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 12 results. Next

A238824 a(1)=0, a(2)=1; thereafter a(n) = A238823(n-1)-2*a(n-1)-a(n-2).

Original entry on oeis.org

0, 1, 1, 3, 7, 17, 43, 105, 262, 643, 1590, 3911, 9643, 23743, 58502, 144099, 355009, 874545, 2154505, 5307663, 13075683, 32212375, 79356454, 195497421, 481615082, 1186475969, 2922926441, 7200734309, 17739268544, 43701326725, 107659793177, 265223778927
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); [0] cat Coefficients(R! -x^2*(-1+x+2*x^2-2*x^3-x^4+x^6-x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1;
t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1;
t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
  • Mathematica
    CoefficientList[Series[-x (- 1 + x + 2 x^2 - 2 x^3 - x^4 + x^6 - x^5)/((1 + x) (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{2,3,-4,-3,2,4,2,-1},{0,1,1,3,7,17,43,105},40] (* Harvey P. Dale, Dec 26 2023 *)

Formula

G.f.: -x^2*(-1+x+2*x^2-2*x^3-x^4+x^6-x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1) ). - R. J. Mathar, Mar 20 2014

A238829 a(n) = A238823(n) - A238826(n).

Original entry on oeis.org

1, 1, 2, 5, 12, 31, 77, 192, 474, 1170, 2881, 7097, 17477, 43050, 106043, 261235, 643552, 1585421, 3905750, 9621993, 23704161, 58396118, 143860974, 354406732, 873093707, 2150897733, 5298813853, 13053818630, 32158552201, 79223751853, 195170567014, 480809724213
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238828.

Programs

  • Magma
    I:=[1,1,2,5,12,31,77]; [n le 7 select I[n] else 3*Self(n-1)-4*Self(n-3)+Self(n-4)+Self(n-5)+3*Self(n-6)-Self(n-7): n in [1..35]]
  • Maple
    g:=proc(n) option remember; local t1;
t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1;
t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1;
t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
h:=n->p(n+3)-p(n+1);
[seq(h(n),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1;
t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
[0,seq(d(n-1)+p(n),n=2..32)]; #A238828
a:=n->g(n)-h(n);
[seq(a(n),n=1..32)]; #A238829
  • Mathematica
    CoefficientList[Series[(1 - x) (2 x^5 + x^4 + x^3 - 2 x^2 - x + 1)/(1 - 3 x + 4 x^3 - x^4 - x^5 - 3 x^6 + x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{3,0,-4,1,1,3,-1},{1,1,2,5,12,31,77},40] (* Harvey P. Dale, Jun 08 2018 *)

Formula

G.f.: -x*(x-1)*(2*x^5+x^4+x^3-2*x^2-x+1) / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7 ). - R. J. Mathar, Mar 20 2014

A238825 a(1)..a(4) = 0,0,0,1; thereafter a(n) = a(n-2)+a(n-3)+2*(d(n-3)+d(n-4)) where d(n) = A238824(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 11, 27, 64, 158, 387, 956, 2355, 5809, 14313, 35272, 86894, 214075, 527368, 1299185, 3200551, 7884653, 19424072, 47851896, 117884841, 290413626, 715444487, 1762523473, 4342040215, 10696772780, 26351885188, 64918818701
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); [0,0,0] cat Coefficients(R! x^4*(x-1)*(x^3+x^2-1) / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7)); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1;
t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1;
t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1;
t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
  • Mathematica
    CoefficientList[Series[x^3 (x - 1) (x^3 + x^2 - 1)/(1 - 3 x + 4 x^3 - x^4 - x^5 - 3 x^6 + x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{3,0,-4,1,1,3,-1},{0,0,0,1,2,5,11,27},40] (* Harvey P. Dale, Aug 17 2025 *)

Formula

G.f.: x^4*(x-1)*(x^3+x^2-1) / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7 ). - R. J. Mathar, Mar 20 2014

A238826 a(n) = p(n+3)-p(n+1), where p(n) = A238825(n).

Original entry on oeis.org

1, 2, 4, 9, 22, 53, 131, 323, 798, 1968, 4853, 11958, 29463, 72581, 178803, 440474, 1085110, 2673183, 6585468, 16223521, 39967243, 98460769, 242561730, 597559646, 1472109847, 3626595728, 8934249307, 22009844973, 54222045921, 133577963318, 329074124992, 810685962909
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238825.

