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 18 results. Next

A134195 Antidiagonal sums of square array A126885.

Original entry on oeis.org

1, 3, 7, 15, 32, 72, 178, 494, 1543, 5373, 20581, 85653, 383494, 1833250, 9301792, 49857540, 281193501, 1663183383, 10286884195, 66365330811, 445598473612, 3107611606908, 22470529228910, 168190079241210, 1301213084182483, 10391369994732593, 85553299734530113
Offset: 0

Views

Author

Gary W. Adamson, Oct 12 2007

Keywords

Comments

Conjecture: partial sums of A104879. - Sean A. Irvine, Jul 14 2022

Examples

			a(4) = 1 + 5 + 11 + 10 + 5 = 32.

Crossrefs

Cf. A126885.

Programs

  • Maxima
    T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$ /* A126885 */
a(n) := sum(T(n - k + 1, k), k, 1, n + 1)$
makelist(a(n), n, 0, 50); /* Franck Maminirina Ramaharo, Jan 26 2019 */

A368296 Square array T(n,k), n >= 2, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 8, 6, 3, 1, 5, 14, 18, 9, 3, 1, 6, 22, 44, 39, 12, 4, 1, 7, 32, 90, 135, 81, 16, 4, 1, 8, 44, 162, 363, 408, 166, 20, 5, 1, 9, 58, 266, 813, 1455, 1228, 336, 25, 5, 1, 10, 74, 408, 1599, 4068, 5824, 3688, 677, 30, 6
Offset: 2

Views

Author

Seiichi Manyama, Dec 20 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  2,  4,   8,   14,   22,    32,    44, ...
  2,  6,  18,   44,   90,   162,   266, ...
  3,  9,  39,  135,  363,   813,  1599, ...
  3, 12,  81,  408, 1455,  4068,  9597, ...
  4, 16, 166, 1228, 5824, 20344, 57586, ...

Links

Crossrefs

Columns k=0..8 give A004526, A002620, A178420, A097137, A097138, A097139, A178719, A178730, A178827.
Cf. A055129, A126885, A368343.

Programs

  • PARI
    T(n, k) = (-((n+1)\2)+sum(j=1, n, j*k^(n-j)))/(k+1);

Formula

T(n,k) = T(n-2,k) + Sum_{j=0..n-2} k^j.
T(n,k) = 1/(k+1) * (-floor((n+1)/2) + Sum_{j=1..n} j*k^(n-j)).
T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k+1)) = Sum_{j=0..n} floor(k^j/(k^2-1)) for k > 1.
T(n,k) = (k+1)*T(n-1,k) - (k-1)*T(n-2,k) - (k+1)*T(n-3,k) + k*T(n-4,k).
G.f. of column k: x^2/((1-x) * (1-k*x) * (1-x^2)).
T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^2-1)) - floor((n+1)/2)) for k > 1.

A014830 a(1)=1; for n > 1, a(n) = 7*a(n-1) + n.

Original entry on oeis.org

1, 9, 66, 466, 3267, 22875, 160132, 1120932, 7846533, 54925741, 384480198, 2691361398, 18839529799, 131876708607, 923136960264, 6461958721864, 45233711053065, 316635977371473, 2216451841600330, 15515162891202330, 108606140238416331, 760242981668914339, 5321700871682400396
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			For n=5, a(5) = 1*15 + 6*20 + 6^2*15 + 6^3*6 + 6^4*1 = 3267. - _Bruno Berselli_, Nov 13 2015

Links

Crossrefs

Row n=7 of A126885.
Cf. A000400, A104712.

Programs

  • Maple
    a:=n->sum((7^(n-j)-1)/6,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007
  • Mathematica
    a[1] = 1; a[n_] := 7*a[n-1]+n; Table[a[n], {n, 10}] (* Zak Seidov, Feb 06 2011 *)
LinearRecurrence[{9, -15, 7}, {1, 9, 66}, 30] (* Harvey P. Dale, Jul 22 2013 *)
  • PARI
    Vec(x/((1 - x)^2*(1 - 7*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020

Formula

a(n) = (7^(n+1) - 6*n - 7)/36. - Rolf Pleisch, Oct 19 2010
a(1)=1, a(2)=9, a(3)=66; for n > 3, a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3). - Harvey P. Dale, Jul 22 2013
a(n) = Sum_{i=0..n-1} 6^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
G.f.: x/((1 - x)^2*(1 - 7*x)). - Colin Barker, Jun 03 2020
E.g.f.: exp(x)*(7*exp(6*x) - 6*x - 7)/36. - Elmo R. Oliveira, Mar 29 2025

A014881 a(1)=1, a(n) = 11*a(n-1) + n.

Original entry on oeis.org

1, 13, 146, 1610, 17715, 194871, 2143588, 23579476, 259374245, 2853116705, 31384283766, 345227121438, 3797498335831, 41772481694155, 459497298635720, 5054470284992936, 55599173134922313, 611590904484145461, 6727499949325600090, 74002499442581601010
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=11 of A126885.

