A005259 -id:A005259 - cp's OEIS frontend

A133370 Primes p such that p does not divide any term of the Apery sequence A005259 .

2, 3, 7, 13, 23, 29, 43, 47, 53, 67, 71, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 199, 223, 229, 239, 263, 269, 277, 281, 311, 313, 317, 337, 349, 353, 359, 373, 383, 389, 397, 401, 409, 421, 449, 457, 461, 467, 479, 487, 491

Offset: 1

Philippe Deléham, Oct 27 2007

nonn

Malik and Straub give arguments suggesting that this sequence is infinite. - N. J. A. Sloane, Aug 06 2017

Robert Price, Table of n, a(n) for n = 1..758
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Amita Malik, Mathematica notebook for generating this sequence and A260793, A291275-A291284
Amita Malik, List of all primes up to 10000 in this sequence and in A260793, A291275-A291284, together with Mathematica code.
E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013.

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

NeverDividesLucasSeqQ[a_, p_] := And @@ Table[Mod[a[n], p]>0, {n, 0, p-1}];
A3[a_, b_, c_, n_ /; n < 0] = 0;
A3[a_, b_, c_, 0] = 1;
A3[a_, b_, c_, n_] := A3[a, b, c, n] = (((2n - 1)(a (n-1)^2 + a (n-1) + b)) A3[a, b, c, n-1] - c (n-1)^3 A3[a, b, c, n-2])/n^3;
A3[a_, b_, c_, d_, n_ /; n < 0] = 0;
Agamma[n_] := A3[17, 5, 1, n];
Select[Range[1000], PrimeQ[#] && NeverDividesLucasSeqQ[Agamma, #]&] (* Jean-François Alcover, Aug 05 2018, copied from Amita Malik's notebook *)

Terms a(16) onwards computed by Amita Malik - N. J. A. Sloane, Aug 21 2017

A178790 The arithmetic mean of (2k+1)A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.

Original entry on oeis.org

1, 8, 127, 2624, 61501, 1552760, 41186755, 1131614720, 31923047665, 919243356008, 26908963456783, 798379043762624, 23954974906866901, 725620080605773592, 22159617936375571627, 681528994326392115200, 21090805673899997148025, 656256696917886135153800

Offset: 1

Zhi-Wei Sun, Jun 14 2010

nonn

Conjecture: the number a(n) = n^(-1)*Sum_{k=0..n-1}(2*k+1)*A_k is always an integer.
We can prove that for any prime p>3 we have a(p)=p (mod p^4).
Conjecture: If p=5,7 (mod 8) is a prime then sum_{k=0}^{p-1}A_k=0 (mod p^2); if p=1,3 (mod 8) is a prime greater than 3 and p=x^2+2y^2 with x,y integers then sum_{k=0}^{p-1}A_k=4x^2-2p (mod p^2).
a(n) is always an integer. The detailed proof can be found in the latest version of arXiv:1006.2776 . - Zhi-Wei Sun, Jun 17 2010

			For n=3 we have a(3)=(A_0+3A_1+5A_2)/3=(1+3*5+5*73)/3=127.

G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2011.

Cf. A005259, A173774, A189766.

List([1..30], n-> Sum([0..n], k -> Binomial(n-1,k)*Binomial(n+k,k) *Binomial(n+k, 2*k+1)* Binomial(2*k,k) )); # G. C. Greubel, Jan 24 2019

[(&+[Binomial(n-1,k)*Binomial(n+k,k)*Binomial(n+k, 2*k+1)* Binomial(2*k,k): k in [0..n-1]]): n in [1..30]]; // G. C. Greubel, Jan 24 2019

G := (-1/2)*(3*x-3+(x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3,2/3],[1],(-1/2)*(x^2-7*x+1)*(x+1)^(-3)*(x^2-34*x+1)^(1/2)+(1/2)*(x^3+30*x^2-24*x+1)*(x+1)^(-3))^2;
ogf := 2*x*G/(1-x)-Int((x+1)*G/(x-1)^2,x);
series(ogf,x=0,25); # Mark van Hoeij, May 07 2013