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); Coefficients(R! -x*(1+x)*(x^3+x^2-1)*(x-1)^2 / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7)); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1;
t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1;
t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1;
t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
[seq(p(n+3)-p(n+1),n=1..32)]; #A238826
  • Mathematica
    CoefficientList[Series[-(1 + x) (x^3 + x^2 - 1) (x - 1)^2/(1 - 3 x + 4 x^3 - x^4 - x^5 - 3 x^6 + x^7), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{3,0,-4,1,1,3,-1},{1,2,4,9,22,53,131},40] (* Harvey P. Dale, Aug 15 2025 *)

Formula

G.f.: -x*(1+x)*(x^3+x^2-1)*(x-1)^2 / ( 1-3*x+4*x^3-x^4-x^5-3*x^6+x^7 ). - R. J. Mathar, Mar 20 2014

A238827 a(n) = 0 for n <= 3; thereafter a(n) = a(n-2)+A238825(n-3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 13, 33, 77, 191, 464, 1147, 2819, 6956, 17132, 42228, 104026, 256303, 631394, 1555488, 3831945, 9440141, 23256017, 57292037, 141140858, 347705663, 856585345, 2110229136, 5198625560, 12807001916, 31550510748, 77725820617, 191480359254, 471718764310, 1162096170669
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238826.

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); [0,0,0,0,0,0] cat Coefficients(R! -x^7*(-1+x^2+x^3) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1;
t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1;
t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1;
t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
[seq(p(n+3)-p(n+1),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1;
t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
  • Mathematica
    CoefficientList[Series[- x^6 (- 1 + x^2 + x^3)/((1 + x) (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{2,3,-4,-3,2,4,2,-1},{0,0,0,0,0,0,1,2,6,13},40] (* Harvey P. Dale, Jun 26 2020 *)

Formula

G.f.: -x^7*(-1+x^2+x^3) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1) ). - R. J. Mathar, Mar 20 2014

A238828 a(0)=0; thereafter a(n) = A238824(n-1)+A238825(n).

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 28, 70, 169, 420, 1030, 2546, 6266, 15452, 38056, 93774, 230993, 569084, 1401913, 3453690, 8508214, 20960336, 51636447, 127208350, 313382262, 772028708, 1901920456, 4685449914, 11542774524, 28436041324, 70053211913, 172578611878
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238827.

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); [0,0] cat Coefficients(R! x^3*(1-2*x^2+2*x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1;
t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1;
t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1;
t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
[seq(p(n+3)-p(n+1),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1;
t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
[0,seq(d(n-1)+p(n),n=2..32)]; #A238828
  • Mathematica
    CoefficientList[Series[x^2 (1 - 2 x^2 + 2 x^5)/((1 + x) (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{2,3,-4,-3,2,4,2,-1},{0,0,1,2,5,12,28,70},40] (* Harvey P. Dale, Aug 29 2023 *)

Formula

G.f.: x^2*(1-2*x^2+2*x^5) / ( (1+x)*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1) ). - R. J. Mathar, Mar 20 2014, adapted to offset Jun 19 2021

Extensions

Offset corrected by N. J. A. Sloane, Jun 16 2021

A238830 a(1)=a(2)=0; thereafter a(n) = a(n-2)+A238828(n-1)+A238827(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 15, 36, 91, 218, 544, 1325, 3281, 8055, 19880, 48930, 120610, 297055, 731922, 1802994, 4441915, 10942602, 26957739, 66410994, 163606230, 403049273, 992926975, 2446110587, 6026082552, 14845470456, 36572353012, 90097307929
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238829.

Programs

  • Maple
    g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1; t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
if n <= 2 then t1[n] else
i(n-2)+b(n-1)+r(n); fi; end proc;
[seq(i(n),n=1..32)]; # A238830
  • Mathematica
    LinearRecurrence[{1,5,-1,-7,-1,6,6,1,-1},{0,0,0,1,2,6,15,36,91},40] (* Harvey P. Dale, Dec 29 2021 *)

Formula

G.f.: x^4*(1+x-x^2+x^5) / ( (x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)*(1+x)^2 ). - R. J. Mathar, Mar 20 2014

A238831 a(n) = 0 if n <= 2; thereafter a(n) = A238827(n) + A238830(n-2).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 8, 21, 49, 124, 295, 735, 1789, 4428, 10874, 26836, 66062, 162838, 401081, 988225, 2434388, 5997403, 14774547, 36397880, 89667011, 220898267, 544190131, 1340632638, 3302695932, 8136311688, 20044096016, 49379354928, 121647818677, 299683787423, 738281805364, 1818783831517
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238830.

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); [0,0,0,0,0] cat Coefficients(R! -x^6*(x-1)*(2*x+1)*(x^2+x+1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1; t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
if n <= 2 then t1[n] else
i(n-2)+b(n-1)+r(n); fi; end proc;
[seq(i(n),n=1..32)]; # A238830
q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
[seq(q(n),n=1..45)]; # A238831
  • Mathematica
    CoefficientList[Series[- x^5 (x - 1) (2 x + 1) (x^2 + x + 1)/((x + 1)^2 (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
  • PARI
    concat([0,0,0,0,0], Vec(-x^6*(x-1)*(2*x+1)*(x^2+x+1)/((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)) + O(x^100))) \\ Colin Barker, Mar 20 2014

Formula

G.f.: -x^6*(x-1)*(2*x+1)*(x^2+x+1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)). - Colin Barker, Mar 20 2014

A238832 a(1)=0; thereafter a(n) = A238824(n-1)+A238830(n-1).

