A068236 -id:A068236 - cp's OEIS frontend

A101096 Third differences of fifth powers (A000584).

1, 29, 150, 390, 750, 1230, 1830, 2550, 3390, 4350, 5430, 6630, 7950, 9390, 10950, 12630, 14430, 16350, 18390, 20550, 22830, 25230, 27750, 30390, 33150, 36030, 39030, 42150, 45390, 48750, 52230, 55830, 59550, 63390, 67350, 71430, 75630, 79950, 84390, 88950

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of the power of 5.

For n>=3 a(n) is equal to the number of functions f:{1,2,3,4,5}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
David J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al.), National Center for Mathematics Education, University of Gothenburg, Sweden, 2003.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883.
Eric Weisstein, Link to section of MathWorld: Eulerian Number.
Eric Weisstein, Link to section of MathWorld: Nexus number.
Eric Weisstein, Link to section of MathWorld: Finite Differences.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A069477.
Third differences of A000584, second differences of A022521, and first differences of A068236.
Cf. A101095 for other sequences related to MagicNKZ.
Cf. A001844.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(x^4+26*x^3+66*x^2+26*x+1)/(1-x)^3)); // G. C. Greubel, Dec 01 2018

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 3, 3}, {k, 0, 34}]
CoefficientList[Series[(-z^4-26z^3-66z^2-26z-1)/(z-1)^3, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
Join[{1,29},Differences[Range[0,40]^5,3]] (* or *) LinearRecurrence[{3,-3,1},{1,29,150,390,750},40] (* Harvey P. Dale, Feb 02 2017 *)

a(n)=if(n>2,60*n^2-180*n+150,28*n-27) \\ Charles R Greathouse IV, Oct 11 2015

[sum([(-1)^j*binomial(3, j)*(k-j+1)^5 for j in range(min(k+2,4))]) for k in range(40)] # Danny Rorabaugh, Apr 27 2015

a(k+1) = MagicNKZ(5,k,3) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n. (Cf. A101095.)
a(n+1) = 30*(1 - 2*n + 2*n^2) for n>2.
a(n+3) = A069477(n). - Vladimir Joseph Stephan Orlovsky, Jun 19 2011
G.f.: x*(x^4+26*x^3+66*x^2+26*x+1)/(1-x)^3. - Colin Barker, Oct 17 2012
Sum_{n>=1} 1/a(n) = (Pi/60)*tanh(Pi/2) + 871/870. - Amiram Eldar, Jan 27 2022

MagicNKZ material edited and SeriesAtLevelR material removed by Danny Rorabaugh, Apr 27 2015

A300656 Triangle read by rows: T(n,k) = 30k^2(n-k)^2 + 1; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 31, 1, 1, 121, 121, 1, 1, 271, 481, 271, 1, 1, 481, 1081, 1081, 481, 1, 1, 751, 1921, 2431, 1921, 751, 1, 1, 1081, 3001, 4321, 4321, 3001, 1081, 1, 1, 1471, 4321, 6751, 7681, 6751, 4321, 1471, 1, 1, 1921, 5881, 9721, 12001, 12001, 9721, 5881, 1921, 1

Offset: 0

Kolosov Petro, Mar 10 2018

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(2, n, k).
Fifth power can be expressed as row sum of triangle T(n, k).
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
--------------------------------------------------------------------------
k=    0     1     2      3      4      5      6      7     8     9    10
--------------------------------------------------------------------------
n=0:  1;
n=1:  1,    1;
n=2:  1,   31,    1;
n=3:  1,  121,  121,     1;
n=4:  1,  271,  481,   271,     1;
n=5:  1,  481, 1081,  1081,   481,     1;
n=6:  1,  751, 1921,  2431,  1921,   751,     1;
n=7:  1, 1081, 3001,  4321,  4321,  3001,  1081,     1;
n=8:  1, 1471, 4321,  6751,  7681,  6751,  4321,  1471,    1;
n=9:  1, 1921, 5881,  9721, 12001, 12001,  9721,  5881, 1921,    1;
n=10: 1, 2431, 7681, 13231, 17281, 18751, 17281, 13231, 7681, 2431,   1;

Kolosov Petro, Rows n = 0..2078 of triangle, flattened.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Various cases of L(m, n, k): A287326(m=1), This sequence (m=2), A300785(m=3). See comments for L(m, n, k).
Row sums give the nonzero terms of A002561.
Cf. A000584, A287326, A007318, A077028, A294317, A068236, A302971, A304042, A002561, A258807, A158558, A094053, A024003, A316349.

