cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A016778 a(n) = (3*n+1)^2.

Original entry on oeis.org

1, 16, 49, 100, 169, 256, 361, 484, 625, 784, 961, 1156, 1369, 1600, 1849, 2116, 2401, 2704, 3025, 3364, 3721, 4096, 4489, 4900, 5329, 5776, 6241, 6724, 7225, 7744, 8281, 8836, 9409, 10000, 10609, 11236, 11881, 12544, 13225, 13924, 14641, 15376, 16129, 16900, 17689
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Paul Curtz, Mar 28 2019: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777:
.
\
100--97--94--91
\ \
49--46--43 88
/ \ \ \
52 16--13 40 85
/ / \ \ \ \
55 19 1 10 37 82
/ / / / / /
58 22 4---7 34 79
\ \ / /
61 25--28--31 76
\ /
64--67--70--73
(End)

Links

Crossrefs

Cf. A000212, A000290, A016777, A016779, A016780, A016781, A214550, A247792, A262178.

Programs

Formula

a(n) = a(n-1) + 3*(6*n-1); a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=16, a(2)=49. - Harvey P. Dale, Mar 03 2013
a(n) = A247792(n) + 6*n. - Miquel Cerda, Oct 23 2016
G.f.: (1 + 13*x + 4*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Oct 23 2016
a(n) = A000212(3*n) + A000212(1+3*n) + A000212(2+3*n). - Paul Curtz, Mar 28 2019
From Amiram Eldar, Nov 12 2020: (Start)
Sum_{n>=0} 1/a(n) = A214550.
Sum_{n>=0} (-1)^n/a(n) = A262178. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: exp(x)*(1 + 15*x + 9*x^2).
a(n) = A000290(A016777(n)) = A016777(n)^2. (End)

A016781 a(n) = (3*n+1)^5.

Original entry on oeis.org

1, 1024, 16807, 100000, 371293, 1048576, 2476099, 5153632, 9765625, 17210368, 28629151, 45435424, 69343957, 102400000, 147008443, 205962976, 282475249, 380204032, 503284375, 656356768, 844596301, 1073741824, 1350125107, 1680700000, 2073071593, 2535525376
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In general the e.g.f. of {(1 + 3*m)^n}{m>=0} is E(n,x) = exp(x)*Sum{m=0..n} A282629(n, m)*x^m, and the o.g.f. is G(n, x) = (Sum_{m=0..n} A225117(n, n-m)*x^m)/(1-x)^(n+1). - Wolfdieter Lang, Apr 02 2017

Links

Crossrefs

Cf. A016777, A016778, A016779, A016780, A225117, A282629.

Programs

  • Magma
    [(3*n+1)^5: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011
  • Mathematica
    (3Range[0,20]+1)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,1024,16807,100000,371293,1048576},30] (* Harvey P. Dale, May 13 2012 *)
  • Maxima
    A016781(n):=(3*n+1)^5$
makelist(A016781(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

Formula

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). - Harvey P. Dale, May 13 2012
From Wolfdieter Lang, Apr 02 2017: (Start)
O.g.f.: (1+1018*x+10678*x^2+14498*x^3+2933*x^4+32*x^5)/(1-x)^6.
E.g.f: exp(x)*(1+1023*x+7380*x^2+8775*x^3+2835*x^4+243*x^5). (End)
a(n) = A000584(A016777(n)). - Michel Marcus, Apr 06 2017
Sum_{n>=0} 1/a(n) = 2*Pi^5/(3^6*sqrt(3)) + 121*zeta(5)/3^5. - Amiram Eldar, Mar 29 2022

A016936 a(n) = (6*n + 2)^4.

Original entry on oeis.org

16, 4096, 38416, 160000, 456976, 1048576, 2085136, 3748096, 6250000, 9834496, 14776336, 21381376, 29986576, 40960000, 54700816, 71639296, 92236816, 116985856, 146410000, 181063936, 221533456, 268435456, 322417936, 384160000, 454371856, 533794816, 623201296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016780, A016933, A016934, A016935.

Programs

  • Magma
    [(6*n+2)^4: n in [0..30]]; // Vincenzo Librandi, May 04 2011
  • Mathematica
    (6*Range[0,30]+2)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{16,4096,38416,160000,456976},30] (* Harvey P. Dale, Aug 22 2012 *)

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Aug 22 2012
From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^4 = A016934(n)^2.
a(n) = 16*A016780(n).
Sum_{n>=0} 1/a(n) = PolyGamma(3, 1/3)/7776. (End)

A016782 a(n) = (3*n+1)^6.

Original entry on oeis.org

1, 4096, 117649, 1000000, 4826809, 16777216, 47045881, 113379904, 244140625, 481890304, 887503681, 1544804416, 2565726409, 4096000000, 6321363049, 9474296896, 13841287201, 19770609664, 27680640625, 38068692544, 51520374361, 68719476736, 90458382169, 117649000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001014, A016777, A016778, A016779, A016780, A016781.

Programs

  • Magma
    [(3*n+1)^6: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011
  • Mathematica
    Table[(3n+1)^6,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,4096,117649,1000000,4826809,16777216,47045881},20] (* Harvey P. Dale, Sep 30 2016 *)

Formula

a(n) = A001014(A016777(n)). - Michel Marcus, Jun 15 2016
From Ilya Gutkovskiy, Jun 15 2016: (Start)
G.f.: (1 + 4089*x + 88998*x^2 + 262438*x^3 + 154113*x^4 + 15177*x^5 + 64*x^6)/(1 - x)^7.
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). (End)
Sum_{n>=0} 1/a(n) = PolyGamma(5, 1/3)/87480. - Amiram Eldar, Mar 29 2022

A016785 a(n) = (3*n + 1)^9.

