A083597 -id:A083597 - cp's OEIS frontend

A002042 a(n) = 7*4^n.

7, 28, 112, 448, 1792, 7168, 28672, 114688, 458752, 1835008, 7340032, 29360128, 117440512, 469762048, 1879048192, 7516192768, 30064771072, 120259084288, 481036337152, 1924145348608, 7696581394432, 30786325577728, 123145302310912, 492581209243648

Offset: 0

N. J. A. Sloane

Subsequence of A000069, the odious numbers. - Reinhard Zumkeller, Aug 26 2007
A rectangular prism with edge lengths 2^n, 2^(n+1) and 2^(n+2) has a surface area 2* (2^n*2^(n+1) + 2^(n+1)*2^(n+2) + 2^n*2^(n+2)) which equals 4*a(n). - J. M. Bergot, Aug 07 2013
x = A306472(n) and y = a(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 3^(6*n+1) = 4*y^3 (see Theorem 2.1 in Chakraborty, Hoque and Sharma). - Stefano Spezia, Feb 18 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Chakraborty, A. Hoque, R. Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4).

First differences of A083597. Bisection of A005009.

Cf. A306472 (37*27^n), A009971 (27^n), A000302 (4^n), A000290 (n^2), A000578 (n^3).

[7*4^n: n in [0..30]]; // Vincenzo Librandi, May 31 2011

7*4^Range[0, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
CoefficientList[Series[7/(1-4x), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 25 2015 *)
NestList[4#&,7,30] (* Harvey P. Dale, Mar 19 2021 *)

a(n)=7<<(2*n) \\ Charles R Greathouse IV, Apr 17 2012

[7*4^n for n in (0..30)] # G. C. Greubel, Feb 18 2019

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n-1), n > 0, with a(0) = 7.
G.f.: 7/(1-4*x). (End)
a(n) = 7*A000302(n). - Michel Marcus, Jun 24 2015
E.g.f.: 7*exp(4*x). - G. C. Greubel, Feb 18 2019

A206374 a(n) = (7*4^n - 1)/3.

Original entry on oeis.org

2, 9, 37, 149, 597, 2389, 9557, 38229, 152917, 611669, 2446677, 9786709, 39146837, 156587349, 626349397, 2505397589, 10021590357, 40086361429, 160345445717, 641381782869, 2565527131477, 10262108525909, 41048434103637, 164193736414549, 656774945658197

Offset: 0

Brad Clardy, Feb 07 2012

First bisection of A062092 and A081253, second bisection of A097163. - Bruno Berselli, Feb 12 2012

Except a(0)=2, this is the 3rd row of table A178415. - Michel Marcus, Apr 13 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002450, A006666, A072197; A002042 (first differences), A178415, A347834.

Magma
```
[(7*4^n-1)/3 : n in [0..30]];
```

Table[(7(4^n) - 1)/3, {n, 0, 24}] (* Alonso del Arte, Feb 11 2012 *)
CoefficientList[Series[(2-x)/(1-5*x+4*x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-4},{2,9},30] (* Vincenzo Librandi, Mar 20 2012 *)

vector(20,n,(7*4^(n-1)-1)/3) \\ Derek Orr, Apr 12 2015

G.f.: (2-x)/(1-5*x+4*x^2). - Bruno Berselli, Feb 12 2012
a(n) = A083597(n)+1. - Bruno Berselli, Feb 12 2012
a(n) = 4*a(n-1)+1 for n>0, a(0)=2. - Bruno Berselli, Oct 22 2015
a(n) = 7*A002450(n) + 2. - Yosu Yurramendi, Jan 24 2017
A006666(a(n)) = 2*n+11 for n > 0. - Juan Miguel Barga Pérez, Jun 18 2020
a(n) = 5*a(n-1) - 4*a(n-2) for n >= 2. - Wesley Ivan Hurt, Jun 30 2020
a(n) = A178415(3, n) = A347834(4, n-1), arrays, for n >= 1.- Wolfdieter Lang, Nov 29 2021

A176965 a(n) = 2^(n-1) - (2^n*(-1)^n + 2)/3.