T:=Flat(List([0..9],n->List([0..n],k->30*k^2*(n-k)^2+1))); # Muniru A Asiru, Oct 24 2018

[[30*k^2*(n-k)^2+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Dec 14 2018

a:=(n,k)->30*k^2*(n-k)^2+1: seq(seq(a(n,k),k=0..n),n=0..9); # Muniru A Asiru, Oct 24 2018

T[n_, k_] := 30 k^2 (n - k)^2 + 1; Column[
Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Apr 12 2020 *)
f[n_]:=Table[SeriesCoefficient[(1 + 26 y + 336 y^2 + 326 y^3 + 31 y^4 + x^2 (1 + 116 y + 486 y^2 + 116 y^3 + y^4) + x (-2 - 82 y - 882 y^2 - 502 y^3 + 28 y^4))/((-1 + x)^3 (-1 + y)^5), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11, 0]] (* Stefano Spezia, Oct 30 2018 *)

t(n, k) = 30*k^2*(n-k)^2+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */ trianglerows(9)

[[30*k^2*(n-k)^2+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018

From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 30 * k^2 * (n-k)^2 + 1.
T(n, k) = 30 * A094053(n,k)^2 + 1.
T(n, k) = A158558((n-k) * k).
T(n+2, k) = 3*T(n+1, k) - 3*T(n, k) + T(n-1, k), for n >= k.
Sum_{k=1..n} T(n, k) = A000584(n).
Sum_{k=0..n-1} T(n, k) = A000584(n).
Sum_{k=0..n} T(n, k) = A002561(n).
Sum_{k=1..n-1} T(n, k) = A258807(n).
Sum_{k=1..n-1} T(n, k) = -A024003(n), n > 1.
Sum_{k=1..r} T(n, k) = A316349(2,r,0)*n^0 - A316349(2,r,1)*n^1 + A316349(2,r,2)*n^2. (End)
G.f.: (1 + 26*y + 336*y^2 + 326*y^3 + 31*y^4 + x^2*(1 + 116*y + 486*y^2 + 116*y^3 + y^4) + x*(-2 - 82*y - 882*y^2 - 502*y^3 + 28*y^4))/((-1 + x)^3*(-1 + y)^5). - Stefano Spezia, Oct 30 2018

A300785 Triangle read by rows: T(n,k) = 140k^3(n-k)^3 - 14k(n-k) + 1; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 127, 1, 1, 1093, 1093, 1, 1, 3739, 8905, 3739, 1, 1, 8905, 30157, 30157, 8905, 1, 1, 17431, 71569, 101935, 71569, 17431, 1, 1, 30157, 139861, 241753, 241753, 139861, 30157, 1, 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923, 1, 1, 71569, 383965, 816229, 1119721, 1119721, 816229, 383965, 71569, 1

Offset: 0

Kolosov Petro, Mar 12 2018

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(3, n, k).
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
--------------------------------------------------------------------
k=   0      1       2       3       4       5       6      7     8
--------------------------------------------------------------------
n=0: 1;
n=1: 1,     1;
n=2: 1,   127,      1;
n=3: 1,  1093,   1093,      1;
n=4: 1,  3739,   8905,   3739,      1;
n=5: 1,  8905,  30157,  30157,   8905,      1;
n=6: 1, 17431,  71569, 101935,  71569,  17431,      1;
n=7: 1, 30157, 139861, 241753, 241753, 139861,  30157,     1;
n=8: 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923,    1;

Muniru A Asiru, Rows n=0..100 of triangle, flattened.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Various cases of L(m, n, k): A287326 (m=1), A300656 (m=2), This sequence (m=3). See comments for L(m, n, k).
Row sums give A258806.
Cf. A000584, A287326, A007318, A077028, A294317, A068236, A300656, A302971, A304042, A001015, A094053, A258808, A024005, A316387.

T:=Flat(List([0..9], n->List([0..n], k->140*k^3*(n-k)^3 - 14*k*(n-k)+1))); # G. C. Greubel, Dec 14 2018

/* As triangle */ [[140*k^3*(n-k)^3-14*k*(n-k)+1: k in [0..n]]: n in [0..10]]; // Bruno Berselli, Mar 21 2018

T:=(n,k)->140*k^3*(n-k)^3-14*k*(n-k)+1: seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Dec 14 2018

T[n_, k_] := 140*k^3*(n - k)^3 - 14*k*(n - k) + 1; Column[
Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* From Kolosov Petro, Apr 12 2020 *)

t(n, k) = 140*k^3*(n-k)^3-14*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */ trianglerows(9)

