A084152 -id:A084152 - cp's OEIS frontend

A084175 Jacobsthal oblong numbers.

0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575, 244335800775, 977343902151, 3909374210503, 15637499638215, 62549992960455

Offset: 0

Paul Barry, May 18 2003

Inverse binomial transform is A001019 doubled up.
Binomial transform is A084177.
Partial sums of A003683.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 11.
Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Except for initial terms, same as A015249 and A084152.

Cf. A001654, A001656, A084158, A084159, A099930.

List([0..30], n-> (2^(2*n+1) -(-2)^n -1)/9); # G. C. Greubel, Sep 21 2019

[(2*4^n-(-2)^n-1)/9: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011

for n from 1 to 25 do print(round(2^n/3)*round(2^(n+1)/3)) od; # Gary Detlefs, Feb 10 2010

Table[(2*4^n -(-2)^n -1)/9, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011, modified by G. C. Greubel, Sep 21 2019 *)
LinearRecurrence[{3,6,-8}, {0,1,3}, 25] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=(2*4^n-(-2)^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

def A084175(n): return ((m:=1<Chai Wah Wu, Apr 25 2025

[gaussian_binomial(n, 2, -2) for n in range(1, 26)] # Zerinvary Lajos, May 28 2009

a(n) = A001045(n)*A001045(n+1).
a(n) = (2*4^n - (-2)^n - 1)/9;
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), a(0)=0, a(1)=1, a(2)=3.
G.f.: x/((1+2*x)*(1-x)*(1-4*x)).
E.g.f.: (2*exp(4*x) - exp(x) - exp(-2*x))/9.
a(n+1) - 4*a(n) = 1, -1, 3, -5, 11, ... = A001045(n+1) signed. - Paul Curtz, May 19 2008
a(n) = round(2^n/3) * round(2^(n+1)/3). - Gary Detlefs, Feb 10 2010
From Peter Bala, Mar 30 2015: (Start)
The shifted o.g.f. A(x) := 1/( (1 + 2*x)*(1 - x)*(1 - 4*x) ) = 1/(1 - 3*x - 6*x^2 + 8*x^3). Hence A(x) == 1/(1 - 3*x + 3*x^2 - x^3) (mod 9) == 1/(1 - x)^3 (mod 9). It follows by Theorem 1 of Heninger et al. that (A(x))^(1/3) = 1 + x + 4*x^2 + 10*x^3 + ... has integral coefficients.
Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} J(3*n)/J(n)*x^n/n ), where J(n) = A001045(n) are the Jacobsthal numbers. Cf. A001656, A099930. (End)

A015249 Gaussian binomial coefficient [ n,2 ] for q = -2.

Original entry on oeis.org

1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575, 244335800775, 977343902151, 3909374210503, 15637499638215

Offset: 2

Olivier Gérard, Dec 11 1999

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

G. C. Greubel, Table of n, a(n) for n = 2..500
Index entries for linear recurrences with constant coefficients, signature (3,6,-8)

Except for initial terms, same as A084152 and A084175.

Join[{a=1,b=3},Table[c=2*b+8*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011 *)
Table[QBinomial[n, 2, -2], {n, 2, 25}] (* G. C. Greubel, Jul 30 2016 *)

a(n)=(4^n - 2 + (-1)^n*2^n)/18 \\ Charles R Greathouse IV, Jul 30 2016

def A015249(n): return ((m:=1<>1|1)//3) # Chai Wah Wu, Apr 25 2025

[gaussian_binomial(n,2,-2) for n in range(2,25)] # Zerinvary Lajos, May 28 2009

G.f.: x^2/((1-x)*(1+2*x)*(1-4*x)).
From Vincenzo Librandi, Mar 20 2011: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) + (-1)^n *2^(n-2), n >= 4.
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), n >= 3. (End)
a(n) = (1/18)*(4^n - 2 + (-1)^n*2^n). - R. J. Mathar, Mar 21 2011
E.g.f.: 2*exp(x)*sinh(3*x/2)^2/9. - Stefano Spezia, Apr 25 2025

A249993 Expansion of 1/((1+x)(1+2x)(1-4x)).

Original entry on oeis.org

1, 1, 11, 29, 147, 525, 2227, 8653, 35123, 139469, 559923, 2235597, 8950579, 35785933, 143176499, 572640461, 2290692915, 9162509517, 36650562355, 146601200845, 586406900531, 2345623407821, 9382502019891, 37529991302349, 150119998763827, 600479927946445

Offset: 0

Alex Ratushnyak, Dec 27 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,10,8).

Cf. A249992.
Cf. A006095, A171477 for g.f. 1/((1-x)*(1-2*x)*(1-4*x)).
Cf. A015249, A084152, A084175 for g.f. 1/((1-x)*(1+2*x)*(1-4*x)).
Cf. A109765 for g.f. 1/((1+x)*(1-2*x)*(1-4*x)).

[(2^(2*n+3) +(-1)^n*(5*2^(n+1)-3))/15: n in [0..40]]; // G. C. Greubel, Oct 10 2022

CoefficientList[Series[1/((1+x)(1+2x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[{1,10,8},{1,1,11},30] (* Harvey P. Dale, Dec 13 2018 *)

Vec(1/((1+x)*(1+2*x)*(1-4*x)) + O(x^40)) \\ Michel Marcus, Dec 28 2014

[(2^(2*n+3) +(-1)^n*(5*2^(n+1)-3))/15 for n in range(41)] # G. C. Greubel, Oct 10 2022

G.f.: 1/((1+x)*(1+2*x)*(1-4*x)).
a(n) = ( 2^(3+2*n) + (5*2^(1+n) - 3)*(-1)^n )/15. Colin Barker, Dec 28 2014
a(n) = a(n-1) + 10*a(n-2) + 8*a(n-3). - Colin Barker, Dec 28 2014
E.g.f.: (1/15)*(10*exp(-2*x) - 3*exp(-x) + 8*exp(4*x)). - G. C. Greubel, Oct 10 2022

A084153 Binomial transform of a Jacobsthal convolution.

Original entry on oeis.org

0, 0, 1, 6, 33, 170, 861, 4326, 21673, 108450, 542421, 2712446, 13562913, 67815930, 339082381, 1695417366, 8477097753, 42385510610, 211927596741, 1059638071086, 5298190530193, 26490953000490, 132454765701501, 662273829905606

Offset: 0

Paul Barry, May 16 2003

Binomial transform of A084152.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-3,-10).

Cf. A084152.

[(5^n -2^(n+1) +(-1)^n)/18: n in [0..40]]; // G. C. Greubel, Oct 10 2022

LinearRecurrence[{6,-3,-10}, {0,0,1}, 41] (* G. C. Greubel, Oct 10 2022 *)

[(5^n -2^(n+1) +(-1)^n)/18 for n in range(41)] # G. C. Greubel, Oct 10 2022

a(n) = (5^n - 2*2^n + (-1)^n)/18.
G.f.: x^2/((1+x)*(1-2*x)*(1-5*x)).
E.g.f.: exp(x)*(exp(2*x) - exp(-x))^2/18 = (exp(5*x) - 2*exp(2*x) + exp(-x))/18.

cp's OEIS Frontend

A084175 Jacobsthal oblong numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A015249 Gaussian binomial coefficient [ n,2 ] for q = -2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Sage

Formula

A249993 Expansion of 1/((1+x)*(1+2*x)*(1-4*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A084153 Binomial transform of a Jacobsthal convolution.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A249993 Expansion of 1/((1+x)(1+2x)(1-4x)).