A096045 -id:A096045 - cp's OEIS frontend

A164346 a(n) = 3 * 4^n.

3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248, 211106232532992, 844424930131968

Offset: 0

Klaus Brockhaus, Aug 13 2009

Binomial transform of A000244 without initial 1.
Second binomial transform of A007283.
Third binomial transform of A010701.
Inverse binomial transform of A005053 without initial 1.
First differences of A024036. - Omar E. Pol, Feb 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 (powers of 4), A000244 (powers of 3), A007283 (3*2^n), A010701 (all 3's), A005053, A002001, A096045, A140660 (3*4^n+1), A002023 (6*4^n), A002063(9*4^n), A056120, A084509.

Magma
```
[ 3*4^n: n in [0..22] ];
```

3 4^Range[0,30]  (* Harvey P. Dale, Mar 11 2011 *)

A164346(n)=3*4^n \\ M. F. Hasler, Jul 28 2015

def A164346(n): return 3<<(n<<1) # Chai Wah Wu, Aug 30 2024

a(n) = 4*a(n-1) for n > 1; a(0) = 3.
G.f.: 3/(1-4*x).
a(n) = A002001(n+1). a(n) = A096045(n)+2. a(n) = A140660(n)-1.
a(n) = A002023(n)/2. a(n) = A002063(n)/3. a(n) = A056120(n+3)/9.
Apparently a(n) = A084509(n+3)/2.
a(n) = A110594(n+1), n>1. - R. J. Mathar, Aug 17 2009
a(n) = 3*A000302(n). - Omar E. Pol, Feb 18 2013
a(n) = A000079(2*n) + A000079(2*n+1). - M. F. Hasler, Jul 28 2015
E.g.f.: 3*exp(4*x). - G. C. Greubel, Sep 15 2017

A096046 a(n) = B(2n,3)/B(2n) (see comment).

Original entry on oeis.org

1, 15, 141, 1275, 11481, 103335, 930021, 8370195, 75331761, 677985855, 6101872701, 54916854315, 494251688841, 4448265199575, 40034386796181, 360309481165635, 3242785330490721, 29185067974416495, 262665611769748461

Offset: 0

Benoit Cloitre, Jun 17 2004

nonn

B(n,p) = Sum_{i=0..n} p^i*Sum_{j=0..i} binomial(n,j)*B(j) where B(k) = k-th Bernoulli number.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A096045, A096047, A096048.

[(1/4)*(7*9^n-3): n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

A096046(n):=(1/4)*(7*9^n-3)$ makelist(A096046(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,3^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

a(n) = (1/4)*(7*9^n - 3).
a(n) = 10*a(n-1) - 9*a(n-2); a(0)=1, a(1)=15.
a(n) = 9*a(n-1) + 6. First differences = 14*A001019(n). - Paul Curtz, Jul 07 2008

A096053 a(n) = (3*9^n - 1)/2.

Original entry on oeis.org

1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681

Offset: 0

Benoit Cloitre, Jun 18 2004

Generalized NSW numbers. - Paul Barry, May 27 2005
Counts total area under elevated Schroeder paths of length 2n+2, where area under a horizontal step is weighted 3. Case r=4 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315. Fifth binomial transform of (1+8x)/(1-16x^2), A107906. - Paul Barry, May 27 2005
Primes in this sequence include: a(2) = 13, a(4) = 1093, a(7) = 797161. Semiprimes in this sequence include: a(3) = 121 = 11^2, a(5) = 9841 = 13 * 757, a(6) = 88573 = 23 * 3851, a(9) = 64570081 = 1871 * 34511, a(10) = 581130733 = 1597 * 363889, a(12) = 47071589413 = 47 * 1001523179, a(19) = 225141952945498681 = 13097927 * 17189128703.
Sum of divisors of 9^n. - Altug Alkan, Nov 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A083420, A096045, A096046, A096047, A096054.

Cf. A107903, A138894 ((5*9^n-1)/4).