[[140*k^3*(n-k)^3 - 14*k*(n-k)+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018

From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1.
T(n, k) = 140*A094053(n, k)^3 + 0*A094053(n, k)^2 - 14*A094053(n, k)^1 + 1.
T(n+3, k) = 4*T(n+2, k) - 6*T(n+1, k) + 4*T(n, k) - T(n-1, k), for n >= k.
Sum_{k=1..n} T(n, k) = A001015(n).
Sum_{k=0..n} T(n, k) = A258806(n).
Sum_{k=0..n-1} T(n, k) = A001015(n).
Sum_{k=1..n-1} T(n, k) = A258808(n).
Sum_{k=1..n-1} T(n, k) = -A024005(n).
Sum_{k=1..r} T(n, k) = -A316387(3, r, 0)*n^0 + A316387(3, r, 1)*n^1 - A316387(3, r, 2)*n^2 + A316387(3, r, 3)*n^3. (End)
G.f.: (1 + 127*x^6*y^3 - 3*x*(1 + y) + 585*x^5*y^2*(1 + y) + 129*x^4*y*(1 + 17*y + y^2) + 3*x^2*(1 + 45*y + y^2) - x^3*(1 - 579*y - 579*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Sep 14 2024

A101098 a(1)=1; thereafter, a(n+1) = 20n^3 + 10n.

Original entry on oeis.org

1, 30, 180, 570, 1320, 2550, 4380, 6930, 10320, 14670, 20100, 26730, 34680, 44070, 55020, 67650, 82080, 98430, 116820, 137370, 160200, 185430, 213180, 243570, 276720, 312750, 351780, 393930, 439320, 488070, 540300, 596130, 655680, 719070, 786420, 857850

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Shells (nexus numbers) of shells of the fifth powers (A000584).

G. C. Greubel, Table of n, a(n) for n = 1..5000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Concatenation([1],List([1..35],n->20*n^3+10*n)); # Muniru A Asiru, Dec 02 2018

[n le 1 select 1 else 10*(n - 1)*(2*(n - 1)^2 + 1): n in [1..50]]; // G. C. Greubel, Dec 01 2018

a:=`if`(n=1,1,20*n^3+10*n): 1,seq(a(n),n=1..35); # Muniru A Asiru, Dec 02 2018

Table[If[n == 1, 1, 10*(n - 1)*(2*(n - 1)^2 + 1)], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)(* modified by G. C. Greubel, Dec 01 2018 *)

my(x='x+O('x^50)); Vec(x + 30*x^2*(1+x)^2/(1-x)^4) \\ G. C. Greubel, Dec 01 2018

s=(x + 30*x^2*(1+x)^2/(1-x)^4).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 01 2018

From R. J. Mathar, Sep 02 2008: (Start)
a(n) = A068236(n-2), n > 1.
G.f.: x + 30*x^2*(1+x)^2/(1-x)^4. (End)

Edited by Ralf Stephan, Dec 16 2004

A254871 Seventh partial sums of fifth powers (A000584).

Original entry on oeis.org

1, 39, 495, 3705, 19995, 85917, 311493, 989235, 2823990, 7383610, 17931498, 40889862, 88304970, 181852230, 359140470, 683363994, 1257722271, 2246496825, 3905261425, 6623425575, 10983195405, 17840105595, 28431558675, 44521334325, 68589834300, 104081944356

Offset: 1

Luciano Ancora, Feb 17 2015

			Second differences:      30, 180,  570,  1320,  2550, ...   (A068236)
First differences:    1, 31, 211,  781,  2101,  4651, ...   (A022521)
------------------------------------------------------------------------
The fifth powers:     1, 32, 243, 1024,  3125,  7776, ...   (A000584)
------------------------------------------------------------------------
First partial sums:   1, 33, 276, 1300,  4425, 12201, ...   (A000539)
Second partial sums:  1, 34, 310, 1610,  6035, 18236, ...   (A101092)
Third partial sums:   1, 35, 345, 1955,  7990, 26226, ...   (A101099)
Fourth partial sums:  1, 36, 381, 2336, 10326, 36552, ...   (A254644)
Fifth partial sums:   1, 37, 418, 2754, 13080, 49632, ...   (A254682)
Sixth partial sums:   1, 38, 456, 3210, 16290, 65922, ...   (A254471)
Seventh partial sums: 1, 39, 495, 3705, 19995, 85917, ... (this sequence)

Luciano Ancora, Table of n, a(n) for n = 1..1000
Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials
Luciano Ancora, Pascal’s triangle and recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000539, A000584, A022521, A101092, A101099, A254471, A254644, A254682, A254869, A254870, A254872.