Apery[n_]:= Sum[Binomial[n+k,k]^2 Binomial[n,k]^2,{k,0,n}]; AA[n_]:= Sum[(2k+1)*Apery[k],{k,0,n-1}]/n; Table[AA[n],{n,1,25}]
Table[Sum[(Binomial[n-1,k]*Binomial[n+k,k]*Binomial[n+k,2*k+1]* Binomial[2*k,k]), {k,0,n-1}], {n,1,30}] (* G. C. Greubel, Jan 24 2019 *)

{a(n) = sum(k=0,n-1, binomial(n-1,k)*binomial(n+k,k)*binomial(n+k, 2*k+1)*binomial(2*k,k))}; \\ G. C. Greubel, Jan 24 2019

[sum(binomial(n-1,k)*binomial(n+k,k)*binomial(n+k, 2*k+1)* binomial(2*k,k) for k in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jan 24 2019

Recursion : (n+2)^3 *(n+3) *(2n+1) *a(n+3) = (n+2) *(2n+1) *(35*n^3+193*n^2+345*n+203) *a(n+2) -(n+1) *(2n+5) *(35*n^3+122*n^2+132*n+40) * a(n+1) +n *(n+1)^3 *(2n+5) *a(n).
a(n) = Sum(k=0..n-1, (binomial(n-1,k)* binomial(n+k,k)* binomial(n+k,2*k+1)*binomial(2*k,k)) ). - Zhi-Wei Sun, Jun 17 2010
a(n) = A189766(n) / n = trace( HilbertMatrix(n)^(-1) )/n. - Richard Penner, Jun 04 2011
a(n) = (1/n)*Sum_{k=0..n-1} (2*k+1)*binomial(n+k,2*k+1)^2*binomial(2*k, k)^2. - Richard Penner, Jun 04 2011
G.f.: 2*x*G/(1-x)-Int((x+1)*G/(x-1)^2,x) where G is the generating function of A005259. - Mark van Hoeij, May 07 2013
a(n) ~ 2^(1/4) * (1 + sqrt(2))^(4*n) / (16*(Pi*n)^(3/2)). - Vaclav Kotesovec, Jan 24 2019

A357567 a(n) = 5A005259(n) - 14A005258(n).

Original entry on oeis.org

-9, -17, 99, 5167, 147491, 3937483, 105834699, 2907476527, 81702447651, 2342097382483, 68273597307599, 2018243113678027, 60365426282638091, 1823553517258576723, 55557712038989195099, 1705170989220937925167, 52672595030914982754851, 1636296525812843554700323

Offset: 0

Peter Bala, Oct 19 2022

Conjectures:
1) a(p) == a(1) (mod p^5) for all primes p >= 5.
2) For r >= 2, a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for all primes p >= 5.
These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259.
From Peter Bala, Oct 25 2022: (Start)
Additional conjectures:
3) the sequence {u(n): n>= 1} defined by u(n) = (3^42)*A005259(n)^25 - (5^25)* A005258(n)^42 also satisfies the congruences in 1) and 2) above.
4) u(n) == 0 (mod n^5) for integer n of the form 3^i*5^j (see A003593). (End)

			a(11) - a(1) = 2018243113678027 + 17 = (2^2)*(3^2)*(11^5)*17*20476637 == 0 (mod 11^5).

Cf. A005258, A005259, A212334, A352655, A357506, A357507, A357508, A357509, A357568, A357569, A357956, A357957, A357958, A357959, A357960.

seq(add(5*binomial(n,k)^2*binomial(n+k,k)^2 - 14*binomial(n,k)^2*binomial(n+k,k), k = 0..n), n = 0..20);

a(n) = 5*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 - 14*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k).

For positive integers n and r, a(n*p^r) == a(n*p^(r-1)) ( mod p^(3*r) ) for all primes p >= 5.

A357956 a(n) = 5A005259(n) - 2A005258(n).

Original entry on oeis.org

3, 19, 327, 6931, 162503, 4072519, 107094207, 2919528211, 81819974343, 2343260407519, 68285241342827, 2018360803903111, 60366625228511423, 1823565812734012639, 55557838850469305327, 1705172303553678726931, 52672608711829111519943, 1636296668756812403477839, 51088496012515356589705107

Offset: 0

Peter Bala, Oct 24 2022

Conjectures:
1) a(p - 1) == 3 (mod p^5) for all primes p >= 5.
2) a(p^r - 1) == a(p^(r-1) - 1) ( mod p^(3*r+3) ) for r >= 2 and for all primes p >= 5.
These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259. Cf. A357567.

