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.

Previous Showing 11-18 of 18 results.

A014882 a(1) = 1, a(n) = 12*a(n-1) + n.

Original entry on oeis.org

1, 14, 171, 2056, 24677, 296130, 3553567, 42642812, 511713753, 6140565046, 73686780563, 884241366768, 10610896401229, 127330756814762, 1527969081777159, 18335628981325924, 220027547775911105, 2640330573310933278, 31683966879731199355, 380207602556774392280
Offset: 1

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Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=12 of A126885.

Programs

  • Magma
    I:=[1, 14, 171]; [n le 3 select I[n] else 14*Self(n-1) - 25*Self(n-2)+ 12*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Maple
    a:=n->sum((12^(n-j)-1^(n-j))/11,j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 12 2007
a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,12]])^n)[2,3]:
seq(a(n), n=1..17);  # Alois P. Heinz, Aug 06 2008
  • Mathematica
    LinearRecurrence[{14, -25, 12}, {1, 14, 171}, 201] (* Vincenzo Librandi, Oct 20 2012 *)

Formula

a(n) = 14*a(n-1) - 25*a(n-2) + 12*a(n-3), with a(1)=1, a(2)=14, a(3)=171. - Vincenzo Librandi, Oct 20 2012
G.f.: x/((1-12*x)*(1-x)^2). - Jinyuan Wang, Mar 11 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: exp(x)*(12*exp(11*x) - 11*x - 12)/121.
a(n) = (12^(n+1) - 11*n - 12)/121. (End)

A014901 a(1)=1, a(n) = 18*a(n-1) + n.

Original entry on oeis.org

1, 20, 363, 6538, 117689, 2118408, 38131351, 686364326, 12354557877, 222382041796, 4002876752339, 72051781542114, 1296932067758065, 23344777219645184, 420205989953613327, 7563707819165039902, 136146740744970718253, 2450641333409472928572, 44111544001370512714315
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=18 of A126885.
Cf. A014935.

Programs

  • Magma
    I:=[1, 20, 363]; [n le 3 select I[n] else 20*Self(n-1) - 37*Self(n-2) + 18*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Mathematica
    LinearRecurrence[{20, -37, 18}, {1, 20, 363}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
nxt[{n_,a_}]:={n+1,18a+n+1}; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Oct 08 2017 *)
  • Maxima
    a[1]:1$
a[2]:20$
a[3]:363$
a[n]:=20*a[n-1]-37*a[n-2]+18*a[n-3]$
A014901(n):=a[n]$
makelist(A014901(n),n,1,30); /* Martin Ettl, Nov 06 2012 */

Formula

a(1)=1, a(2)=20, a(3)=363, a(n) = 20*a(n-1) - 37*a(n-2) + 18*a(n-3). - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-18*x)*(1-x)^2).
E.g.f.: exp(x)*(18*exp(17*x) - 17*x - 18)/289.
a(n) = (18^(n+1) - 17*n - 18)/289. (End)
a(-1-n) = A014935(n)/18^n for all n in Z. - Michael Somos, Mar 29 2025

A014905 a(1)=1, a(n) = 21*a(n-1) + n.

Original entry on oeis.org

1, 23, 486, 10210, 214415, 4502721, 94557148, 1985700116, 41699702445, 875693751355, 18389568778466, 386180944347798, 8109799831303771, 170305796457379205, 3576421725604963320, 75104856237704229736, 1577201980991788824473, 33121241600827565313951, 695546073617378871592990
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=21 of A126885.

Programs

  • Magma
    I:=[1, 23, 486]; [n le 3 select I[n] else 23*Self(n-1) - 43*Self(n-2)+ 21*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 19 2012
  • Mathematica
    LinearRecurrence[{23, -43, 21}, {1, 23, 486}, 20] (* Vincenzo Librandi, Oct 19 2012 *)

Formula

a(1)=1, a(2)=23, a(3)=486, a(n) = 23*a(n-1) - 43*a(n-2) + 21*a(n-3). - Vincenzo Librandi, Oct 19 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-21*x)*(x-1)^2).
E.g.f.: exp(x)*(21*exp(20*x) - 20*x - 21)/400.
a(n) = (21^(n+1) - 20*n - 21)/400. (End)

A014907 a(1)=1, a(n) = 22*a(n-1) + n.

Original entry on oeis.org

1, 24, 531, 11686, 257097, 5656140, 124435087, 2737571922, 60226582293, 1324984810456, 29149665830043, 641292648260958, 14108438261741089, 310385641758303972, 6828484118682687399, 150226650611019122794, 3304986313442420701485, 72709698895733255432688, 1599613375706131619519155
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=22 of A126885.

Programs

  • Magma
    I:=[1, 24, 531]; [n le 3 select I[n] else 24*Self(n-1) - 45*Self(n-2) + 22*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 19 2012
  • Mathematica
    LinearRecurrence[{24, -45, 22}, {1, 24, 531}, 20] (* Vincenzo Librandi, Oct 19 2012 *)
  • Maxima
    a[1]:1$
a[2]:24$
a[3]:531$
a[n]:=24*a[n-1]-45*a[n-2]+22*a[n-3]$
A014907(n):=a[n]$
makelist(A014907(n),n,1,30); /* Martin Ettl, Nov 06 2012 */

Formula

a(1)=1, a(2)=24, a(3)=531, a(n) = 24*a(n-1) - 45*a(n-2) + 22*a(n-3). - Vincenzo Librandi, Oct 19 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-22*x)*(x-1)^2).
E.g.f.: exp(x)*(22*exp(21*x) - 21*x - 22)/441.
a(n) = (22^(n+1) - 21*n - 22)/441. (End)

A014909 a(1)=1, a(n) = 23*a(n-1) + n.

