffrey Shallit, On the worst case of three algorithms for computing the Jacobi symbol, J. Symbolic Comput. 10 (1990), no. 6, 593-610.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-2).

Cf. A079162.

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[ EvenQ[n], a[n - 1] + 2a[n - 2], 2a[n - 1] + a[n - 2]]; Table[a[n], {n, 0, 31}]
LinearRecurrence[{0,5,0,-2},{0,1,1,3},40] (* Harvey P. Dale, Jul 09 2015 *)

Also a(n) = a(n-1) + 2a(n-2) if n is even, else a(n) = 2a(n-1) + a(n-2).
g.f.: -x*(2*x+1)*(x-1)/(1-5*x^2+2*x^4). Simon Plouffe in his 1992 dissertation.
a(2n+1) = A052984(n). [Index corrected by R. J. Mathar, Apr 01 2009]
a(2n) = A107839(n-1). [R. J. Mathar, Apr 01 2009]
a(n) = A109165(n-1)-A109165(n-2). - R. J. Mathar, Jan 13 2025

Extended by Robert G. Wilson v, Dec 29 2002

A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 13, 20, 16, 1, 6, 19, 38, 48, 32, 1, 7, 26, 63, 104, 112, 64, 1, 8, 34, 96, 192, 272, 256, 128, 1, 9, 43, 138, 321, 552, 688, 576, 256, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512, 1, 11, 64, 253, 743, 1683, 2972, 4048

Offset: 0

Clark Kimberling, Feb 25 2012

Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section.
Row sums: (1,3,8,...), even-indexed Fibonacci numbers.
Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers.
v(n,2) = A107839(n), v(n,n) = 2^(n-1), v(n+1,n) = A001792(n),
v(n+2,n) = A049611, v(n+3,n) = A049612.
Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012
Essentially triangle in A049600. - Philippe Deléham, Mar 23 2012

			First five rows:
  1;
  1, 2;
  1, 3,  4;
  1, 4,  8,  8;
  1, 5, 13, 20, 16;
First five polynomials v(n,x):
  1
  1 + 2x
  1 + 3x +  4x^2
  1 + 4x +  8x^2 +  8x^3
  1 + 5x + 13x^2 + 20x^3 + 16x^4
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  4,  0;
  1, 4,  8,  8,  0;
  1, 5, 13, 20, 16,  0;
  1, 6, 19, 38, 48, 32, 0;
Triangle in A049600 begins:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 3,  4;
  0, 1, 4,  8,  8;
  0, 1, 5, 13, 20, 16;
  0, 1, 6, 19, 38, 48, 32;
  ... - _Philippe Deléham_, Mar 23 2012

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened

Cf. A160232, A000045, A049600, A106195.

a208341 n k = a208341_tabl !! (n-1) !! (k-1)
a208341_row n = a208341_tabl !! (n-1)
a208341_tabl = map reverse a106195_tabl
-- Reinhard Zumkeller, Dec 16 2013

T := (n,k) -> hypergeom([n-k+1, -k],[1],-1):
seq(lprint(seq(simplify(T(n,k)),k=0..n)),n=0..7); # Peter Luschny, May 20 2015

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*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[%]   (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208341 *)

T(n,k) = sum(i = 0, k, 2^(k-i)*binomial(n-k,i)*binomial(k,i));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ Michel Marcus, Aug 14 2015

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1.
As DELTA-triangle with 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), 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. - Philippe Deléham, Mar 12 2012
G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - Philippe Deléham, Mar 12 2012
T(n,k) = A106195(n-1,n-k), k = 1..n. - Reinhard Zumkeller, Dec 16 2013
From Peter Bala, Aug 11 2015: (Start)
The following remarks assume the row and column indexing start at 0.
T(n,k) = Sum_{i = 0..k} 2^(k-i)*binomial(n-k,i)*binomial(k,i) = Sum_{i = 0..k} binomial(n-k+i,i)*binomial(k,i).
Riordan array (1/(1 - x), x*(2 - x)/(1 - x)).
O.g.f. 1/(1 - (2*t + 1)*x + t*x^2) = 1 + (1 + 2*t)*x + (1 + 3*t + 4*t^2)*x^2 + ....
Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End)
For kGlen Whitney, Aug 17 2021

New name from Peter Luschny, May 20 2015

Offset corrected by Joerg Arndt, Aug 12 2015

A052913 a(n+2) = 5a(n+1) - 2a(n), with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 18, 82, 374, 1706, 7782, 35498, 161926, 738634, 3369318, 15369322, 70107974, 319801226, 1458790182, 6654348458, 30354161926, 138462112714, 631602239718, 2881086973162, 13142230386374, 59948977985546, 273460429154982, 1247404189803818, 5690100090709126

