A083321 -id:A083321 - cp's OEIS frontend

A242083 a(n) = 3^p - 2^p - 1, where p is prime(n).

4, 18, 210, 2058, 175098, 1586130, 129009090, 1161737178, 94134790218, 68629840493970, 617671248800298, 450283768452043890, 36472994178147530850, 328256958598444055418, 26588814218220014932458, 19383245658672820642055730

Offset: 1

Vincenzo Librandi, May 04 2014

For p>3, all terms are divisible by 42.

Vincenzo Librandi, Table of n, a(n) for n = 1..320

Cf. A000040, A083321, A204768.

Magma
```
[3^p-2^p-1: p in PrimesUpTo(60)];
```

Table[(3^Prime[n] - 2^Prime[n] - 1), {n, 1, 30}]
3^#-2^#-1&/@Prime[Range[20]] (* Harvey P. Dale, Aug 05 2016 *)

a(n) = my(p = prime(n)); 3^p-2^p-1; \\ Michel Marcus, May 05 2014

[3^p-2^p-1 for p in primes(60)] # Bruno Berselli, May 12 2014

a(n) = abs(A083321(A000040(n))). - Michel Marcus, May 05 2014

A332096 Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 0, 1, 0, 1, 1, 7, 18, 28, 25, 0, 1, 8, 0, 7, 1, 1, 15, 64, 158, 271, 317, 126, 45, 17, 59, 14, 2, 15, 3, 0, 2, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 31, 210, 748, 1825, 3351, 4606, 3760, 398, 131, 299, 0, 318, 0, 8

Offset: 1

M. F. Hasler, Jul 20 2020

nonn

It is known (Sprague 1948, cf. A001661) that for any n, only a finite number of positive integers are not the sum of distinct positive n-th powers. Therefore each row is finite, their lengths are given by A332098.
The number of nonzero terms in row n is A332066(n).
The column of the first zero (exact solution m^n = Sum_{x in A} x^n) in each row is given by A030052, unless A030052(n) = A332066(n) + 1 = A332098(n) + 1.

			The table reads:
  n\ m=1   2    3    4     5     6     7     8    9   10   11  12   13
----+--------------------------------------------------------------------------
  1 |  1   1                                                  (A332098(1) = 2.)
  2 |  1   3    4    2     0     1     0     1                (A332098(2) = 8.)
  3 |  1   7   18   28    25     0     1     8    0    7    1
  4 |  1  15   64  158   271   317   126    45   17   59   14   2   15  3  0 ...
  5 |  1  31  210  748  1825  3351  4606  3760  398  131  299   0  318  0  8 ...
The first column is all ones (A000012), since {1..m-1} = {} for m = 1.
The second column is 2^n - 1 = A000225 \ {0}, since {1..m-1} = {1} for m = 2.
The third column is 3^n - 2^n - 1 = |A083321(n)| for n > 1.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.

Cf. A001661, A332098, A332066, A030052, A332097, A078607.

Cf. A000012, A000225, A083321.

A332096(n,m,r=0)={if(r, (m<2||r<2^(n-1)) && return(r-1); my(E, t=1); while(m^n>=r, E=m--); E=abs(r-(m+!!E)^n); for(a=2,m, if(r=m && return(min(E,r-t)); while(m>=t && E, E=min(self()(n,m-1,r-m^n),E); E && E=min(self()(n,m-=1,r),E)); E, m < n/log(2)+1.5, m^n-sum(x=1,m-1,x^n), self()(n,m-1,m^n))}

For all n and m, T(n,m) <= A332097(n) = T(n,m*) with m* = A078607(n).

For m <= m* + 1, T(n,m) = m^n - Sum_{0 < x < m} x^n.

A152692 a(n) = n3^n - n2^n - n*1^n.

Original entry on oeis.org

0, 0, 8, 54, 256, 1050, 3984, 14406, 50432, 172530, 580240, 1926078, 6328128, 20619690, 66732176, 214742070, 687698944, 2193154530, 6968850192, 22073006382, 69714716480, 219623377050, 690291036688, 2165100175014

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

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

Cf. A152691.

