A016758 -id:A016758 - cp's OEIS frontend

A016950 a(n) = (6*n + 3)^6.

729, 531441, 11390625, 85766121, 387420489, 1291467969, 3518743761, 8303765625, 17596287801, 34296447249, 62523502209, 107918163081, 177978515625, 282429536481, 433626201009, 646990183449, 941480149401, 1340095640625, 1870414552161, 2565164201769, 3462825991689

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A016945, A016946, A016947, A016948, A016949.

Subsequence of A001014.

[(6*n+3)^6: n in [0..40]]; // Vincenzo Librandi, May 05 2011

(3 + 6Range[0, 20])^6 (* Harvey P. Dale, Jan 31 2011 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^6 = A016946(n)^3 = A016947(n)^2.
a(n) = 3^6*A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/699840. (End)

A259322 Sum of sixth powers of odd numbers.

Original entry on oeis.org

1, 730, 16355, 134004, 665445, 2437006, 7263815, 18654440, 42792009, 89837890, 175604011, 323639900, 567780525, 955201014, 1550024335, 2437528016, 3728995985, 5567261610, 8132988019, 11651731780, 16401836021, 22723199070, 31026964695

Offset: 1

N. J. A. Sloane, Jun 24 2015

Colin Barker, Table of n, a(n) for n = 1..1000
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000447, A002309, A016758.

f:=n->add((2*i+1)^6,i=0..n);
[seq(f(n),n=0..40)];

Vec(x*(x^6+722*x^5+10543*x^4+23548*x^3+10543*x^2+722*x+1)/(x-1)^8 + O(x^100)) \\ Colin Barker, Jun 29 2015

a(n) = n*(192*n^6-336*n^4+196*n^2-31)/21 \\ Charles R Greathouse IV, Jun 29 2015

a(n) = (n*(-31+196*n^2-336*n^4+192*n^6))/21. - Colin Barker, Jun 29 2015

G.f.: x*(x^6+722*x^5+10543*x^4+23548*x^3+10543*x^2+722*x+1) / (x-1)^8. - Colin Barker, Jun 29 2015

A016746 a(n) = (2*n)^6.

Original entry on oeis.org

0, 64, 4096, 46656, 262144, 1000000, 2985984, 7529536, 16777216, 34012224, 64000000, 113379904, 191102976, 308915776, 481890304, 729000000, 1073741824, 1544804416, 2176782336, 3010936384, 4096000000, 5489031744, 7256313856, 9474296896, 12230590464, 15625000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758.

[(2*n)^6: n in [0..30]]; // Vincenzo Librandi, Sep 05 2011

A016746:=n->(2*n)^6: seq(A016746(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2018

Table[(2*n)^6, {n,0,30}] (* G. C. Greubel, Sep 15 2018 *)

vector(30, n, n--; (2*n)^6) \\ G. C. Greubel, Sep 15 2018

G.f.: 64*x*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1) / (1-x)^7. - R. J. Mathar, May 01 2015
E.g.f.: 64*x*(1 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5)*exp(x). - G. C. Greubel, Sep 15 2018
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi^6/60480.
Sum_{n>=1} (-1)^(n+1)/a(n) = 31*Pi^6/1935360. (End)

A016830 a(n) = (4*n+2)^6.

Original entry on oeis.org

64, 46656, 1000000, 7529536, 34012224, 113379904, 308915776, 729000000, 1544804416, 3010936384, 5489031744, 9474296896, 15625000000, 24794911296, 38068692544, 56800235584, 82653950016, 117649000000, 164206490176, 225199600704, 304006671424, 404567235136, 531441000000

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A016825, A016826, A016827.

(4*Range[0,20]+2)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{64,46656,1000000,7529536,34012224,113379904,308915776},20] (* Harvey P. Dale, Oct 14 2012 *)

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). - Harvey P. Dale, Oct 14 2012
From Amiram Eldar, Jul 07 2022: (Start)
a(n) = A016825(n)^6 = A016826(n)^3 = A016827(n)^2 = 64*A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/61440. (End)

A017118 a(n) = (8*n + 4)^6.

Original entry on oeis.org

4096, 2985984, 64000000, 481890304, 2176782336, 7256313856, 19770609664, 46656000000, 98867482624, 192699928576, 351298031616, 606355001344, 1000000000000, 1586874322944, 2436396322816, 3635215077376, 5289852801024, 7529536000000, 10509215371264, 14412774445056

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A016830, A017113.

[(8*n+4)^6: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2011

(8*Range[0,30]+4)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{4096,2985984,64000000,481890304,2176782336,7256313856,19770609664},30] (* Harvey P. Dale, Jan 07 2016 *)

G.f.: -4096*(1 + 722*x + 10543*x^2 + 23548*x^3 + 10543*x^4 + 722*x^5 + x^6)/(x-1)^7. - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^6.
a(n) = 2^6*A016830(n) = 2^12*A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/3932160. (End)

A017334 a(n) = (10*n + 5)^6.

Original entry on oeis.org

15625, 11390625, 244140625, 1838265625, 8303765625, 27680640625, 75418890625, 177978515625, 377149515625, 735091890625, 1340095640625, 2313060765625, 3814697265625, 6053445140625, 9294114390625, 13867245015625, 20179187015625, 28722900390625, 40089475140625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A017329.

[(10*n+5)^6: n in [0..25]]; // Vincenzo Librandi, Aug 02 2011

(10*Range[0,20]+5)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{15625,11390625,244140625,1838265625,8303765625,27680640625,75418890625},20] (* Harvey P. Dale, Aug 13 2013 *)

G.f.: -15625*(x^6 + 722*x^5 + 10543*x^4 + 23548*x^3 + 10543*x^2 + 722*x + 1)/(x-1)^7. - Colin Barker, Nov 14 2012
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); a(0)=15625, a(1)=11390625, a(2)=244140625, a(3)=1838265625, a(4)=8303765625, a(5)=27680640625, a(6)=75418890625. - Harvey P. Dale, Aug 13 2013
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^6.
a(n) = 5^6 * A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/15000000. (End)

cp's OEIS Frontend

A016950 a(n) = (6*n + 3)^6.

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A259322 Sum of sixth powers of odd numbers.

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A016746 a(n) = (2*n)^6.

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A016830 a(n) = (4*n+2)^6.

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A017118 a(n) = (8*n + 4)^6.

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A017334 a(n) = (10*n + 5)^6.

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