Original entry on oeis.org

1, 25, 578, 13298, 305859, 7034763, 161799556, 3721389796, 85591965317, 1968615202301, 45278149652934, 1041397442017494, 23952141166402375, 550899246827254639, 12670682677026856712, 291425701571617704392, 6702791136147207201033, 154164196131385765623777, 3545776511021872609346890
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=23 of A126885.

Programs

  • Magma
    I:=[1, 25, 578]; [n le 3 select I[n] else 25*Self(n-1)-47*Self(n-2)+23*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 05 2012
  • Mathematica
    Transpose[NestList[{First[#]+1,23Last[#]+First[#]+1}&,{1,1},20]][[2]] (* or *) LinearRecurrence[{25,-47,23},{1,25,578},20] (* Harvey P. Dale, Feb 05 2012 *)

Formula

From Harvey P. Dale, Feb 05 2012: (Start)
a(1)=1, a(2)=25, a(3)=578, a(n) = 25*a(n-1) - 47*a(n-2) + 23*a(n-3).
G.f.: -x/((-1+x)^2*(-1+23*x)). (End)
From Elmo R. Oliveira, Mar 30 2025: (Start)
E.g.f.: exp(x)*(23*exp(22*x) - 22*x - 23)/484.
a(n) = (23^(n+1) - 22*n - 23)/484. (End)

A014913 a(1)=1, a(n) = 24*a(n-1) + n.

Original entry on oeis.org

1, 26, 627, 15052, 361253, 8670078, 208081879, 4993965104, 119855162505, 2876523900130, 69036573603131, 1656877766475156, 39765066395403757, 954361593489690182, 22904678243752564383, 549712277850061545208, 13193094668401477085009, 316634272041635450040234
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=24 of A126885.

Programs

  • Magma
    I:=[1, 26, 627]; [n le 3 select I[n] else 26*Self(n-1) - 49*Self(n-2) + 24*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 19 2012
  • Mathematica
    LinearRecurrence[{26, -49, 24}, {1, 26, 627}, 20] (* Vincenzo Librandi, Oct 19 2012 *)
nxt[{n_,a_}]:={n+1,24a+n+1}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Jun 30 2025 *)

Formula

a(1)=1, a(2)=26, a(3)=627, a(n) = 26*a(n-1) - 49*a(n-2) + 24*a(n-3). - Vincenzo Librandi, Oct 19 2012
From Elmo R. Oliveira, Mar 30 2025: (Start)
G.f.: x/((1-24*x)*(x-1)^2).
E.g.f.: exp(x)*(24*exp(23*x) - 23*x - 24)/529.
a(n) = (24^(n+1) - 23*n - 24)/529. (End)

A014914 a(1)=1, a(n) = 25*a(n-1) + n.

Original entry on oeis.org

1, 27, 678, 16954, 423855, 10596381, 264909532, 6622738308, 165568457709, 4139211442735, 103480286068386, 2587007151709662, 64675178792741563, 1616879469818539089, 40421986745463477240, 1010549668636586931016, 25263741715914673275417, 631593542897866831885443
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=25 of A126885.

Programs

  • Magma
    I:=[1, 27, 678]; [n le 3 select I[n] else 27*Self(n-1) - 51*Self(n-2)+ 25*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 19 2012
  • Mathematica
    LinearRecurrence[{27, -51, 25}, {1, 27, 678}, 20] (* Vincenzo Librandi, Oct 19 2012 *)
nxt[{n_,a_}]:={n+1,25a+n+1}; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Nov 16 2021 *)

Formula

a(1)=1, a(2)=27, a(3)=678, a(n) = 27*a(n-1) - 51*a(n-2) + 25*a(n-3). - Vincenzo Librandi, Oct 19 2012
From Elmo R. Oliveira, Mar 30 2025: (Start)
G.f.: x/((1-25*x)*(x-1)^2).
E.g.f.: exp(x)*(25*exp(24*x) - 24*x - 25)/576.
a(n) = (25^(n+1) - 24*n - 25)/576. (End)

A014900 a(1)=1, a(n) = 17*a(n-1) + n.

Original entry on oeis.org

1, 19, 326, 5546, 94287, 1602885, 27249052, 463233892, 7874976173, 133874594951, 2275868114178, 38689757941038, 657725884997659, 11181340044960217, 190082780764323704, 3231407272993502984, 54933923640889550745, 933876701895122362683, 15875903932217080165630
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=17 of A126885.

Programs

  • Magma
    I:=[1, 19, 326]; [n le 3 select I[n] else 19*Self(n-1) - 35*Self(n-2) + 17*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Maple
    a:=n->sum((17^(n-j)-1)/16,j=0..n): seq(a(n), n=1..16); # Zerinvary Lajos, Jan 05 2007
a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,17]])^n)[2,3]:
seq(a(n), n=1..16);  # Alois P. Heinz, Aug 06 2008
  • Mathematica
    LinearRecurrence[{19, -35, 17}, {1, 19, 326}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
nxt[{n_,a_}]:={n+1,17a+n+1}; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Jun 19 2021 *)

Formula

a(1)=1, a(2)=19, a(3)=326, a(n) = 19*a(n-1) - 35*a(n-2) + 17*a(n-3). - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-17*x)*(1-x)^2).
E.g.f.: exp(x)*(17*exp(16*x) - 16*x - 17)/256.
a(n) = (17^(n+1) - 16*n - 17)/256. (End)
Previous Showing 11-18 of 18 results.

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

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