[n*3^n-n*2^n-n*1^n: n in [0..30]]; // Vincenzo Librandi, Sep 05 2011

Table[n(3^n-2^n-1),{n,0,30}] (* Harvey P. Dale, Oct 20 2013 *)
CoefficientList[Series[-(2 x^2 (-4 + 21 x - 36 x^2 + 21 x^3))/(-1 + 6 x - 11 x^2 + 6 x^3)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 04 2018 *)

vector(30, n, n--; n*(3^n-2^n-1)) \\ G. C. Greubel, Sep 02 2018

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = (-1)^(n+1)*n*A083321(n).
G.f.: 2*x^2*(4-21*x+36*x^2-21*x^3)/((1-x)^2*(1-3*x)^2*(1-2*x)^2). (End)
E.g.f.: x*(3*exp(3*x) - 2*exp(2*x) - exp(x)). - G. C. Greubel, Sep 02 2018

Offset changed from 1 to 0 by Vincenzo Librandi, Sep 05 2011

A178690 Expansion of (exp(3x)-1)(exp(2x)-1)(exp(x)-1).

Original entry on oeis.org

0, 0, 0, 36, 432, 3660, 27000, 185556, 1223712, 7862940, 49653000, 309776676, 1915868592, 11772890220, 71992229400, 438593697396, 2664227115072, 16146540253500, 97676540188200, 590011376299716, 3559691497843152, 21455715437760780, 129219925869401400

Offset: 0

Geoffrey Critzer, Dec 25 2010

a(n) is the number of 3 X n matrices with the following properties:
    i) Each row has at least one nonzero entry.
   ii) Each column has exactly one nonzero entry.
  iii) The nonzero entries in row m, 1 <= m <= 3, are in {1,2,...,m}.
This sequence counts such 3 X n matrices but the results are easily generalized for any such m X n matrix.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-121,372,-508,240).

Cf. A083321, which is essentially the case for m=2.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(3*x)-1)*(Exp(2*x)-1)*(Exp(x)-1) )); [0,0,0] cat [Factorial(n+2)*b[n]: n in [1..m-3]]; // G. C. Greubel, Jan 26 2019

a=Exp[x]-1;b=Exp[2x]-1;c=Exp[3x]-1;Range[0,20]! CoefficientList[Series[a b c,{x,0,20}],x]

concat([0,0,0], Vec(-12*x^3*(20*x^2-18*x+3)/((x-1)*(2*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^30))) \\ Colin Barker, Dec 01 2014

m = 30; T = taylor((exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jan 26 2019

E.g.f.: (exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1).
G.f.: 12*x^3*(3-18*x+20*x^2)/((1-x)*(1-2*x)*(1-4*x)*(1-5*x)*(1-6*x)). - Colin Barker, Nov 30 2014
For n > 0, a(n) = 1 + 2^n - 4^n - 5^n + 6^n. - Vaclav Kotesovec, Dec 01 2014
a(n) = 18*a(n-1) - 121*a(n-2) + 372*a(n-3) - 508*a(n-4) + 240*a(n-5). - Vaclav Kotesovec, Dec 01 2014

A332099 Square array T(n,k) = k^n - Sum_{0 < i < k} e(i)*(k-i)^n where e(i) = 1 if the partial sum up to this term would remain <= k^n, or 0 else; n, k >= 1; read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 0, 4, 7, 1, 0, 2, 18, 15, 1, 0, 0, 28, 64, 31, 1, 0, 1, 25, 158, 210, 63, 1, 0, 0, 0, 271, 748, 664, 127, 1, 0, 1, 1, 317, 1825, 3302, 2058, 255, 1, 0, 0, 8, 126, 3351, 10735, 14068, 6304, 511, 1, 0, 2, 0, 45, 4606, 26141, 59425, 58718, 19170, 1023, 1, 0, 0, 19, 47, 3760, 50478, 183111, 318271, 241948, 58024, 2047, 1

Offset: 1

M. F. Hasler, Apr 19 2020

To compute T(n,k), start from k^n, then subtract (progressively strictly) smaller n-th powers whenever possible.

