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 1-7 of 7 results.

A169720 a(n) = (3*2^(n-1)-1)*(3*2^n-1).

Original entry on oeis.org

1, 10, 55, 253, 1081, 4465, 18145, 73153, 293761, 1177345, 4713985, 18865153, 75479041, 301953025, 1207885825, 4831690753, 19327057921, 77308821505, 309236465665, 1236948221953, 4947797606401, 19791199862785, 79164818325505, 316659311050753, 1266637319700481
Offset: 0

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Comments

A subsequence of the triangular numbers A000217.

Links

Crossrefs

Cf. A169721-A169727, A033484, A000217.

Programs

  • Magma
    I:=[1, 10, 55]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012
  • Mathematica
    CoefficientList[Series[(1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -14, 8}, {1, 10, 55}, 30] (* Vincenzo Librandi, Dec 03 2012 *)
  • PARI
    a(n)=polcoeff((1+3*x-x^2)/((1-x)*(1-2*x)*(1-4*x)+x*O(x^n)),n) \\ Paul D. Hanna, Apr 29 2010

Formula

G.f.: (1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)). - Paul D. Hanna, Apr 29 2010
a(n) = A000217(A033484(n)). - Mitch Harris, Dec 02 2012
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Dec 03 2012
a(n) = (3*A169726(n)-1)/2. - L. Edson Jeffery, Dec 03 2012
a(n) = A006095(n+2) +3*A006095(n+1) - A006905(n). - R. J. Mathar, Dec 04 2016

A169721 a(n) = (2*(3*2^(n-1)-1))^2.

Original entry on oeis.org

1, 16, 100, 484, 2116, 8836, 36100, 145924, 586756, 2353156, 9424900, 37724164, 150945796, 603881476, 2415722500, 9663283204, 38653919236, 154617249796, 618472144900, 2473894871044, 9895592067076, 39582393434116, 158329624068100, 633318596935684
Offset: 0

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Comments

A subsequence of the squares (A000290).

Links

Crossrefs

Cf. A169720-A169727.

Programs

  • Magma
    I:=[1,16,100]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]];// Vincenzo Librandi, Dec 04 2012
  • Mathematica
    Table[(2(3*2^(n-1)-1))^2,{n,0,30}] (* Harvey P. Dale, Oct 29 2012 *)
CoefficientList[Series[(1+x)/((1-x)*(1-2*x)), {x, 0, 30}], x]^2 (* Vincenzo Librandi, Dec 04 2012 *)

Formula

a(n) = A033484(n)^2.
G.f.: (1+9*x+2*x^2)/(1-7*x+14*x^2-8*x^3). - Bruno Berselli, Dec 04 2012
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3). - Vincenzo Librandi, Dec 04 2012

A169726 a(n) = 3*2^n*(2^n-1) + 1.

Original entry on oeis.org

1, 7, 37, 169, 721, 2977, 12097, 48769, 195841, 784897, 3142657, 12576769, 50319361, 201302017, 805257217, 3221127169, 12884705281, 51539214337, 206157643777, 824632147969, 3298531737601, 13194133241857, 52776545550337, 211106207367169, 844424879800321
Offset: 0

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Comments

A subsequence of the centered hexagonal numbers A003215.

Links

Crossrefs

Cf. A000079, A003215, A169720-A169727.

Programs

  • Magma
    I:=[1, 7, 37]; [n le 3 select I[n] else 7*Self(n-1) -14*Self(n-2) +8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 04 2012
  • Magma
    [3*2^n*(2^n-1)+1 : n in [0..30]]; // Wesley Ivan Hurt, Sep 14 2014
  • Maple
    A169726:=n->3*2^n*(2^n-1)+1: seq(A169726(n), n=0..30); # Wesley Ivan Hurt, Sep 14 2014
  • Mathematica
    CoefficientList[Series[(-1 - 2*x^2)/((x-1)*(2*x-1)*(4*x-1)), {x, 0, 30}],x] (* Vincenzo Librandi, Dec 04 2012 *)
Table[c=2^n;3c(c-1)+1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{1,7,37},30] (* Harvey P. Dale, Nov 22 2013 *)

Formula

From R. J. Mathar, Apr 26 2010: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: ( -1-2*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). (End)
a(n) = (2*A169720(n)+1)/3. - L. Edson Jeffery, Dec 03 2012

A169722 a(n) = (3*2^(n-1)-1)*(18*2^(n-1)-7).

Original entry on oeis.org

1, 22, 145, 715, 3151, 13207, 54055, 218695, 879751, 3528967, 14135815, 56583175, 226412551, 905809927, 3623559175, 14494875655, 57980780551, 231925678087, 927707824135, 3710841520135, 14843386527751, 59373587005447, 237494429810695, 949977882820615
Offset: 0

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Links

Crossrefs

Cf. A169720-A169727, A033484.

