A096047 -id:A096047 - cp's OEIS frontend

A096045 a(n) = B(2n, 2)/B(2n) (see formula section).

1, 10, 46, 190, 766, 3070, 12286, 49150, 196606, 786430, 3145726, 12582910, 50331646, 201326590, 805306366, 3221225470, 12884901886, 51539607550, 206158430206, 824633720830, 3298534883326, 13194139533310, 52776558133246, 211106232532990, 844424930131966

Offset: 0

Benoit Cloitre, Jun 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000302, A002063, A096046, A096047, A096048.

[3*4^n-2: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

a[n_]:= Sum[2^k*Sum[Binomial[2*n, j]*BernoulliB[j], {j,0,k}], {k,0,2*n}]/BernoulliB[2*n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 14 2015 *)
NestList[4#+6&,1,30] (* Harvey P. Dale, Dec 27 2016 *)

A096045(n):=3*4^n-2$ makelist(A096045(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

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

[3*4^n-2 for n in range(41)] # G. C. Greubel, Jan 22 2023

a(n) = B(2*n, 2)/B(2*n), where B(n, p) = Sum_{i=0..n} p^i * (Sum_{j=0..i} binomial(n,j)*B(j)) with B(k) = k-th Bernoulli number.
a(n) = 3*4^n - 2.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = 4*a(n-1) + 6. First differences give A002063. - Paul Curtz, Jul 07 2008
From G. C. Greubel, Jan 22 2023: (Start)
a(n) = 3*A000302(n) - 2.
G.f.: (1+5*x)/((1-x)*(1-4*x)).
E.g.f.: 3*exp(4*x) - 2*exp(x). (End)

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

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)

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).

A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

Original entry on oeis.org

1, 91, 3751, 138811, 5028751, 181308931, 6529545751, 235085301451, 8463265086751, 304679288612371, 10968470088963751, 394865064451017691, 14215143591303768751, 511745180725868773411, 18422826609078989373751, 663221758853362301815531, 23875983327059668074930751

Offset: 1

Benoit Cloitre, Jun 18 2004

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (50,-553,1800,-1296).

Cf. A083420, A096045, A096046, A096047, A096048, A096053.

a[n_] := 6^(2*n-1) * BernoulliB[2*n, 1/6] / BernoulliB[2*n]; Array[a, 15] (* Amiram Eldar, May 07 2025 *)

a(n)=(1/12)*36^n-(1/6)*9^n-(1/4)*4^n+1/2;

a(n) = (1/12)*(36^n - 2*9^n - 3*4^n+6).
From Colin Barker, May 30 2020: (Start)
G.f.: x*(1 - 6*x)*(1 + 47*x + 36*x^2) / ((1 - x)*(1 - 4*x)*(1 - 9*x)*(1 - 36*x)).
a(n) = 50*a(n-1) - 553*a(n-2) + 1800*a(n-3) - 1296*a(n-4) for n>4. (End)

A096050 Decimal expansion of lim_{n->oo} B(2n,7)/(B(2n)*49^n) (see comment for B(n,k) definition).

Original entry on oeis.org

1, 0, 6, 2, 7, 5, 1, 6, 9, 9, 6, 9, 0, 2, 1, 1, 0, 7, 8, 2, 4, 5, 8, 3, 2, 5, 1, 9, 3, 3, 2, 6, 2, 6, 6, 9, 8, 2, 2, 7, 9, 5, 4, 2, 1, 1, 5, 1, 7, 2, 6, 6, 3, 1, 5, 7, 7, 2, 4, 0, 8, 4, 2, 6, 8, 1, 7, 1, 9, 1, 0, 5, 7, 9, 2, 3, 9, 1, 8, 7, 8, 5, 9, 0, 4, 0, 0, 9, 5, 8, 2, 1, 1, 2, 2, 3, 5, 7, 7, 1, 3, 8, 8, 8, 2

Offset: 1

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. B(2n,p)/B(2n) takes 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. And lim_{n->oo} B(2n,5)/(B(2n)*25^n) = (21-sqrt(5))/16.

