A016791 -id:A016791 - cp's OEIS frontend

A061103 Duplicate of A016791.

8, 125, 512, 1331, 2744, 4913, 8000, 12167, 17576, 24389, 32768, 42875, 54872, 68921, 85184, 103823, 125000, 148877, 175616, 205379, 238328, 274625, 314432, 357911, 405224, 456533, 512000, 571787, 636056, 704969, 778688, 857375, 941192, 1030301, 1124864

Offset: 1

dead

A016779 a(n) = (3*n + 1)^3.

Original entry on oeis.org

1, 64, 343, 1000, 2197, 4096, 6859, 10648, 15625, 21952, 29791, 39304, 50653, 64000, 79507, 97336, 117649, 140608, 166375, 195112, 226981, 262144, 300763, 343000, 389017, 438976, 493039, 551368, 614125, 681472, 753571, 830584, 912673, 1000000, 1092727, 1191016

Offset: 0

N. J. A. Sloane

The inverse binomial transform is 1, 63, 216, 162, 0, 0, 0 (0 continued). R. J. Mathar, May 07 2008

Perfect cubes with digital root 1 in base 10. Proof: perfect cubes are one of (3*s)^3, (3*s+1)^3 or (3*s+2)^3. Digital roots of (3*s)^3 are 0, digital roots of (3*s+1)^3 are 1, and digital roots of (3*s+2)^3 are 8, using trinomial expansion and the multiplicative property of digits roots. - R. J. Mathar, Jul 31 2010

			a(2) = (3*2+1)^3 = 343.
a(6) = (3*6+1)^3 = 6859.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.3.
Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016791, A054966, A226735.

[(3*n+1)^3: n in [0..30]]; // Vincenzo Librandi, May 09 2011

Table[(3n+1)^3,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,64,343,1000},40] (* Harvey P. Dale, Oct 31 2016 *)

a(n)=(3*n+1)^3 \\ Charles R Greathouse IV, Jan 02 2012

Sum_{n>=0} 1/a(n) = 2*Pi^3 / (81*sqrt(3)) + 13*zeta(3)/27.
O.g.f.: (1 + 60*x + 93*x^2 + 8*x^3)/(1 - x)^4. - R. J. Mathar, May 07 2008
E.g.f.: (1 + 63*x + 108*x^2 + 27*x^3)*exp(x). - Ilya Gutkovskiy, Jun 16 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
Sum_{n>=1} (-1)^n/a(n) = A226735. - R. J. Mathar, Feb 07 2024

A016792 a(n) = (3*n+2)^4.

Original entry on oeis.org

16, 625, 4096, 14641, 38416, 83521, 160000, 279841, 456976, 707281, 1048576, 1500625, 2085136, 2825761, 3748096, 4879681, 6250000, 7890481, 9834496, 12117361, 14776336, 17850625, 21381376, 25411681, 29986576, 35153041, 40960000, 47458321, 54700816, 62742241, 71639296

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 (5,-10,10,-5,1).

Cf. A016789, A016790, A016791.

Subsequence of A000583.

[(3*n+2)^4 : n in [0..30]]; // Vincenzo Librandi, Sep 29 2011

Table[(3n+2)^4,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{16,625,4096,14641,38416},30] (* Harvey P. Dale, Aug 02 2018 *)

From Ilya Gutkovskiy, Jun 16 2016: (Start)
G.f.: (16 + 545*x + 1131*x^2 + 251*x^3 + x^4)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^4 = A016790(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(3, 2/3)/486. (End)

A016793 a(n) = (3*n + 2)^5.

Original entry on oeis.org

32, 3125, 32768, 161051, 537824, 1419857, 3200000, 6436343, 11881376, 20511149, 33554432, 52521875, 79235168, 115856201, 164916224, 229345007, 312500000, 418195493, 550731776, 714924299, 916132832, 1160290625, 1453933568, 1804229351, 2219006624, 2706784157, 3276800000

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 (6,-15,20,-15,6,-1).

Cf. A016789, A016790, A016791, A016792.

Subsequence of A000584.

[(3*n+2)^5 : n in [0..30]]; // Vincenzo Librandi, Sep 29 2011

Table[(3n+2)^5,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{32,3125,32768,161051,537824,1419857},30] (* Harvey P. Dale, May 10 2024 *)

From Ilya Gutkovskiy, Jun 16 2016: (Start)
G.f.: (32 + 2933*x + 14498*x^2 + 10678*x^3 + 1018*x^4 + x^5)/(1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). (End)
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^5.
Sum_{n>=0} 1/a(n) = 121*zeta(5)/243 - 2*Pi^5/(729*sqrt(3)). (End)

A016794 a(n) = (3*n + 2)^6.

Original entry on oeis.org

64, 15625, 262144, 1771561, 7529536, 24137569, 64000000, 148035889, 308915776, 594823321, 1073741824, 1838265625, 3010936384, 4750104241, 7256313856, 10779215329, 15625000000, 22164361129, 30840979456, 42180533641, 56800235584, 75418890625, 98867482624, 128100283921

Offset: 0

N. J. A. Sloane

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

Cf. A016789, A016790, A016791, A016792, A016793.

Subsequence of A001014.

A016794:= n-> (3*n+2)^6; seq(A016794(k), k=0..30); # Wesley Ivan Hurt, Nov 05 2013

(3Range[0,20]+2)^6  (* Harvey P. Dale, Mar 04 2011 *)

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^6 = A016790(n)^3 = A016791(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(5, 2/3)/87480. (End)

A118719 Cubes for which the digital root is also a cube.

