A003777 -id:A003777 - cp's OEIS frontend

A000584 Fifth powers: a(n) = n^5.

0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, 11881376, 14348907, 17210368, 20511149

Offset: 0

N. J. A. Sloane

Totally multiplicative sequence with a(p) = p^5 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A059338. The inverse binomial transform yields the (finite) 0, 1, 30, 150, 240, 120, the 5th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Equals sum of odd numbers from n^2*(n-1)+1 (A100104) to n^2*(n+1)-1 (A003777). - Bruno Berselli, Mar 14 2014
a(n) mod 10 = n mod 10. - Reinhard Zumkeller, May 10 2014
Numbers of the form a(n) + a(n+1) + ... + a(n+k) are nonprime for all n, k>=0; this can be proved by the method indicated in the comment in A256581. - Vladimir Shevelev and Peter J. C. Moses, Apr 04 2015

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..500
Brady Haran and Simon Pampena, Fifth Root Trick, Numberphile video (2014)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Partial sums give A000539.
Cf. A000012, A001477, A000290, A000578, A000583, A062392, A022521 (first differences), A008292, A173018, A123125.
Cf. A013663, A162624, A267316.

a000584 = (^ 5)  -- Reinhard Zumkeller, Nov 11 2012

Range[0,50]^5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

makelist(n^5, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

a(n)=n^5 \\ Charles R Greathouse IV, Jul 28 2015

[n**5 for n in range(30)]  # Zerinvary Lajos, Jun 03 2009

G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (x-1)^6. [Simon Plouffe in his 1992 dissertation]
Multiplicative with a(p^e) = p^(5e). - David W. Wilson, Aug 01 2001
E.g.f.: exp(x)*(x+15*x^2+25*x^3+10*x^4+x^5). - Geoffrey Critzer, Jun 12 2013
a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 120. - Ant King, Sep 23 2013
a(n) = n + Sum_{j=0..n-1}{k=1..4}binomial(5,k)*j^(5-k). - Patrick J. McNab, Mar 28 2016
From Kolosov Petro, Oct 22 2018: (Start)
a(n) = Sum_{k=1..n} A300656(n,k).
a(n) = Sum_{k=0..n-1} A300656(n,k). (End)
a(n) = Sum_{k=1..5} Eulerian(5, k)*binomial(n+5-k, 5), with Eulerian(5, k) = A008292(5, k), the numbers 1, 26, 66, 26, 1, for n >= 0. Worpitzki's identity for powers of 5. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - Wolfdieter Lang, Jul 17 2019
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(5) (A013663).
Sum_{n>=1} (-1)^(n+1)/a(n) = 15*zeta(5)/16 (A267316). (End)

More terms from Henry Bottomley, Jun 21 2001

A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 11, 1, 1, 5, 19, 35, 26, 1, 1, 6, 29, 79, 115, 53, 1, 1, 7, 41, 149, 334, 347, 116, 1, 1, 8, 55, 251, 773, 1339, 1075, 236, 1, 1, 9, 71, 391, 1546, 3869, 5434, 3235, 488, 1, 1, 10, 89, 575, 2791, 9281, 19493, 21754, 9787, 983, 1, 1, 11

Offset: 1

Alois P. Heinz, Aug 07 2008

The coefficients of the polynomial of row n are given by the n-th row of triangle A134541; for example row 4 has polynomial -1+k^2+k^3.

			T(3,3) = 11, because 11 words of length <=3 over 3-letter alphabet {a,b,c} are primitive and earlier than others derived by cyclic shifts of the alphabet: a, ab, ac, aab, aac, aba, abb, abc, aca, acb, acc.
Table begins:
  1,   1,    1,     1,     1,      1,      1,       1, ...
  1,   2,    3,     4,     5,      6,      7,       8, ...
  1,   5,   11,    19,    29,     41,     55,      71, ...
  1,  11,   35,    79,   149,    251,    391,     575, ...
  1,  26,  115,   334,   773,   1546,   2791,    4670, ...
  1,  53,  347,  1339,  3869,   9281,  19543,   37367, ...
  1, 116, 1075,  5434, 19493,  55936, 137191,  299510, ...
  1, 236, 3235, 21754, 97493, 335656, 960391, 2396150, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A000012, A085945, A320087, A320088, A320089, A320090, A320091, A320092, A320093, A320094.
Rows n=1-4 give: A000012, A000027, A028387, A003777.
Main diagonal gives A320095.
Cf. A143325, A143326, A134541, A008683.