Cf. A096045, A096046, A096047, A096048, A096049.

RealDigits[x/.FindRoot[ 1728x^3-6192x^2+7368x-2911==0,{x,1}, WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Feb 19 2012 *)

solve(q=1,1.1,1728*q^3-6192*q^2+7368*q-2911)

Limit_{n->oo} B(2n, 7)/(B(2n)*49^n) = 1.0627516996902110782... is the smallest root of 1728*X^3 - 6192*X^2 + 7368*X - 2911 = 0.

A096051 Decimal expansion of lim_{n->oo} B(2n,8)/(B(2n)*64^n) (see comment for B(n,k) definition).

Original entry on oeis.org

1, 0, 4, 1, 8, 4, 1, 8, 8, 8, 4, 0, 1, 9, 2, 1, 7, 8, 2, 2, 2, 8, 4, 5, 0, 8, 0, 5, 4, 1, 3, 5, 9, 2, 9, 9, 4, 3, 8, 7, 8, 8, 0, 5, 8, 0, 3, 3, 0, 2, 1, 7, 9, 9, 4, 7, 7, 3, 0, 9, 4, 3, 0, 4, 4, 2, 9, 2, 3, 3, 3, 9, 4, 3, 9, 5, 5, 6, 3, 7, 8, 2, 9, 3, 9, 2, 5, 8, 0, 3, 3, 2, 6, 2, 3, 1, 1, 3, 1, 6, 2, 3, 3, 2, 1

Offset: 1

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) is the k-th Bernoulli number.

			1.04184188840192178222845080541359299438788058033021...

Cf. A027641, A027642, A096045, A096046, A096047, A096048, A096049, A096050.

RealDigits[(16 - Sqrt[2])/14, 10, 100][[1]] (* Amiram Eldar, May 08 2022 *)

PARI
```
(16-sqrt(2))/14
```

Equals (16-sqrt(2))/14.

A096052 Decimal expansion of lim_{n->oo} B(2n,5)/(B(2n)*25^n) (see comment for B(n,k) definition).

Original entry on oeis.org

1, 1, 7, 2, 7, 4, 5, 7, 5, 1, 4, 0, 6, 2, 6, 3, 1, 4, 3, 9, 7, 4, 4, 2, 6, 6, 4, 5, 7, 0, 4, 2, 9, 5, 2, 3, 5, 2, 8, 4, 9, 6, 1, 3, 5, 2, 5, 2, 4, 2, 7, 9, 6, 4, 2, 2, 3, 3, 0, 6, 8, 9, 2, 2, 1, 6, 1, 8, 4, 2, 4, 4, 2, 1, 4, 7, 6, 3, 7, 1, 9, 3, 7, 8, 6, 5, 9, 9, 0, 9, 9, 4, 7, 6, 3, 2, 6, 1, 0, 7, 8, 1, 4, 4, 0

Offset: 1

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) is the k-th Bernoulli number.

			1.17274575140626314397442664570429523528496135252427...

Cf. A027641, A027642, A096045, A096046, A096047, A096048, A096049, A096050, A096051.

RealDigits[(21 - Sqrt[5])/16, 10, 100][[1]] (* Amiram Eldar, May 08 2022 *)

PARI
```
(21-sqrt(5))/16
```

Equals (21-sqrt(5))/16.

cp's OEIS Frontend

A096045 a(n) = B(2*n, 2)/B(2*n) (see formula section).

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

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

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A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

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A096050 Decimal expansion of lim_{n->oo} B(2n,7)/(B(2n)*49^n) (see comment for B(n,k) definition).

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A096051 Decimal expansion of lim_{n->oo} B(2n,8)/(B(2n)*64^n) (see comment for B(n,k) definition).

A096045 a(n) = B(2n, 2)/B(2n) (see formula section).