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Main diagonal of the array: m(1,j)=3^(j-1), m(i,1)=1; m(i,j) = m(i-1,j) + m(i,j-1): 1 3 9 27 81 ... / 1 4 13 40 ... / 1 5 18 58 ... / 1 6 24 82 ... - Benoit Cloitre, Aug 05 2002
a(n) is also the number of 3 X n matrices of integers for which the upper-left hand corner is a 1, the rows and columns are weakly increasing, and two adjacent entries differ by at most 1. - Richard Stanley, Jun 06 2010
a(n) is the number of compositions of n when there are 4 types of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010
If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, or A162909/A162910, or A071766/A229742, or A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n>=0), and the products numerator*denominator, term by term, are summed at each level n, then the resulting sequence of integers is a(n). - Yosu Yurramendi, May 23 2015
Number of 1’s in the substitution system {0 -> 110, 1 -> 11110} at step n from initial string "1" (1 -> 11110 -> 11110111101111011110110 -> ...) . - Ilya Gutkovskiy, Apr 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 894
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-2).

Cf. A007482 (inverse binomial transform).

a:=[1,4];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 16 2019

I:=[1,4]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 24 2015

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!((1-x)/(1-5*x+2*x^2))); // Marius A. Burtea, Oct 16 2019

spec := [S,{S=Sequence(Union(Prod(Sequence(Z),Union(Z,Z)),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1-5*x+2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

Transpose[NestList[{Last[#],5Last[#]-2First[#]}&, {1,4},20]][[1]] (* Harvey P. Dale, Mar 12 2011 *)
LinearRecurrence[{5, -2}, {1, 4}, 25] (* Jean-François Alcover, Jan 08 2019 *)

Vec((1-x)/(1-5*x+2*x^2) + O(x^30)) \\ Michel Marcus, Mar 05 2015

def A052913_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-5*x+2*x^2)).list()
A052913_list(30) # G. C. Greubel, Oct 16 2019

G.f.: (1-x)/(1-5*x+2*x^2).
a(n) = Sum_{alpha=RootOf(1 - 5*z + 2*z^2)} (1/17)*(3+alpha)*alpha^(-1-n).
a(n) = ((17+3*sqrt(17))/34)*((5+sqrt(17))/2)^n + ((17-3*sqrt(17))/34)*((5-sqrt(17))/2)^n. - N. J. A. Sloane, Jun 03 2002
a(n) = A107839(n) - A107839(n-1). - R. J. Mathar, May 21 2015
a(n) = 2*A020698(n-1), n>1. - R. J. Mathar, Nov 23 2015
E.g.f.: (1/17)*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 16 2019
a(n) = 3*A107839(n-1) + (-1)^n*A152594(n) with A107839(-1) = 0. - Klaus Purath, Jul 29 2020

Typo in definition corrected by Bruno Berselli, Jun 07 2010

A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.

Original entry on oeis.org

1, 1, 3, 3, 6, 8, 5, 21, 25, 21, 11, 48, 101, 90, 55, 21, 123, 290, 414, 300, 144, 43, 282, 850, 1416, 1551, 954, 377, 85, 657, 2255, 4671, 6109, 5481, 2939, 987, 171, 1476, 5883, 13986, 22374, 24300, 18585, 8850, 2584, 341, 3303, 14736, 40320, 74295, 97713

Offset: 1

Clark Kimberling, Dec 31 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1     3
   3     6    8
   5    21    25    21
  11    48   101    90    55
  21   123   290   414   300  144
  43   282   850  1416  1551  954    377
  85   657  2255  4671  6109  5481  2939  987
Row 4 represents the polynomial p(4,x) = 5 + 21 x + 25 x^2 + 21 x^3, so (T(4,k)) = (5,21,25,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A001045 (column 1); A001906 (p(n,n-1)); A001076 (row sums), (p(n,1)); A077985 (alternating row sums), (p(n,-1)); A186446 (p(n,2)), A107839, (p(n,-2)); A190989, (p(n,3)); A023000, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 2 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where p(1,x) = 1, p(2,x) = 1 + 3 x, u = p(2,x), and v = 2 - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(9 + 6 x + 5 x^2), b = (1/2) (3 x + 1 - 1/k), c = (1/2) (3 x + 1 + 1/k).