[n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(-21+49*n +56*n^2+14*n^3+n^4)/3991680: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) ((-21 + 49 n + 56 n^2 + 14 n^3 + n^4)/3991680), {n, 23}] (* or *)
CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^13, {x, 0, 22}], x]

vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(-21 + 49*n + 56*n^2 + 14*n^3 + n^4)/3991680) \\ Derek Orr, Feb 19 2015

G.f.: (- x - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(-21 + 49*n + 56*n^2 + 14*n^3 + n^4)/3991680.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^5.

A069473 First differences of (n+1)^6-n^6 (A022522).

Original entry on oeis.org

62, 602, 2702, 8162, 19502, 39962, 73502, 124802, 199262, 303002, 442862, 626402, 861902, 1158362, 1525502, 1973762, 2514302, 3159002, 3920462, 4812002, 5847662, 7042202, 8411102, 9970562, 11737502, 13729562, 15965102, 18463202

Offset: 0

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 26 2002

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001014, A022522, A069474, A069475, A069476, A068236.

[30*n^4+120*n^3+210*n^2+180*n+62: n in [0..30]]; // Bruno Berselli, Feb 25 2015

Differences[Table[(n + 1)^6 - n^6, {n, 0, 30}]] (* Harvey P. Dale, Dec 27 2011 *)

a(n) = 30*n^4 + 120*n^3 + 210*n^2 + 180*n + 62.

G.f.: 2*(31 + 146*x + 156*x^2 + 26*x^3 + x^4)/(1 - x)^5. [Bruno Berselli, Feb 25 2015]

Offset changed from 1 to 0 and added a(0)=62 by Bruno Berselli, Feb 25 2015

A069477 a(n) = 60n^2 + 180n + 150.

Original entry on oeis.org

390, 750, 1230, 1830, 2550, 3390, 4350, 5430, 6630, 7950, 9390, 10950, 12630, 14430, 16350, 18390, 20550, 22830, 25230, 27750, 30390, 33150, 36030, 39030, 42150, 45390, 48750, 52230, 55830, 59550, 63390, 67350, 71430, 75630, 79950, 84390, 88950, 93630, 98430, 103350

Offset: 1

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

First differences of A068236, successive differences of (n+1)^5 - n^5 (A022521).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001844, A022521, A068236, A069477, A069894, A101096.

[30*(2*n^2 + 6*n + 5): n in [1..40]]; // Vincenzo Librandi, Nov 23 2011

Table[30 (2 n^2 + 6 n + 5), {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
LinearRecurrence[{3,-3,1},{390,750,1230},40] (* Harvey P. Dale, Apr 06 2012 *)

a(n)=60*n^2+180*n+150 \\ Charles R Greathouse IV, Nov 23 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=390, a(2)=750, a(3)=1230. - Harvey P. Dale, Apr 06 2012
Sum_{n>=1} 1/a(n) = (Pi/60)*tanh(Pi/2) - 1/25. - Amiram Eldar, Jan 27 2022
From Elmo R. Oliveira, Feb 08 2025: (Start)
G.f.: 30*x*(5*x^2 - 14*x + 13)/(1-x)^3.
E.g.f.: 30*(exp(x)*(2*x^2 + 8*x + 5) - 5).
a(n) = 30*A001844(n+1) = 15*A069894(n+1). (End)

A069478 First differences of A069477, successive differences of (n+1)^5 - n^5.

Original entry on oeis.org

360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A022521, A068236, A069477.

[120*(n+2): n in [1..50]]; // G. C. Greubel, Nov 11 2018

Differences[#[[2]]-#[[1]]&/@Partition[Range[50]^5,2,1],3] (* Harvey P. Dale, Jan 28 2013 *)
120(Range[50] +2) (* G. C. Greubel, Nov 11 2018 *)

vector(50, n, 120*(n+2)) \\ G. C. Greubel, Nov 11 2018

a(n) = 120*(n+2).
From Chai Wah Wu, Nov 11 2018: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 2.
G.f.: x*(-240*x + 360)/(x - 1)^2. (End)
E.g.f.: 120*(-2 + (2+x)*exp(x)). - G. C. Greubel, Nov 11 2018

Corrected by T. D. Noe, Mar 14 2008

A353890 a(n) is the period of the binary sequence {b(m)} defined by b(m) = 1 if (m+1)^n - m^n and (m+2)^n - 2*(m+1)^n + m^n are coprime, 0 otherwise.