Peter Bala, Some conjectural supercongruences for the Apéry numbers.

Cf. A005258, A005259, A212334, A352655, A357567, A357568, A357569, A357957, A357958, A357959, A357960.

seq(add(5*binomial(n,k)^2*binomial(n+k,k)^2 - 2*binomial(n,k)^2* binomial(n+k,k), k = 0..n), n = 0..20);
# Alternatively:
a := n -> 5*hypergeom([-n, -n, 1 + n, 1 + n], [1, 1, 1], 1) - 2*hypergeom([1 + n, -n, -n], [1, 1], 1): seq(simplify(a(n)), n = 0..18); # Peter Luschny, Nov 01 2022

a(n) = 5*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 - 2*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k).
a(n*p^r - 1) == a(n*p^(r-1) - 1) ( mod p^(3*r) ) for positive integers n and r and for all primes p >= 5.
a(n) = 5*hypergeom([-n, -n, 1 + n, 1 + n], [1, 1, 1], 1) - 2*hypergeom([1 + n, -n, -n], [1, 1], 1). - Peter Luschny, Nov 01 2022

A178791 The arithmetic mean of (2k+1)(-1)^k*A_k (k=0,...,n-1), where A_0, A_1,... are Apéry numbers given by A005259.

Original entry on oeis.org

1, -7, 117, -2441, 57449, -1453635, 38609845, -1061792695, 29973352185, -863536596143, 25288254409373, -750531594051981, 22525211241191881, -682459907754004723, 20845409947239778533, -641211780685502724425

Offset: 1

Zhi-Wei Sun, Jun 14 2010

sign

On Jun 14 2010, Zhi-Wei Sun conjectured that the number a(n) = n^{-1}*Sum_{k=0..n-1} (2*k+1)*(-1)^k*A_k is always an integer and that a(p) = p(p/3) (mod p^3) for any prime p>3. He also formulated the following conjecture: If p=1 (mod 3) is a prime and p = x^2 + 3y^2 with x,y integers then Sum_{k=0..p-1} (-1)^k*A_k = 4*x^2 - 2*p (mod p^2); if p=2 (mod 3) is a prime then Sum_{k=0..p-1 }(-1)^k*A_k=0 (mod p^2).

			For n=3 we have a(3) = (A_0 - 3A_1 + 5A_2)/3 = (1 - 3*5 + 5*73)/3 = 117.

G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A178790, A005259, A173774.

G := (-1/2)*(3*x-3+(x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3, 2/3], [1], (-1/2)*(x^2-7*x+1)*(x+1)^(-3)*(x^2-34*x+1)^(1/2)+(1/2)*(x^3+30*x^2 -24*x+1)*(x+1)^(-3))^2;
ogf := 2*x*G/(x+1)+Int((x-1)*G/(x+1)^2,x);
series(ogf, x=0, 25);
series(-subs(x=-x,%), x=0, 25); # Mark van Hoeij, May 07 2013

Apery[n_]:= Sum[Binomial[n+k,k]^2Binomial[n,k]^2,{k,0,n}]; AA[n_]:= Sum[(2k+1)(-1)^k*Apery[k],{k,0,n-1}]/n; Table[AA[n],{n,25}]

A(n) = sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2); \\ A005259
a(n) = sum(k=0, n-1, (2*k+1)*(-1)^k*A(k))/n; \\ Michel Marcus, Jan 24 2019

G.f.: apart from the minus signs (just replace x by -x) the generating function is 2*x*G/(x+1) + Int((x-1)*G/(x+1)^2, x) where G is the generating function of A005259. - Mark van Hoeij, May 07 2013

a(n) ~ -(-1)^n * 2^(3/4) * (1 + sqrt(2))^(4*n) / (24 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jan 24 2019

A357959 a(n) = 5A005259(n-1) + 2A005258(n).

Original entry on oeis.org

11, 63, 659, 9727, 187511, 4304943, 109312739, 2941124607, 82033399631, 2345394917563, 68306797052879, 2018580243252847, 60368874298729631, 1823588997226603663, 55558079041172790659, 1705174802761490321407, 52672634815976274443711, 1636296942340074307669443

Offset: 1

Peter Bala, Oct 25 2022

Conjectures:
1) a(p) == a(1) (mod p^5) for all primes p >= 5 (checked up to p = 271).
2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for r >= 2, and for all primes p >= 5.
These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259. Cf. A357959.
There is also a product version of these conjectures:
3) the sequence {u(n): n >= 1} defined by u(n) = A005259(n-1)^5 * A005258(n)^6 also satisfies the congruences in 1) and 2) above. See A357960.