Programs

  • Magma
    I:=[1,22,145]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 04 2012
  • Mathematica
    LinearRecurrence[{7, -14, 8}, {1, 22, 145}, 30] (* Vincenzo Librandi, Dec 04 2012 *)
  • Maxima
    makelist(coeff(taylor((1+15*x+5*x^2)/(1-7*x+14*x^2-8*x^3), x, 0, n), x, n), n, 0, 23); /* Bruno Berselli, Dec 04 2012 */

Formula

G.f.: (1+15*x+5*x^2)/(1-7*x+14*x^2-8*x^3). - Bruno Berselli, Dec 04 2012
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3). - Vincenzo Librandi, Dec 04 2012

A169723 3^(n-1)*(2*3^(n-1)+3)+1.

Original entry on oeis.org

6, 28, 190, 1540, 13366, 118828, 1065070, 9572500, 86113126, 774900028, 6973745950, 62762650660, 564860667286, 5083736439628, 45753599258830, 411782307236020, 3706040506843846, 33354363786753628, 300189271756259710, 2701703438832768580
Offset: 1

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Comments

A subsequence of the triangular numbers A000217.

Links

Crossrefs

Cf. A169720-A169727, A100702.

Programs

  • Magma
    I:=[6,28,190]; [n le 3 select I[n] else 13*Self(n-1)-39*Self(n-2)+27*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012
  • Mathematica
    LinearRecurrence[{13, -39, 27}, {6, 28, 190}, 50]  (* or *) CoefficientList[Series[(-6 + 50 x - 60 x^2)/((x - 1) (3 x - 1) (9 x - 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Dec 03 2012 *)

Formula

G.f.: x*(-6+50*x-60*x^2)/((x-1)*(3*x-1)*(9*x-1)). - Vincenzo Librandi, Dec 03 2012
a(n) = 13*a(n-1)-39*a(n-2)+27*a(n-3). - Vincenzo Librandi, Dec 03 2012
a(n) = 2*9^(n-1)+3^n+1. - Bruno Berselli, Dec 05 2012

A169724 (2*3^(n-1)+1)^2.

Original entry on oeis.org

9, 49, 361, 3025, 26569, 237169, 2128681, 19140625, 172213129, 1549760689, 13947373801, 125524947025, 1129720271689, 10167469690609, 91507188951721, 823564585774225, 7412080927594249, 66708727315226929, 600378542737678441, 5403406875341014225
Offset: 1

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Comments

A subsequence of the squares A000290.
Essentially equal to A052919(n)^2.

Links

Crossrefs

Cf. A169720-A169727, A100702.

Programs

  • Magma
    I:=[9, 49, 361]; [n le 3 select I[n] else 13*Self(n-1) - 39*Self(n-2) + 27*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012
  • Mathematica
    CoefficientList[Series[(-9 + 68 x - 75 x^2)/((x - 1) (3 x - 1) (9 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 03 2012 *)
LinearRecurrence[{13,-39,27},{9,49,361},20] (* Harvey P. Dale, Apr 23 2024 *)

Formula

a(n)= +13*a(n-1) -39*a(n-2) +27*a(n-3). G.f.: x*( -9+68*x-75*x^2 ) / ( (x-1)*(3*x-1)*(9*x-1) ). [R. J. Mathar, Apr 26 2010]

Extensions

G.f. adapted to the offset by Vincenzo Librandi, Dec 03 2012

A169725 a(n) = 3^(n-1)*(6*3^(n-1) + 5) + 1.

Original entry on oeis.org

12, 70, 532, 4510, 39772, 355510, 3192292, 28708750, 258313132, 2324621350, 20921001652, 188287243390, 1694579876092, 15251202941590, 137260778644612, 1235346864312430, 11118121348344652, 100063090843700230, 900567813719097172, 8105110311849259870
Offset: 1

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Links

Crossrefs

Cf. A169720-A169727, A100702.

Programs

  • Magma
    I:=[12, 70, 532]; [n le 3 select I[n] else 13*Self(n-1) -39*Self(n-2) +27*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012
  • Maple
    A169725 := proc(n)
    3^(n-1)*(6*3^(n-1)+5)+1 ;
end proc: # R. J. Mathar, Jun 02 2016
  • Mathematica
    Table[3^(n-1) (6 3^(n - 1) + 5) + 1, {n, 20}] (* or *) LinearRecurrence[{13, -39, 27}, {12, 70, 532}, 20] (* Harvey P. Dale, Aug 10 2011 *)
CoefficientList[Series[(-12 + 86 x - 90 x^2)/((x - 1) (3 x - 1) (9 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 03 2012 *)

Formula

From R. J. Mathar, Apr 26 2010: (Start)
a(n) = 13*a(n-1) - 39*a(n-2) + 27*a(n-3).
G.f.: x*( -12 + 86*x - 90*x^2 ) / ( (x-1)*(3*x-1)*(9*x-1) ). (End)
E.g.f.: (2*exp(9*x) + 5*exp(3*x) + 3*exp(x) - 10)/3. - Stefano Spezia, Dec 25 2021

Extensions

G.f. adapted to the offset by Vincenzo Librandi, Dec 03 2012
Showing 1-7 of 7 results.

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