A116415 -id:A116415 - 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

A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 15, 10, 1, 1, 9, 28, 35, 15, 1, 1, 11, 45, 84, 70, 21, 1, 1, 13, 66, 165, 210, 126, 28, 1, 1, 15, 91, 286, 495, 462, 210, 36, 1, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1

Offset: 0

Clark Kimberling

Row sums are odd-indexed Fibonacci numbers.
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - Emeric Deutsch, May 31 2004
Diagonal sums are A052535. - Paul Barry, Jan 21 2005
Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry, Jan 21 2005
Coefficients of Morgan-Voyce polynomial b(n,x); e.g., b(3,x)=x^3+5x^2+6x+1. See A172431 for coefficients of Morgan-Voyce polynomial B(n,x). - Clark Kimberling, Feb 13 2010
T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. - Emanuele Munarini, Apr 07 2011
Roots of signed n-th polynomials are chaotic with respect to the operation (-2, x^2), with cycle lengths A003558(n). Example: starting with a root to x^3 - 5x^2 + 6x - 1 = 0; (2 + 2*cos(2*Pi/N) = 3.24697... = A116415; we obtain the trajectory (3.24697...-> 1.55495...-> 0.198062...; the 3 roots to the polynomial with cycle length 3 matching A003558(3) = 3. The operation (-2, x^2) is the reversal of the well known chaotic operation (x^2 - 2) [Kappraff, Adamson, 2004] starting with seed 2*cos(2*Pi/N). Check: given 2*cos(2*Pi/7) = 1.24697..., we obtain the 3-cycle using (x^2 - 2): (1.24697...-> -0.445041...-> 1.801937...; where the terms in either set are intermediate terms in the other, irrespective of sign. - Gary W. Adamson, Sep 22 2011
A054142 is jointly generated with A172431 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section of A172431. - Clark Kimberling, Mar 09 2012
Subtriangle of the triangle given by (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012
The o.g.f. for row n of the array A(n, k) = binomial(2*n-k,k), k >= 0, n >= 0 is G(n,x) = Sum_{k=0..n} T(n, k)*x^k + (-x)^(2*n+1) * c(-x)^(2*n+1) / sqrt(1-4*(-x)), for n >= 0. Here c(x) is the o.g.f. of A000108 (Catalan). For powers of c(x) see the W. Lang link in A115139. For the alternating sign case replace x by -x. - Wolfdieter Lang, Sep 12 2016
Multiplying the n-th diagonal by A001147(n) generates A001497. - Tom Copeland, Oct 04 2016

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  6,   1;
  1,  7, 15,  10,   1;
  1,  9, 28,  35,  15,   1;
  1, 11, 45,  84,  70,  21,   1;
  1, 13, 66, 165, 210, 126,  28,  1;
  1, 15, 91, 286, 495, 462, 210, 36, 1; ...
...
(0, 1, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3,  1;
  0, 1, 5,  6,  1;
  0, 1, 7, 15, 10,  1;
  0, 1, 9, 28, 35, 15, 1. _Philippe Deléham_, Apr 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..495
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Molly Fenn and Eric Sommers, A transitivity result for ad-nilpotent ideals in type A, arXiv:2101.04091 [math.RT], 2021.
J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444 See Eq. 2.27. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004).

These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
Cf. A000108, A003558, A027989, A052535, A054142, A076756, A084938, A085478, A098435, A115139, A172431, A172991, A188648.
Cf. A001147, A001497.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n-k,k) ))); # G. C. Greubel, Aug 01 2019

[Binomial(2*n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T:=(n,k)->binomial(2*n-k,k): seq(seq(T(n,k), k=0..n), n=0..11);

Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] (* Emanuele Munarini, Apr 07 2011 *)

create_list(binomial(2*n-k,k),n,0,10,k,0,n); /* Emanuele Munarini, Apr 07 2011 */

T(n,k)=if(n<0,0,polcoeff(charpoly(matrix(n,n,i,j,-min(i,j))),k))

[[binomial(2*n-k,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f.: (1-t*z)/((1-t*z)^2-z). - Emeric Deutsch, May 31 2004
Column k has g.f.: (Sum_{j=0..k+1} binomial(k+1, 2j)*x^j)*x^k/(1-x)^(k+1). - Paul Barry, Jun 22 2005
Recurrence: T(n+2,k+2) = T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k). - Emanuele Munarini, Apr 07 2011
T(n, k) = binomial(2*n-k, k) = A085478(n, n-k), for n >= 0, k = 0..n. - Wolfdieter Lang, Mar 25 2020

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A052961 Expansion of (1 - 3x) / (1 - 5x + 3*x^2).

