A105082 -id:A105082 - cp's OEIS frontend

A048696 Generalized Pellian with second term equal to 9.

1, 9, 19, 47, 113, 273, 659, 1591, 3841, 9273, 22387, 54047, 130481, 315009, 760499, 1836007, 4432513, 10701033, 25834579, 62370191, 150574961, 363520113, 877615187, 2118750487, 5115116161, 12348982809, 29813081779, 71975146367

Offset: 0

Barry E. Williams

Binomial transform of 5,6,10,12,20,24,40. - Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
Binomial transform of A164587. Inverse binomial transform of A164298. - Klaus Brockhaus, Aug 17 2009
For n > 0: a(n) = A105082(n) - A105082(n-1). - Reinhard Zumkeller, Dec 15 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A001333, A000129, A048654, A048655.

a048696 n = a048696_list !! n
a048696_list = 1 : 9 : zipWith (+)
               a048696_list (map (2 *) $ tail a048696_list)
-- Reinhard Zumkeller, Dec 15 2013

[ n le 2 select 8*n-7 else 2*Self(n-1)+Self(n-2): n in [1..28] ]; // Klaus Brockhaus, Aug 17 2009

with(combinat): a:=n->7*fibonacci(n, 2)+fibonacci(n+1, 2): seq(a(n), n=0..25); # Zerinvary Lajos, Apr 04 2008

a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{8},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{2,1},{1,9},30] (* Harvey P. Dale, Apr 20 2012 *)

a[0]:1$
a[1]:9$
a[n]:=2*a[n-1]+a[n-2]$
A048696(n):=a[n]$
makelist(A048696(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=9.
a(n) = ((4*sqrt(2)+1)(1+sqrt(2))^n - (4*sqrt(2)-1)(1-sqrt(2))^n)/2.
G.f.: (1+7*x)/(1 - 2*x - x^2). - Philippe Deléham, Nov 03 2008

A048772 Partial sums of A048696.

Original entry on oeis.org

1, 10, 29, 76, 189, 462, 1121, 2712, 6553, 15826, 38213, 92260, 222741, 537750, 1298249, 3134256, 7566769, 18267802, 44102381, 106472572, 257047533, 620567646, 1498182833

Offset: 0

Barry E. Williams

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A001333, A000129, A048694-A048696, A105082.

a048772 n = a048772_list !! n
a048772_list = scanl1 (+) a048696_list
-- Reinhard Zumkeller, Dec 15 2013

Accumulate[LinearRecurrence[{2,1},{1,9},30]] (* or *) LinearRecurrence[ {3,-1,-1},{1,10,29},30] (* Harvey P. Dale, Apr 20 2012 *)

a(n)=([0,1,0; 0,0,1; -1,-1,3]^n*[1;10;29])[1,1] \\ Charles R Greathouse IV, Feb 10 2017

a(n)=2*a(n-1)+a(n-2)+8; a(0)=1, a(1)=10.
a(n)=[ {(9+5*sqrt(2))(1+sqrt(2))^n - (9-5*sqrt(2))(1-sqrt(2))^n}/2*sqrt(2) ]-4.
a(0)=1, a(1)=10, a(2)=29, a(n)=3*a(n-1)-a(n-2)-a(n-3). - Harvey P. Dale, Apr 20 2012
G.f. ( 1+7*x ) / ( (x-1)*(x^2+2*x-1) ). a(n)=A048739(n)+7*A048739(n-1). - R. J. Mathar, Nov 08 2012

A382448 Triangle read by rows, defined by the two-variable g.f. (x^3y^2 + x^3y + 1)/(1 - x^2y - xy - x).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 8, 15, 8, 1, 1, 10, 29, 29, 10, 1, 1, 12, 47, 73, 47, 12, 1, 1, 14, 69, 149, 149, 69, 14, 1, 1, 16, 95, 265, 371, 265, 95, 16, 1, 1, 18, 125, 429, 785, 785, 429, 125, 18, 1, 1, 20, 159, 649, 1479, 1941, 1479, 649, 159, 20, 1

Offset: 0

F. Chapoton, Mar 26 2025

Every row is symmetric.

			Triangle begins:
  [0] [1]
  [1] [1,  1]
  [2] [1,  3,   1]
  [3] [1,  6,   6,   1]
  [4] [1,  8,  15,   8,   1]
  [5] [1, 10,  29,  29,  10,  1]
  [6] [1, 12,  47,  73,  47, 12,    1]
  [7] [1, 14,  69, 149, 149, 69,   14,   1]
  [8] [1, 16,  95, 265, 371, 265,  95,  16, 1]
  [9] [1, 18, 125, 429, 785, 785, 429, 125, 18, 1]

Similar to A008288, A103450, A382436 and A382444. Row sums are A105082.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (x^3*y^2 + x^3*y + 1)/(1 - x^2*y - x*y - x)
[list(u) for u in list(gf.O(10))]

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A048696 Generalized Pellian with second term equal to 9.

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A048772 Partial sums of A048696.

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A382448 Triangle read by rows, defined by the two-variable g.f. (x^3y^2 + x^3y + 1)/(1 - x^2y - xy - x).

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Sage

cp's OEIS Frontend

A048696 Generalized Pellian with second term equal to 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

Formula

A048772 Partial sums of A048696.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A382448 Triangle read by rows, defined by the two-variable g.f. (x^3*y^2 + x^3*y + 1)/(1 - x^2*y - x*y - x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

A382448 Triangle read by rows, defined by the two-variable g.f. (x^3y^2 + x^3y + 1)/(1 - x^2y - xy - x).