Original entry on oeis.org

1, 262144, 40353607, 1000000000, 10604499373, 68719476736, 322687697779, 1207269217792, 3814697265625, 10578455953408, 26439622160671, 60716992766464, 129961739795077, 262144000000000, 502592611936843, 922190162669056, 1628413597910449, 2779905883635712
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784.

Programs

  • Magma
    [(3*n+1)^9 : n in [0..20]]; // Vincenzo Librandi, Sep 28 2011
  • Maple
    A016785:=n->(3*n+1)^9; seq(A016785(k), k=0..100); # Wesley Ivan Hurt, Nov 05 2013
  • Mathematica
    Table[(3*n+1)^9, {n,0,100}] (* Wesley Ivan Hurt, Nov 05 2013 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,262144,40353607,1000000000,10604499373,68719476736,322687697779,1207269217792,3814697265625,10578455953408},100] (* Harvey P. Dale, Aug 17 2014 *)

Formula

From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016777(n)^9 = A016779(n)^3.
Sum_{n>=0} 1/a(n) = 1618*Pi^9/(55801305*sqrt(3)) + 9841*zeta(9)/3^9. (End)

A016784 a(n) = (3*n+1)^8.

Original entry on oeis.org

1, 65536, 5764801, 100000000, 815730721, 4294967296, 16983563041, 54875873536, 152587890625, 377801998336, 852891037441, 1785793904896, 3512479453921, 6553600000000, 11688200277601, 20047612231936, 33232930569601, 53459728531456, 83733937890625
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001016, A016777, A016778, A016779, A016780, A016781, A016782, A016783.

Programs

Formula

a(n)= A001016(A016777(n)). - Michel Marcus, Jun 15 2016
G.f.: (1 + 65527*x + 5175013*x^2 + 50476003*x^3 + 117758659*x^4 + 77404933*x^5 + 13270807*x^6 + 388321*x^7 + 256*x^8)/(1 - x)^9. - Ilya Gutkovskiy, Jun 16 2016
Sum_{n>=0} 1/a(n) = PolyGamma(7, 1/3)/33067440. - Amiram Eldar, Mar 29 2022

A016787 a(n) = (3*n + 1)^11.

Original entry on oeis.org

1, 4194304, 1977326743, 100000000000, 1792160394037, 17592186044416, 116490258898219, 584318301411328, 2384185791015625, 8293509467471872, 25408476896404831, 70188843638032384, 177917621779460413, 419430400000000000, 929293739471222707, 1951354384207722496
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785, A016786.
Subsequence of A008455.

Programs

  • Magma
    [(3*n+1)^11: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011
  • Mathematica
    Table[(3*n + 1)^11, {n, 0, 30}] (* Amiram Eldar, Mar 30 2022 *)

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016777(n)^11.
Sum_{n>=0} 1/a(n) = 7388*Pi^11/(2511058725*sqrt(3)) + 88573*zeta(11)/177147. (End)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12). - Wesley Ivan Hurt, Apr 12 2023

A016786 a(n) = (3*n+1)^10.

Original entry on oeis.org

1, 1048576, 282475249, 10000000000, 137858491849, 1099511627776, 6131066257801, 26559922791424, 95367431640625, 296196766695424, 819628286980801, 2064377754059776, 4808584372417849, 10485760000000000, 21611482313284249, 42420747482776576, 79792266297612001
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008454, A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785.

Programs

  • Magma
    [(3*n+1)^10: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011
  • Mathematica
    Table[(3n+1)^10,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,1048576,282475249,10000000000,137858491849,1099511627776,6131066257801,26559922791424,95367431640625,296196766695424,819628286980801},20] (* Harvey P. Dale, May 14 2019 *)

Formula

a(n) = A008454(A016777(n)). - Michel Marcus, Jun 15 2016
Sum_{n>=0} 1/a(n) = PolyGamma(9, 1/3)/21427701120. - Amiram Eldar, Mar 29 2022

A016788 a(n) = (3*n+1)^12.

Original entry on oeis.org

1, 16777216, 13841287201, 1000000000000, 23298085122481, 281474976710656, 2213314919066161, 12855002631049216, 59604644775390625, 232218265089212416, 787662783788549761, 2386420683693101056, 6582952005840035281, 16777216000000000000, 39959630797262576401
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008456, A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785, A016786, A016787.
Subsequence of A008456.

Programs

Formula

a(n) = A008456(A016777(n)). - Michel Marcus, Jun 16 2016
Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/3)/21213424108800. - Amiram Eldar, Mar 30 2022

A017200 a(n) = (9*n+3)^4.

Original entry on oeis.org

81, 20736, 194481, 810000, 2313441, 5308416, 10556001, 18974736, 31640625, 49787136, 74805201, 108243216, 151807041, 207360000, 276922881, 362673936, 466948881, 592240896, 741200625, 916636176
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000583 (n^4), A016780 ((3n+1)^4), A017197 (9n+3).

Programs

Formula

a(n) = A000583(A017197(n)). - Michel Marcus, Nov 06 2015
a(n) = 81*A016780(n). - Michel Marcus, Nov 06 2015
From Ilya Gutkovskiy, Jun 16 2016: (Start)
G.f.: 81*(1 + 251*x + 1131*x^2 + 545*x^3 + 16*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)
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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