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 23, 58, 141, 353, 861, 2134, 5236, 12924, 31798, 78382, 193029, 475619, 1171600, 2886427, 7110657, 17517598, 43154977, 106314193, 261908415, 645221312, 1589525242, 3915853416, 9646844896, 23765351096, 58546797181, 144232146189, 355321086856, 875346302897, 2156447153427, 5312485264678
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238831.

Programs

  • Maple
    g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1; t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
if n <= 2 then t1[n] else
i(n-2)+b(n-1)+r(n); fi; end proc;
[seq(i(n),n=1..32)]; # A238830
q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
[seq(q(n),n=1..45)]; # A238831
e:=n-> if n<=1 then 0 else d(n-1)+i(n-1); fi;
[seq(e(n),n=1..45)]; # A238832
  • PARI
    concat([0,0], Vec(x^3*(2*x^5+2*x^4+x^3-2*x^2+1)/((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)) + O(x^100))) \\ Colin Barker, Mar 20 2014

Formula

G.f.: x^3*(2*x^5+2*x^4+x^3-2*x^2+1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)). - Colin Barker, Mar 20 2014

A238833 a(n) = n-1 for n <= 2; thereafter a(n) = A238824(n-2) + A238832(n-1).

Original entry on oeis.org

0, 1, 0, 2, 2, 7, 16, 40, 101, 246, 615, 1504, 3724, 9147, 22567, 55541, 136884, 337128, 830628, 2046145, 5040932, 12418320, 30593281, 75367352, 185670647, 457405836, 1126836394, 2776001211, 6838779857, 16847579205, 41504619640, 102248123906, 251891939366, 620544865783, 1528734638988, 3766092860744
Offset: 1

Views

Author

N. J. A. Sloane, Mar 08 2014

Keywords

Links

Crossrefs

Cf. A238823-A238832.

Programs

  • Magma
    m:=40; R:=LaurentSeriesRing(RationalField(), m); [0] cat Coefficients(R! -x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*x^2+x-1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1))); // Vincenzo Librandi, Mar 21 2014
  • Maple
    g:=proc(n) option remember; local t1; t1:=[2,3,6,14,34,84,208,515];
if n <= 7 then t1[n] else
3*g(n-1)-4*g(n-3)+g(n-4)+g(n-5)+3*g(n-6)-g(n-7); fi; end proc;
[seq(g(n),n=1..32)]; # A238823
d:=proc(n) option remember; global g; local t1; t1:=[0,1];
if n <= 2 then t1[n] else
g(n-1)-2*d(n-1)-d(n-2); fi; end proc;
[seq(d(n),n=1..32)]; # A238824
p:=proc(n) option remember; global d; local t1; t1:=[0,0,0,1];
if n <= 4 then t1[n] else
p(n-2)+p(n-3)+2*(d(n-3)+d(n-4)); fi; end proc;
[seq(p(n),n=1..32)]; # A238825
h:=n->p(n+3)-p(n+1); [seq(h(n),n=1..32)]; #A238826
r:=proc(n) option remember; global p; local t1; t1:=[0,0,0,0];
if n <= 4 then t1[n] else
r(n-2)+p(n-3); fi; end proc;
[seq(r(n),n=1..32)]; # A238827
b:=n-> if n=1 then 0 else d(n-1)+p(n); fi; [seq(b(n),n=1..32)]; #A238828
a:=n->g(n)-h(n); [seq(a(n),n=1..32)]; #A238829
i:=proc(n) option remember; global b,r; local t1; t1:=[0,0];
if n <= 2 then t1[n] else
i(n-2)+b(n-1)+r(n); fi; end proc;
[seq(i(n),n=1..32)]; # A238830
q:=n-> if n<=2 then 0 else r(n)+i(n-2); fi;
[seq(q(n),n=1..45)]; # A238831
e:=n-> if n<=1 then 0 else d(n-1)+i(n-1); fi;
[seq(e(n),n=1..45)]; # A238832
j:=n-> if n<=2 then n-1 else d(n-2)+e(n-1); fi;
[seq(j(n),n=1..45)]; # A238833
  • Mathematica
    CoefficientList[Series[- x (x^8 + 2 x^7 + x^6 - 2 x^5 - 2 x^4 - x^3 + 3 x^2 + x - 1)/((x + 1)^2 (x^7 - 3 x^6 - x^5 - x^4 + 4 x^3 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{1,5,-1,-7,-1,6,6,1,-1},{0,1,0,2,2,7,16,40,101,246},40] (* Harvey P. Dale, Jul 23 2021 *)
  • PARI
    concat(0, Vec(-x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*x^2+x-1)/((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)) + O(x^100))) \\ Colin Barker, Mar 20 2014

Formula

G.f.: -x^2*(x^8+2*x^7+x^6-2*x^5-2*x^4-x^3+3*x^2+x-1) / ((x+1)^2*(x^7-3*x^6-x^5-x^4+4*x^3-3*x+1)). - Colin Barker, Mar 20 2014
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