Programs

  • Magma
    I:=[1, 13, 146]; [n le 3 select I[n] else 13*Self(n-1) - 23*Self(n-2)+ 11*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Maple
    a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,11]])^n)[2,3]:
seq(a(n), n=1..17);  # Alois P. Heinz, Aug 06 2008
  • Mathematica
    LinearRecurrence[{13, -23, 11}, {1, 13, 146}, 20] (* Vincenzo Librandi, Oct 20 2012 *)

Formula

a(n) = 13*a(n-1) - 23*a(n-2) + 11*a(n-3), with a(1)=1, a(2)=13, a(3)=146. - Vincenzo Librandi, Oct 20 2012
G.f.: x/((1-11*x)*(1-x)^2). - Jinyuan Wang, Mar 11 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: exp(x)*(11*exp(10*x) - 10*x - 11)/100.
a(n) = (11^(n+1) - 10*n - 11)/100. (End)

A014896 a(1) = 1, a(n) = 13*a(n-1) + n.

Original entry on oeis.org

1, 15, 198, 2578, 33519, 435753, 5664796, 73642356, 957350637, 12445558291, 161792257794, 2103299351334, 27342891567355, 355457590375629, 4620948674883192, 60072332773481512, 780940326055259673, 10152224238718375767, 131978915103338884990, 1715725896343405504890
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=13 of A126885.

Programs

  • Magma
    I:=[1, 15, 198]; [n le 3 select I[n] else 15*Self(n-1) - 27*Self(n-2)+ 13*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Maple
    a:=n->sum((13^(n-j)-1)/12,j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 05 2007
a:= n-> (Matrix([[1,0,1],[1,1,1],[0,0,13]])^n)[2,3]:
seq(a(n), n=1..17);  # Alois P. Heinz, Aug 06 2008
  • Mathematica
    LinearRecurrence[{15, -27, 13}, {1, 15, 198}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
  • Maxima
    a[1]:1$
a[2]:15$
a[3]:198$
a[n]:=15*a[n-1]-27*a[n-2]+13*a[n-3]$
A014896(n):=a[n]$ makelist(A014896(n),n,1,30); /* Martin Ettl, Nov 07 2012 */

Formula

a(n) = 15*a(n-1) - 27*a(n-2) + 13*a(n-3), with a(1)=1, a(2)=15, a(3)=198. - Vincenzo Librandi, Oct 20 2012
G.f.: x/((1-13*x)*(1-x)^2). - Jinyuan Wang, Mar 11 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: exp(x)*(13*exp(12*x) - 12*x - 13)/144.
a(n) = (13^(n+1) - 12*n - 11)/144. (End)

A014897 a(1)=1, a(n) = 14*a(n-1) + n.

Original entry on oeis.org

1, 16, 227, 3182, 44553, 623748, 8732479, 122254714, 1711566005, 23961924080, 335466937131, 4696537119846, 65751519677857, 920521275490012, 12887297856860183, 180422169996042578, 2525910379944596109, 35362745319224345544, 495078434469140837635, 6931098082567971726910
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=14 of A126885.

Programs

  • Magma
    I:=[1, 16, 227]; [n le 3 select I[n] else 16*Self(n-1) - 29*Self(n-2) + 14*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Mathematica
    LinearRecurrence[{16, -29, 14}, {1, 16, 227}, 20] (* Vincenzo Librandi, Oct 20 2012 *)

Formula

a(1)=1, a(2)=16, a(3)=227, a(n) = 16*a(n-1) - 29*a(n-2) + 14*a(n-3). - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-14*x)*(1-x)^2).
E.g.f.: exp(x)*(14*exp(13*x) - 13*x - 14)/169.
a(n) = (14^(n+1) - 13*n - 14)/169. (End)

A014898 a(1)=1, a(n) = 15*a(n-1) + n.

Original entry on oeis.org

1, 17, 258, 3874, 58115, 871731, 13075972, 196139588, 2942093829, 44131407445, 661971111686, 9929566675302, 148943500129543, 2234152501943159, 33512287529147400, 502684312937211016, 7540264694058165257, 113103970410872478873, 1696559556163087183114, 25448393342446307746730
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=15 of A126885.

Programs

  • Magma
    I:=[1, 17, 258]; [n le 3 select I[n] else 17*Self(n-1) - 31*Self(n-2) + 15*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Mathematica
    LinearRecurrence[{17, -31, 15}, {1, 17, 258}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
nxt[{n_,a_}]:={n+1,15a+n+1}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Jun 15 2025 *)

Formula

a(n) = 17*a(n-1) - 31*a(n-2) + 15*a(n-3); a(1)=1, a(2)=17, a(3)=258. - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-15*x)*(1-x)^2).
E.g.f.: exp(x)*(15*exp(14*x) - 14*x - 15)/196.
a(n) = (15^(n+1) - 14*n - 15)/196. (End)

A014899 a(n) = (16^(n+1) - 15*n - 16)/225.