[(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

Table[(3*9^n - 1)/2, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

a(n)=(3*9^n-1)/2 \\ Charles R Greathouse IV, Sep 28 2015

vector(30, n, n--; sigma(9^n)) \\ Altug Alkan, Nov 10 2015

From Paul Barry, May 27 2005: (Start)
G.f.: (1+3*x)/(1-10*x+9*x^2);
a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;
a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)
a(n-1) = (-9^n/3)*B(2n,1/3)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.
a(n) = 10*a(n-1) - 9*a(n-2).
a(n) = 9*a(n-1) + 4. - Vincenzo Librandi, Nov 01 2011
a(n) = A000203(A001019(n)). - Altug Alkan, Nov 10 2015
a(n) = A320030(3^n-1). - Nathan M Epstein, Jan 02 2019

Edited by N. J. A. Sloane, at the suggestion of Andrew S. Plewe, Jun 15 2007

A096047 a(n)=B(2n,4)/B(2n) (see comment).

Original entry on oeis.org

1, 22, 346, 5482, 87466, 1398442, 22370986, 357919402, 5726644906, 91626056362, 1466015853226, 23456249457322, 375299974539946, 6004799525530282, 96076792140049066, 1537228673167043242, 24595658766377724586

Offset: 0

Benoit Cloitre, Jun 17 2004

B(n,p)=sum(i=0,n,p^i*sum(j=0,i,binomial(n,j)*B(j))) where B(k)=k-th Bernoulli number

Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A096045, A096046, A096048.

LinearRecurrence[{21,-84,64},{1,22,346},20] (* Harvey P. Dale, Oct 13 2016 *)

a[0]:1$ a[1]:22$ a[2]:346$ a[n]:=(1/3)*(4*16^n+4^n-2)$ A096047(n):=a[n]$ makelist(A096047(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,4^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

a(n)=(1/3)*(4*16^n+4^n-2); a(0)=1, a(1)=22, a(2)=346 and a(n)=21*a(n-1)-84*a(n-2)+64*a(n-3)

A096048 a(n)=B(2n,6)/B(2n) (see comment).

Original entry on oeis.org

1, 42, 1446, 51486, 1848966, 66524910, 2394568086, 86201542014, 3103229527206, 111716029897998, 4021774981740726, 144783880503964062, 5212219528644719046, 187639901505929327406, 6755036440486736068566

Offset: 0

Benoit Cloitre, Jun 17 2004

nonn

B(n,p)=sum(i=0,n,p^i*sum(j=0,i,binomial(n,j)*B(j))) where B(k)=k-th Bernoulli number

Cf. A096045, A096046, A096047.

a[0]:1$ a[1]:42$ a[2]:1446$ a[3]:51486$ a[n]:=(1/10)*(11*36^n+2*9^n+3*4^n-6)$ A096048(n):=a[n]$ makelist(A096048(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,6^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

a(n)=(1/10)*(11*36^n+2*9^n+3*4^n-6); a(0)=1, a(1)=42, a(2)=1446, a(3)=51486 and a(n)=50*a(n-1)-553*a(n-2)+1800*a(n-3)-1296*a(n-4)

A140660 a(n) = 3*4^n + 1.

Original entry on oeis.org

4, 13, 49, 193, 769, 3073, 12289, 49153, 196609, 786433, 3145729, 12582913, 50331649, 201326593, 805306369, 3221225473, 12884901889, 51539607553, 206158430209, 824633720833, 3298534883329, 13194139533313, 52776558133249

Offset: 0

Paul Curtz, Jul 10 2008

nonn

An Engel expansion of 4/3 to the base 4 as defined in A181565, with the associated series expansion 4/3 = 4/4 + 4^2/(4*13) + 4^3/(4*13*49) + 4^4/(4*13*49*193) + .... Cf. A199115. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A181565, A199115.