Original entry on oeis.org

1, 1, 5, 11, 91, 1247, 3485, 263017, 852841, 1241058127, 74966255, 243641132605417, 181556731572385303, 718802057694183783881, 6582662048285, 943422576750791493013356207217, 487331778345355477261, 607088607861933740557075591887834842297

Offset: 2

Samuel Harkness, May 09 2022

nonn

For any n, consecutive n-th powers will never share a divisor > 1, so now consider the second differences. Specifically, each m > 0, define the binary sequence {b(m)} as follows: b(m) = 1 if the first difference (m+1)^n - m^n and the second difference (m+2)^n - 2*(m+1)^n + m^n are coprime, 0 otherwise. I conjecture that {b(m)} is periodic with period a(n).
If m^n mod p == (m+1)^n mod p == (m+2)^n mod p, then p is in the prime factorization of a(n).
All primes p >= 5 belong to a prime factorization for a(n). p will always belong to the prime factorization of n=p-1 due to Fermat's Little Theorem.
I conjecture that the greatest prime factor for any prime n >= 5 is phi(2^n+1)/2 + 1 = Jacobsthal(n). n*A069925 + 1 = A001045(n).
I conjecture that all prime factors "f" are f=n*k+1, unless n is composite, in which case additionally all prime factors for any divisor of n will also be included in the prime factorization for a(n).

			For n=2 and n=3, the first and second differences are coprime for all m. Each of their sequences {b(m)} consist only of 1's, which can be described trivially as [1] with a period of 1, so a(2) = a(3) = 1.
For n > 3, the first and second differences are coprime for some m values, but not for all. Each repeating periodic sequence {b(m)} begins at m=1, and can be used to predict what b(m) will be at any higher m value for that power n.
n=4 has the 5-term repeating sequence, beginning at m=1:
  [0 0 1 1 1], so a(4) = 5.
The sequence is repeating, so for example, f(41)..f(45) is also [0 0 1 1 1].
n=5 has the 11-term repeating sequence
  [1 1 0 1 1 0 1 1 1 1 1]
so a(5) = 11.
n=6 has the 91-term repeating sequence
  [0 0 0 0 0 0 1 0 0 0 0 1 1
   1 0 0 0 0 0 1 1 0 0 0 0 1
   1 1 0 0 0 0 1 1 1 0 0 0 0
   1 1 1 0 0 0 0 1 1 1 0 0 0
   0 1 1 1 0 0 0 0 1 1 1 0 0
   0 0 1 1 0 0 0 0 0 1 1 1 0
   0 0 0 1 0 0 0 0 0 0 1 1 1]
so a(6) = 91.
The period for higher n values has yet to be found. If they exist, it seems they would be quite large given the large expansion from 5, 11, to 91.
Example: the 233rd term in the sequence of values for n=6 is calculated by using m=233 and n=6. Define the first difference for the 233rd term as 234^6 - 233^6 = 4164782373647. The second difference for the 233rd term is 235^6 - 2*234^6 + 233^6 = 89948228762. The terms 4164782373647 and 89948228762 share a common factor, so the 233rd term of the sequence for 6th powered terms is denoted 0 (not coprime). Because the 6th powered terms repeat their tendency of being coprime or not every 91 terms, we could instead look at 233 mod 91 = 51, and from the table for n=6 above, the 51st term is 0.

Samuel Harkness, MATLAB program

Cf. A005408, A007395, A003215, A008588, A005917, A005914, A022521, A068236, A022522, A069473, A069925, A001045, A002587.

MATLAB
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a(7)-a(19) from Jon E. Schoenfield, May 10 2022

cp's OEIS Frontend

A101096 Third differences of fifth powers (A000584).

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A300656 Triangle read by rows: T(n,k) = 30*k^2*(n-k)^2 + 1; n >= 0, 0 <= k <= n.

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A300785 Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1; n >= 0, 0 <= k <= n.

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A101098 a(1)=1; thereafter, a(n+1) = 20*n^3 + 10*n.

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A254871 Seventh partial sums of fifth powers (A000584).

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A069473 First differences of (n+1)^6-n^6 (A022522).

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A300656 Triangle read by rows: T(n,k) = 30k^2(n-k)^2 + 1; n >= 0, 0 <= k <= n.

A300785 Triangle read by rows: T(n,k) = 140k^3(n-k)^3 - 14k(n-k) + 1; n >= 0, 0 <= k <= n.

A101098 a(1)=1; thereafter, a(n+1) = 20n^3 + 10n.

A069477 a(n) = 60n^2 + 180n + 150.