Original entry on oeis.org

1, 0, 6, 2, 26, 10, 106, 42, 426, 170, 1706, 682, 6826, 2730, 27306, 10922, 109226, 43690, 436906, 174762, 1747626, 699050, 6990506, 2796202, 27962026, 11184810, 111848106, 44739242, 447392426, 178956970, 1789569706, 715827882, 7158278826

Offset: 1

Roger L. Bagula, Apr 29 2010

The ratio a(n+1)/a(n) approaches 10 for even n and 2/5 for odd n as n->infinity.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Merger of A020988 (even n) and A020989 (odd n).

List([1..30], n-> (3*2^(n-1) -(-2)^n -2)/3); # G. C. Greubel, Dec 28 2019

[(3*2^(n-1) -(-2)^n -2)/3: n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq( (3*2^(n-1) -(-2)^n -2)/3, n=1..30); # G. C. Greubel, Dec 28 2019

a[n_]:= a[n]= 2^(n-1)*If[n==1, 1, a[n-1]/2 +(-1)^(n-1)*Sqrt[(5 +4*(-1)^(n-1) )]/2]; Table[a[n], {n,30}]
LinearRecurrence[{1,4,-4}, {1,0,6}, 30] (* G. C. Greubel, Dec 28 2019 *)

vector(30, n, (3*2^(n-1) -(-2)^n -2)/3 ) \\ G. C. Greubel, Dec 28 2019

[(3*2^(n-1) -(-2)^n -2)/3 for n in (1..30)] # G. C. Greubel, Dec 28 2019

From R. J. Mathar, Apr 30 2010: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
G.f.: x*(1 - x + 2*x^2)/( (1-x)*(1+2*x)*(1-2*x) ). (End)
a(n) = A087231(n), n > 2. - R. J. Mathar, May 03 2010
a(2n-1) = A061547(2n), a(2n) = A061547(2n-1), n > 0. - Yosu Yurramendi, Dec 23 2016
a(n+1) = 2*A096773(n), n > 0. - Yosu Yurramendi, Dec 30 2016
a(2n-1) = A020989(n-1), a(2n) = A020988(n-1), n > 0. - Yosu Yurramendi, Jan 03 2017
a(2n-1) = (A083597(n-1) + A000302(n-1))/2, a(2n) = (A083597(n-1) - A000302(n-1))/2, n > 0. - Yosu Yurramendi, Mar 04 2017
a(n+2) = 4*a(n) + 2, a(1) = 1, a(2) = 0, n > 0. - Yosu Yurramendi, Mar 07 2017
a(n) = (-16 + (9 - (-1)^n) * 2^(n - (-1)^n))/24. - Loren M. Pearson, Dec 28 2019
E.g.f.: (3*exp(2*x) - 4*exp(x) + 3 - 2*exp(-2*x))/6. - G. C. Greubel, Dec 28 2019
a(n) = (2^n*5^(n mod 2) - 4)/6. - Heinz Ebert, Jun 29 2021

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

A221049 Expansion of (1+2x+3x^2-x^3)/((1-x)(1+x)(1-2x)(1+2*x)).

Original entry on oeis.org

1, 2, 8, 9, 36, 37, 148, 149, 596, 597, 2388, 2389, 9556, 9557, 38228, 38229, 152916, 152917, 611668, 611669, 2446676, 2446677, 9786708, 9786709, 39146836, 39146837, 156587348, 156587349, 626349396, 626349397, 2505397588, 2505397589, 10021590356, 10021590357

Offset: 0

Philippe Deléham, Apr 14 2013

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -4).

Cf. A083597, A206374.

LinearRecurrence[{0, 5, 0, -4}, {1, 2, 8, 9}, 40] (* T. D. Noe, Apr 17 2013 *)

a(n) = a(n-1)*4 if n even, a(n) = a(n-1)+1 if n odd.
a(2n) = (7*4^n-4)/3 = A083597(n).
a(2n+1) = (7*4^n-1)/3 = A206374(n).
a(n) = 5*a(n-2) - 4*a(n-4) with a(0)=1, a(1)=2, a(2)=8, a(3)=9.

cp's OEIS Frontend

A002042 a(n) = 7*4^n.

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A206374 a(n) = (7*4^n - 1)/3.

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A176965 a(n) = 2^(n-1) - (2^n*(-1)^n + 2)/3.

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A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

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A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A221049 Expansion of (1+2*x+3*x^2-x^3)/((1-x)*(1+x)*(1-2*x)*(1+2*x)).

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Formula

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

A221049 Expansion of (1+2x+3x^2-x^3)/((1-x)(1+x)(1-2x)(1+2*x)).