[3*4^n+1: n in [0..30] ]; // Vincenzo Librandi, May 23 2011

LinearRecurrence[{5,-4}, {4,13}, 50] (* or *) CoefficientList[Series[ (7*x-4)/((1-x)*(4*x-1)), {x,0,50}], x] (* G. C. Greubel, Sep 15 2017 *)

x='x+O('x^50); Vec((7*x-4)/((1-x)*(4*x-1))) \\ G. C. Greubel, Sep 15 2017

a(n) = A002001(n+1) + 1.
a(n) = 4*a(n-1) - 3.
First differences: a(n+1) - a(n) = A002063(n).
a(n+k) - a(n) = 3*(4^k - 1)*A000302(n) = 9*A002450(k)*A000302(n).
a(n) = A140529(n) - A096045(n).
O.g.f.: (7*x - 4)/((1 - x)*(4*x - 1)). - R. J. Mathar, Jul 14 2008
From G. C. Greubel, Sep 15 2017: (Start)
E.g.f.: 3*exp(4*x) + exp(x).
a(n) = 5*a(n-1) - 4*a(n-2). (End)

Edited and extended R. J. Mathar, Jul 14 2008

A140683 a(n) = 3(-1)^(n+1)2^n - 1.

Original entry on oeis.org

-4, 5, -13, 23, -49, 95, -193, 383, -769, 1535, -3073, 6143, -12289, 24575, -49153, 98303, -196609, 393215, -786433, 1572863, -3145729, 6291455, -12582913, 25165823, -50331649, 100663295, -201326593, 402653183, -805306369, 1610612735, -3221225473

Offset: 0

Paul Curtz, Jul 11 2008

sign

Alternated reading of negative of A140660 and A140529.
The binomial transform yields -4 followed by the negative of A140657.
The inverse binomial transform yields essentially a signed version of A000244. - R. J. Mathar, Aug 02 2008

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

[3*(-1)^(n+1)*2^n-1: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

Table[3(-1)^(n+1)2^n-1,{n,0,40}] (* or *) LinearRecurrence[{-1,2},{-4,5},40] (* Harvey P. Dale, May 26 2011 *)

a(2n) = -A140660(n). a(2n+1) = A140529(n).
a(n+1) - a(n) = (-1)^n*A005010(n). a(2n) + a(2n+1) = A096045(n).
a(n) = A140590(n+1) - 2*A140590(n).
O.g.f: (4-x)/((x-1)(2x+1)). - R. J. Mathar, Aug 02 2008
a(n) = -a(n-1) + 2*a(n-2); a(0)=-4, a(1)=5. - Harvey P. Dale, May 26 2011

Edited and extended by R. J. Mathar, Aug 02 2008

A140657 Powers of 2 with 3 alternatingly added and subtracted.

Original entry on oeis.org

4, -1, 7, 5, 19, 29, 67, 125, 259, 509, 1027, 2045, 4099, 8189, 16387, 32765, 65539, 131069, 262147, 524285, 1048579, 2097149, 4194307, 8388605, 16777219, 33554429, 67108867, 134217725, 268435459, 536870909, 1073741827, 2147483645, 4294967299, 8589934589

Offset: 0

Paul Curtz, Jul 10 2008

Jean-François Alcover and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 101 terms from Jean-François Alcover)
Index entries for linear recurrences with constant coefficients, signature (1, 2).

Cf. A000079, A007283, A010734, A033999, A096045, A140683.

[2^n+3*(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

LinearRecurrence[{1,2},{4,-1},40] (* or *) Total/@Partition[Riffle[ Table[ 2^n, {n,0,40}],{3,-3}],2] (* Harvey P. Dale, Nov 13 2014 *)
CoefficientList[Series[(4 - 5 x) / ((1 + x) (1 - 2 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Jan 14 2015 *)

a(2n) = A000079(2n+1) + 3, a(2n+1) = A000079(2n+2) - 3.
a(n+1) - 2*a(n) = -9*A033999(n) = (-1)^(n+1)*A010734.
a(n) + a(n+1) = 3^*2^n = A007283(n).
a(2n) + a(2n+1) = A096045(n) + 2.
a(-n) = -A140683(n)/2^n.
O.g.f.: (4-5*x)/((1-2*x)(1+x)). - R. J. Mathar, Jul 29 2008
a(n) = 2^n+3*(-1)^n. - R. J. Mathar , Jul 29 2008

Edited and extended by R. J. Mathar, Jul 29 2008

4 inserted as first term and formulas accordingly updated by Jean-François Alcover, Jan 14 2015

A259713 a(n) = 32^n - 2(-1)^n.