Original entry on oeis.org

1, 2, 7, 29, 124, 533, 2293, 9866, 42451, 182657, 785932, 3381689, 14550649, 62608178, 269388943, 1159120181, 4987434076, 21459809837, 92336746957, 397304305274, 1709511285499, 7355643511673, 31649683701868, 136181487974321, 585958388766001, 2521247479907042

Offset: 0

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

a(n) is the number of tilings of a 2 X n rectangle using integer dimension tiles at least one of whose dimensions is 1, so allowable dimensions are 1 X 1, 1 X 2, 1 X 3, 1 X 4, ..., and 2 X 1. - David Callan, Aug 27 2014
a(n+1) counts closed walks on K_2 containing one loop on the index vertex and four loops on the other vertex. Equivalently the (1,1)entry of A^(n+1) where the adjacency matrix of digraph is A=(1,1;1,4). - _David Neil McGrath, Nov 05 2014
A production matrix for the sequence is M =
  1, 1, 0, 0, 0, 0, 0, ...
  1, 0, 4, 0, 0, 0, 0, ...
  1, 0, 0, 4, 0, 0, 0, ...
  1, 0, 0, 0, 4, 0, 0, ...
  1, 0, 0, 0, 0, 4, 0, ...
  1, 0, 0, 0, 0, 0, 4, ...
  ...
Take powers of M and extract the upper left term, getting the sequence starting (1, 1, 2, 7, 29, 124, ...). - Gary W. Adamson, Jul 22 2016
From Gary W. Adamson, Jul 29 2016: (Start)
The sequence is N=1 in an infinite set obtained from matrix powers of [(1,N); (1,4)], extracting the upper left terms.
The infinite set begins:
N=1  (A052961):  1,  2,  7,  29   124,  533,   2293, ...
N=2  (A052984):  1,  3, 13,  59,  269, 1227,   5597, ...
N=3  (A004253):  1,  4, 19,  91,  436, 2089,  10009, ...
N=4  (A000351):  1,  5, 25, 125,  625, 3125,  15625, ...
N=5  (A015449):  1,  6, 31, 161,  836, 4341,  22541, ...
N=6  (A124610):  1,  7, 37, 199, 1069, 5743,  30853, ...
N=7  (A111363):  1,  8, 43, 239, 1324, 7337,  40653, ...
N=8  (A123270):  1,  9, 49, 281, 1601, 9129,  52049, ...
N=9  (A188168):  1, 10, 55, 325, 1900, 11125, 65125, ...
N=10 (A092164):  1, 11, 61, 371, 2221, 13331, 79981, ...
... (End)

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1032
Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698, March 2025.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Column k=2 of A254414.

Cf. A000351, A004253, A015449, A052984, A092164, A111363, A123270, A124610, A188168.

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

I:=[1,2]; [n le 2 select I[n] else 5*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 17 2014

spec:= [S,{S = Sequence(Union(Prod(Sequence(Union(Z,Z,Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size = n), n = 0..20);
seq(coeff(series((1-3*x)/(1-5*x+3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 23 2019

CoefficientList[Series[(1-3x)/(1-5x+3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-3},{1,2},30] (* Harvey P. Dale, Nov 23 2013 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+3*x^2)) \\ G. C. Greubel, Oct 23 2019

def A052961_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-5*x+3*x^2)).list()
A052961_list(30) # G. C. Greubel, Oct 23 2019

G.f.: (1-3*x)/(1-5*x+3*x^2).
a(n) = 5*a(n-1) - 3*a(n-2), with a(0) = 1, a(1) = 2.
a(n) = Sum_{alpha=RootOf(1-5*z+3*z^2)} (-1 + 9*alpha)*alpha^(-1-n)/13.
E.g.f.: (1 + sqrt(13) + (sqrt(13)-1) * exp(sqrt(13)*x)) / (2*sqrt(13) * exp(((sqrt(13)-5)*x)/2)). - Vaclav Kotesovec, Feb 16 2015
a(n) = A116415(n) - 3*A116415(n-1). - R. J. Mathar, Feb 27 2019

A368150 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) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