Original entry on oeis.org

0, 1, 8, 64, 125, 343, 512, 1000, 1331, 2197, 2744, 4096, 4913, 6859, 8000, 10648, 12167, 15625, 17576, 21952, 24389, 29791, 32768, 39304, 42875, 50653, 54872, 64000, 68921, 79507, 85184, 97336, 103823, 117649, 125000, 140608, 148877

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

All cubes have a digital root 1,8 or 9. (except for the number 0) So this sequence contains all cubes with a digital root which is not 9.

This sequence is 0 union A016779 union A016791.

			64 is in the sequence because (1) it is a cube and (2) the digital root 1 is also a cube.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000578, A002117, A010888, A116978.

[0] cat [(6*n+(-1)^n-9)^3 div 64: n in [2..37]];  // Bruno Berselli, May 05 2011

Join[{0}, Table[(3*k + {1, 2})^3, {k, 0, 15}] // Flatten] (* Amiram Eldar, Dec 19 2020 *)

a010888(n)=if(n, (n-1)%9+1)
lista(nn) = {for (n=0, nn, if (ispower(a010888(n^3), 3), print1(n^3, ", ")););} \\ Michel Marcus, Feb 18 2015

a(n) = (floor(3*n/2)-2)^3 for n >= 2. - Nathaniel Johnston, May 05 2011
G.f.: x^2*(1+7*x+53*x^2+40*x^3+53*x^4+7*x^5+x^6)/((1+x)^3*(1-x)^4).  a(n) = A001651(n-1)^3 for n>1. - Bruno Berselli, May 05 2011
Sum_{n>=2} 1/a(n) = 26*zeta(3)/27. - Amiram Eldar, Dec 19 2020

A016796 a(n) = (3*n + 2)^8.

Original entry on oeis.org

256, 390625, 16777216, 214358881, 1475789056, 6975757441, 25600000000, 78310985281, 208827064576, 500246412961, 1099511627776, 2251875390625, 4347792138496, 7984925229121, 14048223625216, 23811286661761, 39062500000000, 62259690411361, 96717311574016, 146830437604321

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795.

Subsequence of A001016.

(3 Range[0, 20] + 2)^8 (* Harvey P. Dale, Jan 24 2011 *)

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^6 = A016790(n)^3 = A016791(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(7, 2/3)/33067440. (End)

A016798 a(n) = (3*n + 2)^10.

Original entry on oeis.org

1024, 9765625, 1073741824, 25937424601, 289254654976, 2015993900449, 10240000000000, 41426511213649, 141167095653376, 420707233300201, 1125899906842624, 2758547353515625, 6278211847988224, 13422659310152401, 27197360938418176, 52599132235830049

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795.

Subsequence of A008454.

(3*Range[0,20]+2)^10 (* Harvey P. Dale, Nov 28 2014 *)

From  Harvey P. Dale, Nov 28 2014: (Start)
G.f.: -(1/((x-1)^11))(x^10+1048565*x^9+270940968*x^8+6950443776*x^7+ 43221615834*x^6+86805830970*x^5+61387794480*x^4+14663204952*x^3+ 966376269*x^2+9754361*x+1024).
a(n) = 59049*n^10 + 393660*n^9 + 1180980*n^8 + 2099520*n^7 + 2449440*n^6 + 1959552*n^5 + 1088640*n^4 + 414720*n^3 + 103680*n^2 + 15360*n + 1024. [corrected by Amiram Eldar, Mar 31 2022] (End)
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^10 = A016790(n)^5 = A016793(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(9, 2/3)/21427701120. (End)
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, Dec 31 2023

A016797 a(n) = (3*n + 2)^9.

Original entry on oeis.org

512, 1953125, 134217728, 2357947691, 20661046784, 118587876497, 512000000000, 1801152661463, 5429503678976, 14507145975869, 35184372088832, 78815638671875, 165216101262848, 327381934393961, 618121839509504, 1119130473102767, 1953125000000000, 3299763591802133

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795, A016796.

Subsequence of A001017.

Table[(3n+2)^9,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

G.f.: (512 + 1948005*x + 114709518*x^2 + 1103599596*x^3 + 2887100154*x^4 + 2388954618*x^5 + 608260290*x^6 + 37732212*x^7 + 262134*x^8 + x^9)/(1 - x)^10. - Ilya Gutkovskiy, Jun 16 2016
From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016789(n)^9.
Sum_{n>=0} 1/a(n) = 9841*zeta(9)/19683 - 1618*Pi^9/(55801305*sqrt(3)). (End)

A016799 a(n) = (3*n + 2)^11.

Original entry on oeis.org

2048, 48828125, 8589934592, 285311670611, 4049565169664, 34271896307633, 204800000000000, 952809757913927, 3670344486987776, 12200509765705829, 36028797018963968, 96549157373046875, 238572050223552512, 550329031716248441, 1196683881290399744, 2472159215084012303

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795, A016796, A016797, A016798.

Subsequence of A008455.

(3Range[0,20]+2)^11 (* Harvey P. Dale, Apr 25 2011 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016789(n)^11.
Sum_{n>=0} 1/a(n) = 88573*zeta(11)/177147 - 7388*Pi^11/(2511058725*sqrt(3)). (End)

cp's OEIS Frontend

A061103 Duplicate of A016791.

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A016779 a(n) = (3*n + 1)^3.

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Magma

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A016792 a(n) = (3*n+2)^4.

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Magma

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A016793 a(n) = (3*n + 2)^5.

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A016794 a(n) = (3*n + 2)^6.

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A118719 Cubes for which the digital root is also a cube.

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Magma

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A016796 a(n) = (3*n + 2)^8.

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A016798 a(n) = (3*n + 2)^10.

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