Original entry on oeis.org

1, 8, 10, 26, 46, 98, 190, 386, 766, 1538, 3070, 6146, 12286, 24578, 49150, 98306, 196606, 393218, 786430, 1572866, 3145726, 6291458, 12582910, 25165826, 50331646, 100663298, 201326590, 402653186, 805306366, 1610612738, 3221225470, 6442450946, 12884901886

Offset: 0

Paul Curtz, Jul 03 2015

Inverse binomial transform of 3^n, with 3 (second term) excluded.

a(n) mod 9 gives A010689.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000244, A005010, A007283, A010689, A096045, A140788, A182460, A201630.

[3*2^n-2*(-1)^n: n in [0..40]]; // Vincenzo Librandi, Jul 04 2015

Table[3 2^n - 2 (-1)^n, {n, 0, 50}] (* Vincenzo Librandi, Jul 04 2015 *)
LinearRecurrence[{1,2},{1,8},40] (* Harvey P. Dale, Aug 19 2020 *)

Vec(-(7*x+1)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jul 03 2015

a(n) = a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(1)=8.
a(n) = 2*a(n-1) - 6*(-1)^n for n>0, a(0)=1.
a(4n+2) = 10*A182460(n); a(2n) = A096045(n), a(2n+1) = A140788(n).
a(n) = 3*A014551(n+1) - A201630(n).
a(n+2) - a(n) = a(n) + a(n+1) = A005010(n).
G.f.: -(7*x+1) / ((x+1)*(2*x-1)). - Colin Barker, Jul 03 2015

Typo in data fixed by Colin Barker, Jul 03 2015

A096049 a(n) = [B(2n,5)/B(2n)] ( [x] = floor(x), see comment for B(n,k) definition ).

Original entry on oeis.org

1, 31, 745, 18397, 458545, 11455304, 286331664, 7157976493, 178947452208, 4473674081283, 111841775707840, 2796043915880138, 69901094917491465, 1747527354316971026, 43688183741551848165, 1092204592811481165247, 27305114815741345242261, 682627870365123204281633

Offset: 0

Benoit Cloitre, Jun 17 2004

nonn

B(n,p) = Sum_{i=0..n} p^i*(Sum_{j=0..i} binomial(n,j)*B(j)) where B(k)=k-th Bernoulli number. B(2n,p)/B(2n) take integer values for all n if p=1,2,3,4,6. p=5 is the smallest integer for which B(2n,5)/B(2n) is not always integer valued.

Cf. A096045, A096046, A096047, A096048.

a(n)=floor(sum(i=0,2*n,5^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n))

a(n) = floor((1/16)*(21-sqrt(5))*25^n + (1/8)*sqrt(5)*((25/4)^n+(25/9)^n-(25/16)^n) - (1/16)*(5-sqrt(5)) + (1/4)*sqrt(5)*(25/36)^n).

cp's OEIS Frontend

A164346 a(n) = 3 * 4^n.

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A096046 a(n) = B(2n,3)/B(2n) (see comment).

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A096053 a(n) = (3*9^n - 1)/2.

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A096047 a(n)=B(2n,4)/B(2n) (see comment).

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A096048 a(n)=B(2n,6)/B(2n) (see comment).

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A140660 a(n) = 3*4^n + 1.

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A140683 a(n) = 3*(-1)^(n+1)*2^n - 1.

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A140683 a(n) = 3(-1)^(n+1)2^n - 1.

A259713 a(n) = 32^n - 2(-1)^n.