Original entry on oeis.org

1, 1, 3, 2, 6, 8, 3, 15, 25, 21, 5, 30, 76, 90, 55, 8, 60, 188, 324, 300, 144, 13, 114, 439, 948, 1251, 954, 377, 21, 213, 961, 2529, 4207, 4527, 2939, 987, 34, 390, 2026, 6246, 12606, 17154, 15646, 8850, 2584, 55, 705, 4136, 14640, 34590, 56970, 65840

Offset: 1

Clark Kimberling, Dec 25 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
   2    6    8
   3   15   25    21
   5   30   76    90     55
   8   60  188   324    300   144
  13  114  439   948   1251   954   377
  21  213  961  2529   4207  4527  2939   987
Row 4 represents the polynomial p(4,x) = 3 + 15*x + 25*x^2 + 21*x^3, so (T(4,k)) = (3,15,25,21), 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. A000045 (column 1); A001906 (p(n,n-1)); A000302 (row sums), (p(n,1)); A122803 (alternating row sums), (p(n,-1)); A190972 (p(n,2)), A116415, (p(n,-2)); A190990, (p(n,3)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368151.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 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 = 1 - x^2.

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

A116414 Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

Original entry on oeis.org

1, 4, 1, 13, 8, 1, 40, 42, 12, 1, 121, 184, 87, 16, 1, 364, 731, 496, 148, 20, 1, 1093, 2736, 2454, 1040, 225, 24, 1, 3280, 9844, 11064, 6170, 1880, 318, 28, 1, 9841, 34448, 46738, 32624, 13015, 3080, 427, 32, 1, 29524, 118101, 188208, 158724, 79044, 24381, 4704

Offset: 0

Paul Barry, Feb 13 2006

Row sums are A116415. Diagonal sums are A007070. First column is A003462(n+1). Product of A007318 and A116412.

Subtriangle of triangle given by (0, 4, -3/4, 3/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

			Triangle begins
    1;
    4,   1;
   13,   8,   1;
   40,  42,  12,   1;
  121, 184,  87,  16,  1;
  364, 731, 496, 148, 20, 1;
Triangle T(n,k), 0 <= k <= n, given by (0, 4, -3/4, 3/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1;
  0,   1;
  0,   4,   1;
  0,  13,   8,   1;
  0,  40,  42,  12,   1;
  0, 121, 184,  87,  16,  1;
  0, 364, 731, 496, 148, 20, 1;
  ... - _Philippe Deléham_, Jan 18 2012

Michael De Vlieger, Table of n, a(n) for n = 0..11324 (rows 0 <= n <= 150)
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A003462, A116412, A110441.

With[{n = 10}, DeleteCases[#, 0] & /@ Rest@ CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 4 x + 3 x^2 - x y), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Riordan array (1/(1-4x+3x^2), x/(1-4x+3x^2)); number triangle T(n,k) = Sum_{j=0..n} binomial(n-j,k)*binomial(k+j,j)*3^j.
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(2,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Oct 31 2013
G.f.: (1-4*x+3*x^2)/(1-4*x+3*x^2-x*y). - Philippe Deléham, Oct 31 2013
From Peter Bala, Oct 07 2019: (Start)
O.g.f.: 1/(1 - 4*x + 3*x^2 - x*y) = 1 + (4 + y)*x + (13 + 8*y + y^2)*x^2 + ....
Recurrence for row polynomials: R(n,y) = (4 + y)*R(n-1,y) - 3*R(n-2,y) with R(0,y) = 1 and R(1,y) = 4 + y.
The row reverse polynomial y^n*R(n,1/y) is equal to the numerator polynomial of the finite continued fraction 1 + y/(1 + 3*y/(1 + ... + y/(1 + 3*y/(1)))) (with 2*n partial numerators). Cf. A110441. (End)

A010903 Pisot sequence E(3,13): a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

Original entry on oeis.org

3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134

Offset: 0

Simon Plouffe

According to Boyd (Acta Arithm. 32 (1977) p 89), quoting Pisot, every E(3,.) sequence satisfies a linear recurrence of at most order 3. Here this is easily derived from the first terms of the sequence. - R. J. Mathar, May 26 2008
A010920 coincides with this sequence for at least the first 32600 terms and probably more. - R. J. Mathar, May 26 2008
For n >= 1, a(n-1) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See pp. 11-13, 24.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A010920.