			Examples of supercongruences:
a(13) - a(1) = 60368874298729631 - 11 = (2^2)*3*5*(13^5)*131*20685869 == 0 (mod 13^5).
a(5^2) - a(5) = 51292638914356604042099497031437511 - 187511 = (2^4)*3*(5^10)* 37*72974432287*40526706713533 == 0 (mod 5^10).

Cf. A005258, A005259, A212334, A352655, A357567, A357956, A356957, A357958, A357960.

seq( add( 5*binomial(n-1,k)^2*binomial(n+k-1,k)^2 + 2*binomial(n,k)^2* binomial(n+k,k), k = 0..n ), n = 1..20);

a(n) = 5*Sum_{k = 0..n-1} binomial(n-1,k)^2*binomial(n+k-1,k)^2 + 2*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k).

a(n*p^r) == a(n*p^(r-1)) ( mod p^(3*r) ) for positive integers n and r and for all primes p >= 5.

A248142 Least positive integer m such that m + n divides A(m) + A(n), where A(.) is given by A005259.

Original entry on oeis.org

1, 1, 7, 2238, 5, 9, 3, 3, 1, 2484, 2, 2, 26, 12, 24, 5, 41, 32, 14, 3, 29, 29, 6, 15, 30, 7, 8, 37, 21, 5, 44, 18, 5, 16, 39, 34, 8, 1, 6, 5, 17, 8, 26, 6, 865, 39, 8, 13, 16, 781, 356, 35, 184, 65, 30, 139, 18, 25, 16, 123

Offset: 1

Zhi-Wei Sun, Oct 02 2014

nonn

Conjecture: a(n) exists for any n > 0.

			 a(3) = 7 since 3 + 7 = 10 divides A(3) + A(7) = 1445 + 584307365 = 584308810.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2853

Cf. A005259, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248137, A248139.

A[0]=1;A[1]=5
A[n_]:=((2n-1)(17*n(n-1)+5)*A[n-1]-(n-1)^3*A[n-2])/n^3
Do[m=1; Label[aa]; If[Mod[A[m]+A[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

A357507 a(n) = A005259(n)^5 * (A005259(n-1))^7.

Original entry on oeis.org

3125, 161958718203125, 69598400094777710760545478125, 514885225734532980507136994998009584838203125, 15708056924221066705174364772957342407662356116035885781253125, 1125221282019374727979322420623179115437017599670596496532725068048858642578125

Offset: 1

Peter Bala, Oct 01 2022

The Apéry numbers A(n) = A005259(n) satisfy the supercongruences A(p) == 5 (mod p^3) and A(p-1) == 1 (mod p^3) for all primes p >= 5 (see, for example, Straub, Introduction). It follows that a(p) == 3125 (mod p^3) for all primes p >= 5. We conjecture that, in fact, the stronger congruence a(p) == 3125 (mod p^5) holds for all primes p >= 3 (checked up to p = 251). Compare with the congruence A(p) + 7*A(p-1) == 12 (mod p^5) conjectured to hold for all primes p >= 5. See A212334.

Conjecture: a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for r >= 2 and all primes p >= 5. - Peter Bala, Oct 26 2022

A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).

Cf. A005259, A212334, A339946, A352655, A357506, A357508, A357509, A357567, A357568, A357569, A357956, A357957, A357958, A357959.

A005259 := n -> add(binomial(n,k)^2*binomial(n+k,k)^2, k = 0..n):
seq(A005259(n)^5 * A005259(n-1)^7, n = 1..10);

A357958 a(n) = 5A005259(n) + 14A005258(n-1).

Original entry on oeis.org

39, 407, 7491, 167063, 4112539, 107461667, 2923006251, 81853622423, 2343591359499, 68288538877907, 2018394003648391, 60366962358086243, 1823569260750104179, 55557874330437332267, 1705172670555862322491, 52672612525369663916183

Offset: 1

Peter Bala, Oct 25 2022

Conjectures:
1) a(p) == a(1) (mod p^5) for all primes p >= 5 (checked up to p = 271).
2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for r >= 2 and for all primes p >= 3.
These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259. Cf. A357959.
There is also a product version of these conjectures:
3) the sequence {u(n): n>= 1} defined by u(n) = A005259(n)^25 * A005258(n-1)^14 conjecturally satisfies the congruences in 1) and 2) above.