with(numtheory):
f1:= proc (n) option remember; unapply(k^(n-1)
        -add(f1(d)(k), d=divisors(n) minus {n}), k)
     end:
g1:= proc(n) option remember; unapply(add(f1(j)(x), j=1..n), x) end:
T:= (n, k)-> g1(n)(k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

t[n_, k_] := Sum[k^(d-1)*MoebiusMu[j/d], {j, 1, n}, {d, Divisors[j]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 13 2013 *)

T(n,k) = Sum_{j=1..n} Sum_{d|j} k^(d-1) * mu(j/d).
T(n,k) = Sum_{j=1..n} A143325(j,k).
T(n,k) = A143326(n,k) / k.

A106230 Least k > 0 for n > 0 such that (n^2 + 1)(k^2) + (n^2 + 1)k + 1 = j^2 where j sequence = A106229.

Original entry on oeis.org

3, 8, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, 624, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443, 1520, 1599, 1680, 1763, 1848, 1935, 2024, 2115, 2208, 2303, 2400, 2499

Offset: 1

Pierre CAMI, Apr 26 2005

For (n^2 + 1)*(k^2) + (n^2 +1)*k + 1 = j^2 there is a sequence k(i,n) with a recurrence.
For n=1, k(1,1) = 0, k(2,1) = 3, k(i,1) = 6*k(i-1,1) + 2 - k(i-2,1).
For n=2, k(1,2) = 1, k(2,2) = 19, k(i,2) = 18*k(i-1,2) + 8 - k(i-2,2).
For n>2, k(1,n) = 0, k(2,n) = n^2 - 2*n, k(3,n) = n^2 + 2*n, k(4,n) = (4*n^2 + 2)*k(2,n) + 2*n^2 then k(i,n) = (4*n^2 + 2)*k(i-2,n) + 2*n^2 - k(i-4,n). As i increases the ratio j(i,n)/k(i,n) tends to sqrt(n^2 + 1).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003777, A005563, A005899, A106229.

CoefficientList[Series[(-3 + z + 12*z^2 - 20*z^3 + 8*z^4)/(-1 + z)^3, {z, 0, 60}], z] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
LinearRecurrence[{3,-3,1},{3,8,3,8,15},60] (* Harvey P. Dale, Jul 05 2022 *)

x='x+O('x^50); Vec((3-x-12*x^2+20*x^3-8*x^4)/(1-x)^3) \\ G. C. Greubel, May 11 2017

a(n)=n*if(n>2, n-2, n+2) \\ Charles R Greathouse IV, Oct 19 2022

For n > 2, a(n) = n^2 - 2*n.
a(n) = A005563(n-2), n>2. - R. J. Mathar, Aug 28 2008
G.f.: (3 - x - 12*x^2 + 20*x^3 - 8*x^4)/(1 - x)^3. - G. C. Greubel, May 11 2017

A028295 a(n) = n^6 - (883/60)n^5 + (157/3)n^4 + (2155/12)n^3 - (4570/3)n^2 + (42767/15)*n - 967.

Original entry on oeis.org

133, 1903, 10561, 38015, 106461, 252737, 533397, 1030505, 1858149, 3169675, 5165641, 8102491, 12301949, 18161133, 26163389, 36889845, 51031685, 69403143, 92955217, 122790103, 160176349, 206564729, 263604837, 333162401, 417337317, 518482403, 639222873

Offset: 6

R. K. Guy

Old name was: "Number of stacks of n pikelets, distance 6 flips from a well-ordered stack".