LinearRecurrence[{5, -3}, {3, 13}, 24] (* Jean-François Alcover, Oct 22 2019 *)
nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; NestList[nxt,{3,13},30][[;;,1]] (* Harvey P. Dale, Mar 29 2025 *)

Vec((3-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Jul 27 2016

a(n) = 5*a(n-1) - 3*a(n-2) = 3*A116415(n) - 2*A116415(n-1). - R. J. Mathar, May 26 2008
O.g.f.: (3-2*x)/(1-5*x+3*x^2). - R. J. Mathar, May 26 2008
a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-11+3*sqrt(13)) + (5+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13). - Colin Barker, Nov 26 2016

A109114 a(n) = 5a(n-1) - 3a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 27, 117, 504, 2169, 9333, 40158, 172791, 743481, 3199032, 13764717, 59226489, 254838294, 1096512003, 4718045133, 20300689656, 87349312881, 375844495437, 1617174538542, 6958339206399, 29940172416369, 128825844462648

Offset: 0

Emeric Deutsch, Jun 19 2005

nonn

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 302, P_{11}).

Index entries for linear recurrences with constant coefficients, signature (5,-3).

a[0]:=1: a[1]:=6: for n from 2 to 26 do a[n]:=5*a[n-1]-3*a[n-2] od: seq(a[n],n=0..26);

LinearRecurrence[{5,-3},{1,6},30] (* Harvey P. Dale, Dec 03 2012 *)

a(n) = ((sqrt(13) + 7)*((5 + sqrt(13))/2)^n + (sqrt(13) - 7)*((5 - sqrt(13))/2)^n)/(2*sqrt(13)).
G.f.: (1+z)/(1 - 5z + 3z^2).
a(n) = A116415(n)+A116415(n-1). - R. J. Mathar, Jul 26 2022

A228569 Binomial transform of A006497.

Original entry on oeis.org

2, 5, 19, 80, 343, 1475, 6346, 27305, 117487, 505520, 2175139, 9359135, 40270258, 173273885, 745558651, 3207971600, 13803182047, 59391995435, 255550431034, 1099576168865, 4731229551223, 20357419249520, 87593407593931, 376894780221095, 1621693678323682

Offset: 0

R. J. Mathar, Nov 10 2013

Michael De Vlieger, Table of n, a(n) for n = 0..1577
P. Bhadouria, D. Jhala and B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence B_3.
Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

LinearRecurrence[{5,-3},{2,5},30] (* Harvey P. Dale, Sep 09 2017 *)

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

a(n) = 2*A116415(n)-5*A116315(n-1) for n>0.

A106732 Expansion of -3x/(1 - 5x + 3*x^2).

Original entry on oeis.org

0, -3, -15, -66, -285, -1227, -5280, -22719, -97755, -420618, -1809825, -7787271, -33506880, -144172587, -620342295, -2669193714, -11484941685, -49417127283, -212630811360, -914902674951, -3936620940675, -16938396678522, -72882120570585

Offset: 0

Roger L. Bagula, May 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A116415.

[n le 2 select -3*(1+(-1)^n)/2 else 5*Self(n-1) - 3*Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 10 2021

a[0]:=0: a[1]:=-3: for n from 2 to 22 do a[n]:=5*a[n-1]-3*a[n-2] od: seq(a[n], n=0..30);

LinearRecurrence[{5,-3}, {0,-3}, 31] (* G. C. Greubel, Sep 10 2021 *)
CoefficientList[Series[-3x/(1-5x+3x^2),{x,0,30}],x] (* Harvey P. Dale, Jan 29 2025 *)

[-3^((n+1)/2)*chebyshev_U(n-1, 5/(2*sqrt(3))) for n in (0..30)] # G. C. Greubel, Sep 10 2021

G.f.: -3*x/(1 - 5*x + 3*x^2).
a(n) = 5*a(n-1) - 3*a(n-2), a(0) = 0, a(1) = -3.
a(n) = -3*A116415(n-1), n>0.
a(n) = -3^((n+1)/2)*ChebyshevU(n-1, 5/(2*sqrt(3))). - G. C. Greubel, Sep 10 2021

Edited by N. J. A. Sloane, Apr 30 2006

New name by G. C. Greubel, Sep 10 2021

cp's OEIS Frontend

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

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A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

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A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

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A052961 Expansion of (1 - 3*x) / (1 - 5*x + 3*x^2).

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A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 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 - x^2.

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A116414 Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

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

A052961 Expansion of (1 - 3x) / (1 - 5x + 3*x^2).

A368150 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) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - x^2.

A109114 a(n) = 5a(n-1) - 3a(n-2), a(0)=1, a(1)=6.

A106732 Expansion of -3x/(1 - 5x + 3*x^2).