A110441 Triangular array formed by the Mersenne numbers.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 15, 23, 9, 1, 31, 72, 48, 12, 1, 63, 201, 198, 82, 15, 1, 127, 522, 699, 420, 125, 18, 1, 255, 1291, 2223, 1795, 765, 177, 21, 1, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1

Offset: 0

Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005

This sequence factors A038255 into a product of Riordan arrays.
Subtriangle of the triangle given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012
From Peter Bala, Jul 22 2014: (Start)
Let M denote the lower unit triangular array A130330 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)
For 1<=k<=n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01 and 02. - Milan Janjic, Dec 20 2016
The convolution triangle of the Mersenne numbers. - Peter Luschny, Oct 09 2022

			Triangle starts:
   1;
   3,  1;
   7,  6,  1;
  15, 23,  9,  1;
  31, 72, 48, 12,  1;
(0, 3, -2/3, 2/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1
  0,  1
  0,  3,  1
  0,  7,  6,  1
  0, 15, 23,  9,  1
  0, 31, 72, 48, 12, 1. - _Philippe Deléham_, Mar 19 2012
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
/ 1          \/1         \/1        \      / 1       \
| 3  1       ||0  1      ||0 1      |      | 3  1    |
| 7  3 1     ||0  3 1    ||0 0 1    |... = | 7  6 1  |
|15  7 3 1   ||0  7 3 1  ||0 0 3 1  |      |15 23 9 1|
|31 15 7 3 1 ||0 15 7 3 1||0 0 7 3 1|      |...      |
|...         ||...       ||...      |      |...      | - _Peter Bala_, Jul 22 2014

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, J. Int. Seq. 8 (2005), #05.3.7.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A000225, A130330, A206306, A116414.

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - (3 + y) x + 2 x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Riordan array M(n, k): (1/(1-3z+2z^2), z/(1-3z+2z^2)). Leftmost column M(n, 0) is the Mersenne numbers A000225, first column is A045618, second column is A055582, row sum is A007070 and diagonal sum is even-indexed Fibonacci numbers A001906.
T(n,k) = Sum_{j=0..n} C(j+k,k)C(n-j,k)2^(n-j-k). - Paul Barry, Feb 13 2006
From Philippe Deléham, Mar 19 2012: (Start)
G.f.: 1/(1-(3+y)*x+2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -2*T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000225(n+1), A007070(n), A107839(n), A154244(n), A186446(n), A190975(n+1), A190979(n+1), A190869(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7 respectively. (End)
Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (2^(i+1) - 1)*T(n-i,k). - Peter Bala, Jul 22 2014
From Peter Bala, Oct 07 2019: (Start)
Recurrence for row polynomials: R(n,x) = (3 + x)*R(n-1,x) - 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.
The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 + 2*x/(1 + ... + x/(1 + 2*x/(1)))) (with 2*n partial numerators). Cf. A116414. (End)

A359993 Array read by antidiagonals: T(m,n) is the number of connected spanning subgraphs in the grid graph P_m X P_n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 23, 23, 1, 1, 105, 431, 105, 1, 1, 479, 7857, 7857, 479, 1, 1, 2185, 142625, 555195, 142625, 2185, 1, 1, 9967, 2587279, 38757695, 38757695, 2587279, 9967, 1, 1, 45465, 46929343, 2698167665, 10286937043, 2698167665, 46929343, 45465, 1

Offset: 1

Andrew Howroyd, Jan 28 2023

Also T(m,n) except when m = n = 0 is the number of connected edge covers in the m X n grid graph.

			Table starts:
=================================================================
m\n| 1    2       3          4             5                6
---+-------------------------------------------------------------
1  | 1    1       1          1             1                1 ...
2  | 1    5      23        105           479             2185 ...
3  | 1   23     431       7857        142625          2587279 ...
4  | 1  105    7857     555195      38757695       2698167665 ...
5  | 1  479  142625   38757695   10286937043    2711895924889 ...
6  | 1 2185 2587279 2698167665 2711895924889 2692324030864335 ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 1..465
Eric Weisstein's World of Mathematics, Grid Graph

Rows 1..4 are A000012, A107839(n-1), A158453, A359991.
Main diagonal is A359992.
Cf. A116469 (spanning trees), A287151 (connected induced subgraphs), A286912 (edge covers), A359990 (edge cuts), A360194 (spanning forests).

T(m,n) = T(n,m).

cp's OEIS Frontend

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A052984 a(n) = 5*a(n-1) - 2*a(n-2) for n>1, with a(0) = 1, a(1) = 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A005824 a(n) = 5*a(n-2) - 2*a(n-4), with initial terms 0,1,1,3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A208341 Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

A367297 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

A052984 a(n) = 5a(n-1) - 2a(n-2) for n>1, with a(0) = 1, a(1) = 3.

A005824 a(n) = 5a(n-2) - 2a(n-4), with initial terms 0,1,1,3.

A052913 a(n+2) = 5a(n+1) - 2a(n), with a(0) = 1, a(1) = 4.

A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.