			Examples of supercongruences:
a(13) - a(1) = 1823569260750104179 - 39 = (2^2)*5*7*(13^5)*35081444357 == 0 (mod 13^5).
a(7^2) - a(7) = (2^3)*(7^9)* 10412078726049425470554760052126170543547100055154203726400782433 == 0 (mod 7^9).

Cf. A005258, A005259, A212334, A352655, A357567, A357956, A357957, A357959, A357960.

seq( add( 5*binomial(n,k)^2*binomial(n+k,k)^2 + 14*binomial(n-1,k)^2* binomial(n+k-1,k), k = 0..n ), n = 1..20);

a(n) = 5*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 + 14*Sum_{k = 0..n-1} binomial(n-1,k)^2*binomial(n+k-1,k).

a(p^r) == a(p^(r-1)) ( mod p^(3*r) ) for positive integer r and for all primes p >= 5.

A357957 a(n) = A005259(n)^5 - A005258(n)^2.

Original entry on oeis.org

0, 3116, 2073071232, 6299980938881516, 39141322964380888600000, 368495989505416178203682748116, 4552312485541626792249211584618373944, 68109360474242016374599574592870648425552876, 1174806832391451114413440151405736019461523615095744

Offset: 0

Peter Bala, Oct 24 2022

Conjectures:
1) a(p - 1) == 0 (mod p^5) for all primes p >= 5 (checked up to p = 271).
2) a(p^r - 1) == a(p^(r-1) - 1) ( mod p^(3*r+3) ) for r >= 2 and all primes p >= 5.
These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259.
3) Put u(n) = A005259(n)^5 / A005258(n)^2. Then u(p^r - 1) == u(p^(r-1) - 1) ( mod p^(3*r+3) ) for r >= 2 and all primes p >= 5.

			a(7) = 4552312485541626792249211584618373944 = (2^3)*(3^3)*(7^5)*29*107* 404116272977592231282158029 == 0 (mod 7^5).

Cf. A005258, A005259, A212334, A352655, A357567, A357568, A357569, A357956, A357958, A357959, A357960.

seq(add(binomial(n,k)^2*binomial(n+k,k)^2, k = 0..n)^5 - add(binomial(n,k)^2*binomial(n+k,k), k = 0..n)^2, n = 0..20);
# Alternatively:
a := n -> hypergeom([-n, -n, 1 + n, 1 + n], [1, 1, 1], 1)^5 - hypergeom([1 + n, -n, -n], [1, 1], 1)^2: seq(simplify(a(n)), n=0..8); # Peter Luschny, Nov 01 2022

a(n) = ( Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 )^5 - ( Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k) )^2.
a(n*p^r - 1) == a(n*p^(r-1) - 1) ( mod p^(3*r) ) for positive integers n and r and for all primes p >= 5.
a(n) = hypergeom([-n, -n, 1 + n, 1 + n], [1, 1, 1], 1)^5 - hypergeom([1 + n, -n, -n], [1, 1], 1)^2. - Peter Luschny, Nov 01 2022

cp's OEIS Frontend

A133370 Primes p such that p does not divide any term of the Apery sequence A005259 .

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A178790 The arithmetic mean of (2*k+1)*A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.

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A357567 a(n) = 5*A005259(n) - 14*A005258(n).

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A357956 a(n) = 5*A005259(n) - 2*A005258(n).

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A178791 The arithmetic mean of (2k+1)(-1)^k*A_k (k=0,...,n-1), where A_0, A_1,... are Apéry numbers given by A005259.

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A357959 a(n) = 5*A005259(n-1) + 2*A005258(n).

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A248142 Least positive integer m such that m + n divides A(m) + A(n), where A(.) is given by A005259.

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A178790 The arithmetic mean of (2k+1)A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.

A357567 a(n) = 5A005259(n) - 14A005258(n).

A357956 a(n) = 5A005259(n) - 2A005258(n).

A357959 a(n) = 5A005259(n-1) + 2A005258(n).

A357958 a(n) = 5A005259(n) + 14A005258(n-1).