A144843 -id:A144843 - cp's OEIS frontend

A016129 Expansion of 1/((1-2x)(1-6*x)).

1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, 90698752, 544194560, 3265171456, 19591036928, 117546237952, 705277460480, 4231664828416, 25389989101568, 152339934871552, 914039609753600, 5484237659570176, 32905425959518208, 197432555761303552

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Row sums of A100851.
Cf. A071951, A089278, A089500, A129467, A144843.
Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).

[(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // Vincenzo Librandi, Oct 09 2011

Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* Harvey P. Dale, Jan 15 2015 *)

Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,8,12) for n in range(1, 31)] # Zerinvary Lajos, Apr 23 2009

[(6^n - 2^n)/4 for n in range(1,31)] # Zerinvary Lajos, Jun 04 2009

a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - Wolfdieter Lang, Nov 07 2003
From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)
G.f.: 1/((1-2*x)*(1-6*x)).
E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.
a(n) = (6^(n+1) - 2^(n+1))/4. (End)
a(n)^2 = A144843(n+1). - Philippe Deléham, Nov 26 2008
a(n) = 8*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 01 2009
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A176794 Triangle read by rows: T(n, k) = 2^n(q^k - 1)(q^(n - k) - 1) + 1, where q = 3.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 129, 129, 1, 1, 833, 1025, 833, 1, 1, 5121, 6657, 6657, 5121, 1, 1, 30977, 40961, 43265, 40961, 30977, 1, 1, 186369, 247809, 266241, 266241, 247809, 186369, 1, 1, 1119233, 1490945, 1610753, 1638401, 1610753, 1490945, 1119233, 1

Offset: 0

Roger L. Bagula, Apr 26 2010

			Triangle begins as:
  1;
  1,       1;
  1,      17,       1;
  1,     129,     129,       1;
  1,     833,    1025,     833,       1;
  1,    5121,    6657,    6657,    5121,       1;
  1,   30977,   40961,   43265,   40961,   30977,       1;
  1,  186369,  247809,  266241,  266241,  247809,  186369,       1;
  1, 1119233, 1490945, 1610753, 1638401, 1610753, 1490945, 1119233,     1;

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

Cf. A000012 (q=1), A176793 (q=2), this sequence (q=3), A176795 (q=4).

Cf. A144843.

f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
A176794:= func< n,k | f(n,k,3) >;
[A176794(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 03 2024

T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten

def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n
def A176794(n,k): return f(n,k,3)
flatten([[A176794(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Oct 03 2024

T(n, k) = 1 - (f(n+1, 2*k+1, q) - f(n+1, 1, q)) - (f(n+1, 2*n-2*k+1, q) - f(n+1, 2*n+1, q)), where f(n, k, q) = 2^(n-1) * q^((k-1)/2), and q = 3.
From G. C. Greubel, Oct 02 2024: (Start)
T(n, k) = 2^n*(3^k - 1)*(3^(n-k) - 1) + 1.
T(2*n, n) = 1 + 4^n*(3^n - 1)^2 = 1 + 16*A144843(n).
Sum_{k=0..n} T(n, k) = 2^n*(n + 2 + (n-2)*3^n) + (n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/4)*(1 + (-1)^n)*(2 + 2^n - 6^n). (End)

Edited by G. C. Greubel, Oct 03 2024

cp's OEIS Frontend

A016129 Expansion of 1/((1-2x)(1-6*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Sage

Formula

A176794 Triangle read by rows: T(n, k) = 2^n(q^k - 1)(q^(n - k) - 1) + 1, where q = 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

cp's OEIS Frontend

A016129 Expansion of 1/((1-2*x)*(1-6*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Sage

Formula

A176794 Triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A016129 Expansion of 1/((1-2x)(1-6*x)).

A176794 Triangle read by rows: T(n, k) = 2^n(q^k - 1)(q^(n - k) - 1) + 1, where q = 3.