G. C. Greubel, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A003777, A003878, A028294.

[(60*n^6 -883*n^5 +3140*n^4 +10775*n^3 -91400*n^2 +171068*n -58020)/60: n in [6..46]]; // G. C. Greubel, Jan 03 2024

(* Codes from Robert G. Wilson v, Jul 29 2018: Start *)
a[n_]:= n^6 - (883/60)*n^5 + (157/3)*n^4 + (2155/12)*n^3 - (4570/3)*n^2 + (42767/15)*n - 967; Table[a[n], {n,6,36}]
CoefficientList[ Series[x^6 (3x^6 -2x^5 -187x^4 +604x^3 -33x^2 -972x - 133)/(x-1)^7, {x,0,36}], x]
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {133,1903,10561,38015,106461, 252737,533397}, 36]
(* End *)

[(60*n^6 -883*n^5 +3140*n^4 +10775*n^3 -91400*n^2 +171068*n -58020)/60 for n in range(6,47)] # G. C. Greubel, Jan 03 2024

G.f.: x^6*(133 + 972*x + 33*x^2 - 604*x^3 + 187*x^4 + 2*x^5 - 3*x^6) / (1-x)^7. - R. J. Mathar, Jun 21 2011

E.g.f.: (1/5!)*(116040 - 69480*x - 30540*x^2 - 2340*x^3 + 95*x^4 + 3*x^5 - (116040 - 185520*x + 96960*x^2 - 25880*x^3 + 3580*x^4 - 34*x^5 - 120*x^6)*exp(x)). - G. C. Greubel, Jan 03 2024

Entry revised by N. J. A. Sloane, Jun 15 2014

a(17)-a(32) from Robert G. Wilson v, Jul 29 2018

A106229 Least j > 1 for n > 0 such that j^2 = (n^2 + 1)(k^2) + (n^2 + 1)k + 1 where k sequence = A106230.

Original entry on oeis.org

5, 19, 11, 35, 79, 149, 251, 391, 575, 809, 1099, 1451, 1871, 2365, 2939, 3599, 4351, 5201, 6155, 7219, 8399, 9701, 11131, 12695, 14399, 16249, 18251, 20411, 22735, 25229, 27899, 30751, 33791, 37025, 40459, 44099, 47951, 52021, 56315, 60839, 65599, 70601, 75851

Offset: 1

Pierre CAMI, Apr 26 2005

For j^2 = (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1, there is a sequence j(i,n) with a recurrence.
For n=1, j(1,1) = 1, j(2,1) = 5, j(i,1) = 6*j(i-1,1) - j(i-2,1).
For n=2, j(1,2) = 1, j(2,2) = 19, j(i,2) = 18*j(i-1,2) - j(i-2,2).
For n>2, j(1,n) = 1, j(2,n) = n^3 - 2*n^2 + n - 1, j(3,n) = n^3 + 2*n^2 + n + 1, j(4,n) = (4*n^2 + 2)*j(2,n) + 1 then j(i,n) = (4*n^2+2)*j(i-2,n) - j(i-4,n).

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A003777, A005563, A005899, A106230.

LinearRecurrence[{4,-6,4,-1},{5,19,11,35,79,149},43] (* Georg Fischer, Oct 25 2020 *)

a(n) = if(n<3, 14*n-9, n^3-2*n^2+n-1); \\ Jinyuan Wang, Apr 07 2020

For n > 2, a(n) = n^3 - 2*n^2 + n - 1.

More terms from Jinyuan Wang, Apr 07 2020

cp's OEIS Frontend

A000584 Fifth powers: a(n) = n^5.

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A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

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A106230 Least k > 0 for n > 0 such that (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1 = j^2 where j sequence = A106229.

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A028295 a(n) = n^6 - (883/60)*n^5 + (157/3)*n^4 + (2155/12)*n^3 - (4570/3)*n^2 + (42767/15)*n - 967.

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A106229 Least j > 1 for n > 0 such that j^2 = (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1 where k sequence = A106230.

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A106230 Least k > 0 for n > 0 such that (n^2 + 1)(k^2) + (n^2 + 1)k + 1 = j^2 where j sequence = A106229.

A028295 a(n) = n^6 - (883/60)n^5 + (157/3)n^4 + (2155/12)n^3 - (4570/3)n^2 + (42767/15)*n - 967.

A106229 Least j > 1 for n > 0 such that j^2 = (n^2 + 1)(k^2) + (n^2 + 1)k + 1 where k sequence = A106230.