Original entry on oeis.org

0, 1, 18, 291, 4660, 74565, 1193046, 19088743, 305419896, 4886718345, 78187493530, 1250999896491, 20015998343868, 320255973501901, 5124095576030430, 81985529216486895, 1311768467463790336, 20988295479420645393, 335812727670730326306, 5373003642731685220915
Offset: 0

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=16 of A126885.

Programs

  • Magma
    I:=[0, 1, 18]; [n le 3 select I[n] else 18*Self(n-1) - 33*Self(n-2) + 16*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Maple
    a:=n->sum((16^(n-j)-1)/15,j=0..n): seq(a(n), n=1..16); # Zerinvary Lajos, Jan 05 2007
n0:=20: tabl:=array(1..n0-1): for n from 0 to n0 do: tabl[n+1]:=(4^(2*n+2) - 15*n - 16)/225:od:print( tabl): # Michel Lagneau, Apr 26 2010
  • Mathematica
    s=0;lst={};Do[AppendTo[lst,s+=s+=s+=s+=s+=n],{n,5!}];lst/16 (* Vladimir Joseph Stephan Orlovsky, Oct 20 2009 *)
Table[(16^(n+1)-15*n-16)/225,{n,0,20}] (* Harvey P. Dale, Dec 20 2010 *)
LinearRecurrence[{18, -33, 16}, {0, 1, 18}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
  • Maxima
    A014899(n):=(16^(n+1)-15*n-16)/225$ makelist(A014899(n),n,0,30); /* Martin Ettl, Nov 07 2012 */
  • PARI
    a(n)=(16^(n+1)-15*n)\225 \\ Charles R Greathouse IV, May 15 2013

Formula

a(n) = 16*a(n-1) + n = 18*a(n-1) - 33*a(n-2) + 16*a(n-3).
G.f.: x/((1-16*x)*(x-1)^2). - R. J. Mathar, Apr 29 2010
E.g.f.: exp(x)*(16*exp(15*x) - 15*x - 16)/225. - Elmo R. Oliveira, Mar 31 2025

Extensions

a(0) added by R. J. Mathar, Apr 29 2010

A014903 a(1)=1, a(n) = 19*a(n-1) + n.

Original entry on oeis.org

1, 21, 402, 7642, 145203, 2758863, 52418404, 995949684, 18923044005, 359537836105, 6831218886006, 129793158834126, 2466070017848407, 46855330339119747, 890251276443275208, 16914774252422228968, 321380710796022350409, 6106233505124424657789, 116018436597364068498010
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=19 of A126885.

Programs

  • Magma
    I:=[1, 21, 402]; [n le 3 select I[n] else 21*Self(n-1) - 39*Self(n-2) + 19*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Mathematica
    LinearRecurrence[{21, -39, 19}, {1, 21, 402}, 20] (* Vincenzo Librandi, Oct 20 2012 *)

Formula

a(1)=1, a(2)=21, a(3)=402, a(n) = 21*a(n-1) - 39*a(n-2) + 19*a(n-3). - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-19*x)*(1-x)^2).
E.g.f.: exp(x)*(19*exp(18*x) - 18*x - 19)/324.
a(n) = (19^(n+1) - 18*n - 19)/324. (End)

A014904 a(1)=1, a(n) = 20*a(n-1) + n.

Original entry on oeis.org

1, 22, 443, 8864, 177285, 3545706, 70914127, 1418282548, 28365650969, 567313019390, 11346260387811, 226925207756232, 4538504155124653, 90770083102493074, 1815401662049861495, 36308033240997229916, 726160664819944598337, 14523213296398891966758, 290464265927977839335179
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Links

Crossrefs

Row n=20 of A126885.

Programs

  • Magma
    I:=[1, 22, 443]; [n le 3 select I[n] else 22*Self(n-1) - 41*Self(n-2) + 20*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012
  • Mathematica
    LinearRecurrence[{22, -41, 20}, {1, 22, 443}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
  • Maxima
    a[1]:1$
a[2]:22$
a[3]:443$
a[n]:=22*a[n-1]-41*a[n-2]+20*a[n-3]$
A014904(n):=a[n]$
makelist(A014904(n),n,1,30); /* Martin Ettl, Nov 06 2012 */
  • PARI
    Vec(x/((1-20*x)*(x-1)^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 05 2024

Formula

From R. J. Mathar, Jul 15 2010: (Start)
G.f.: x/((1-20*x)*(x-1)^2).
a(n) = 22*a(n-1) - 41*a(n-2) + 20*a(n-3). (End)
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: exp(x)*(20*exp(19*x) - 19*x - 20)/361.
a(n) = (20^(n+1) - 19*n - 20)/361. (End)
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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