A144472 -id:A144472 - cp's OEIS frontend

A145593 Inverse binomial transform of A144472.

-1, 3, 4, -7, 24, -47, 104, -207, 424, -847, 1704, -3407, 6824, -13647, 27304, -54607, 109224, -218447, 436904, -873807, 1747624, -3495247, 6990504, -13981007, 27962024, -55924047, 111848104, -223696207, 447392424, -894784847

Offset: 0

Paul Curtz, Oct 14 2008

sign

From the third entry on, the last digits have period 2.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2, 1, 2).

LinearRecurrence[{-2,1,2},{-1,3,4},30] (* Harvey P. Dale, May 11 2024 *)

G.f.: (1-x-11x^2)/((1+2x)(1+x)(x-1)). - R. J. Mathar, Oct 24 2008

a(n)=(5*2^n/3-9/2)*(-1)^n+11/6. - R. J. Mathar, Oct 24 2008

Extended by R. J. Mathar, Oct 24 2008

A005015 a(n) = 11*2^n.

Original entry on oeis.org

11, 22, 44, 88, 176, 352, 704, 1408, 2816, 5632, 11264, 22528, 45056, 90112, 180224, 360448, 720896, 1441792, 2883584, 5767168, 11534336, 23068672, 46137344, 92274688, 184549376, 369098752, 738197504, 1476395008, 2952790016, 5905580032, 11811160064

Offset: 0

N. J. A. Sloane

The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005
11 times powers of 2. - Omar E. Pol, Dec 16 2008
A144472 = -1,2,9,13,31,57,.... a(n) = A144472(n+1)+A144472(n+2). Also a(n) = A144472(n+3)-A144472(n+1). A144472(n+1) is a Jacobsthal sequence from 2 and 9: A144472(n+3) = A144472(n+2)+2*A144472(n+1). Note a(n) mod 9 = period 6: repeat 2,4,8,7,5,1 = A153130(n+1). - Paul Curtz, Jan 06 2009

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

Row sums of (10, 1)-Pascal triangle A093645.

Cf. A000079, A144472, A153130.

[11*2^n: n in [0..40]]; // Vincenzo Librandi, Aug 14 2011

11*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,11,30] (* Harvey P. Dale, Jun 11 2021 *)

a(n)=11<Charles R Greathouse IV, Oct 07 2015

G.f.: 11/(1-2*x).
a(n) = 2*a(n-1), n>0; a(0)=11. - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*11. - Omar E. Pol, Dec 16 2008
E.g.f.: 11*exp(2*x). - Elmo R. Oliveira, Aug 16 2024

A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

Original entry on oeis.org

1, 0, 3, 2, 9, 12, 31, 54, 117, 224, 459, 906, 1825, 3636, 7287, 14558, 29133, 58248, 116515, 233010, 466041, 932060, 1864143, 3728262, 7456549, 14913072, 29826171, 59652314, 119304657, 238609284, 477218599, 954437166, 1908874365

Offset: 0

Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000

Growth of happy bug population in GCSE math course work assignment.
The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. - Wolfdieter Lang, Jun 25 2010
With offset 1: a(n) = -2^n*Sum_{k=0..n} k^p*q^k for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013
From Paul Curtz, Nov 02 2021 (Start)
a(n-2) difference table (from 0, 0, a(n)):
    0    0    1    0    3    2    9    12    31    54  ...
    0    1   -1    3   -1    7    3    19    23    63  ...
    1   -2    4   -4    8   -4   16     4    40    44  ...
   -3    6   -8   12  -12   20  -12    36     4    84  ...
    9  -14   20  -24   32  -32   48   -32    80     0  ...
  -23   34  -44   56  -64   80  -80   112   -80   176  ...
   57  -78  100 -120  144 -160  192  -192   256  -192  ...
   ... .
The signature is valid for every row.
a(n-2) + a(n-1) = A001045(n).
a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).
a(n-2) + a(n+3) = see A144472(n+1).
Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).
First subdiagonal: -A036895(n) = -2*A001787(n).
Main diagonal: A001787(n) = -first and -third upper diagonals.
Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Iwan Duursma, Xiao Li, and Hsin-Po Wang, Multilinear Algebra for Distributed Storage, arXiv:2006.08911 [cs.IT], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A232603, A232604.

Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.

CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *)

c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013

G.f.: 1 / (1-3*x^2-2*x^3).
With offset 1: a(1)=1; a(n) = 2*a(n-1) - (-1)^n*n; a(n) = (1/9)*(2^(n+1) - (-1)^n*(3*n+2)). - Benoit Cloitre, Nov 02 2002
a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - Paul Barry, Nov 24 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - Paul Barry, Oct 16 2004
a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = ( 2^(n+2) + (-1)^n*(3*n+5) )/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. - Wolfdieter Lang, Jun 25 2010
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)
G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012
E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019

More terms from James Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000

A021069 Decimal expansion of 1/65.

Original entry on oeis.org

0, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5

Offset: 0

N. J. A. Sloane

Without the leading 0 also the decimal expansion of 2/13.

			0.0153846153846...  - _Natan Arie Consigli_, Sep 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..10000
Christian Táfula, Knights are 24/13 times faster than the king, arXiv:2405.19589 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A010734, A020806, A021017, A021095, A128834, A130806, A144472.

Magma
```
1./65; // G. C. Greubel, Jan 15 2018
```

Digits:=100: evalf(1/65); # Wesley Ivan Hurt, Sep 20 2016

Join[ConstantArray[0, Abs@ Last@ #], First@ #] &@ RealDigits[N[1/65, 120]] (* Michael De Vlieger, Sep 06 2016 *)

1./65. /* Michel Marcus, Aug 22 2016 */

Equals 2 - 24/13. See Táfula link. - Michel Marcus, May 31 2024

G.f.: x*(1 + 4*x - 2*x^2 + 6*x^3)/((1 - x)*(1 + x)*(1 - x + x^2)). - Stefano Spezia, Apr 30 2025

cp's OEIS Frontend

A145593 Inverse binomial transform of A144472.

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A005015 a(n) = 11*2^n.

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A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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A021069 Decimal expansion of 1/65.

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Magma

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Mathematica

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Formula

A053088 a(n) = 3a(n-2) + 2a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.