Since we subtract the smaller n-th powers in a greedy way, T(n,k) may be nonzero even if k^n is a sum of smaller n-th powers: cf. rows of A332065 for these k.

			The square array starts
  n\k: 1   2    3     4      5      6     7     8     9    10    11    12    13
  --+----------------------------------------------------------------------------
  1 |  1   1    0     0      0      0     0     0     0     0     0     0     0
  2 |  1   3    4     2      0      1     0     1     0     2     0     2     0
  3 |  1   7   18    28     25      0     1     8     0    19    15    18     0
  4 |  1  15   64   158    271    317    126   45    47    59   191    61    285
  5 |  1  31  210   748   1825   3351   4606  3760  398   131   702     0   1229
  6 |  1  63  664  3302  10735  26141  50478 77324 84477 21595 55300 29603  2093
  (...)
Columns 1, 2, 3: A000012, A000225, |A083321|, cf. FORMULA.
We have T(2,10) = 10^2 - 9^2 - 4^2 - 1 = 2, because we first have to subtract 9^2 = 81, even though 10 is in row 2 of A332065 since 10^2 - 8^2 - 6^2 = 0.

Cf. A030052 (least k such that k^n = sum of distinct n-th powers).
Cf. A332065 (all k such that k^n is a sum of distinct n-th powers).
Cf. A332101 (least k such that k^n <= sum of all smaller n-th powers).

A332099(n,k,t=k^n)={while(k&&t,t-=(k=min(sqrtnint(t,n),k-1))^n);t}

T(n,k) > 0  for  k < A030052(n), and  T(n,k) = 0  for  k = A030052(n).
T(n,k) = k^n - Sum_{0 < m < k} m^k  for  k < A332101(n).
T(n,1) = 1 = A000012(n);  T(n,2) = 2^n - 1 = A000225(n);
T(n,3) = 3^n - 2^n - 1 = |A083321(n)|.

A362316 Expansion of e.g.f (exp(x)-1)(exp(2x)-1).

Original entry on oeis.org

0, 0, 4, 18, 64, 210, 664, 2058, 6304, 19170, 58024, 175098, 527344, 1586130, 4766584, 14316138, 42981184, 129009090, 387158344, 1161737178, 3485735824, 10458256050, 31376865304, 94134790218, 282412759264, 847255055010, 2541798719464, 7625463267258, 22876524019504, 68629840493970

Offset: 0

Enrique Navarrete, Apr 16 2023

Number of ternary strings with at least one 0 or one 1 and at least one 2.

			The 4 strings for n=4 are 12, 21, 02, 20.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A001550, A053152, A083321 (alternates signs).

LinearRecurrence[{6,-11,6},{0,0,4,18},30] (* Paolo Xausa, Aug 07 2023 *)

a(n) = 3^n - 2^n - 1, n>0; a(0)=0.

a(n) = 2*A053152(n).

cp's OEIS Frontend

A242083 a(n) = 3^p - 2^p - 1, where p is prime(n).

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A332096 Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.

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A152692 a(n) = n*3^n - n*2^n - n*1^n.

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A178690 Expansion of (exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1).

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Author

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Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A332099 Square array T(n,k) = k^n - Sum_{0 < i < k} e(i)*(k-i)^n where e(i) = 1 if the partial sum up to this term would remain <= k^n, or 0 else; n, k >= 1; read by falling antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A362316 Expansion of e.g.f (exp(x)-1)*(exp(2*x)-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A152692 a(n) = n3^n - n2^n - n*1^n.

A178690 Expansion of (exp(3x)-1)(exp(2x)-1)(exp(x)-1).

A362316 Expansion of e.g.f (exp(x)-1)(exp(2x)-1).