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.

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A140142 a(1)=1, a(n)=a(n-1)+n^0 if n odd, a(n)=a(n-1)+ n^4 if n is even.

Original entry on oeis.org

1, 17, 18, 274, 275, 1571, 1572, 5668, 5669, 15669, 15670, 36406, 36407, 74823, 74824, 140360, 140361, 245337, 245338, 405338, 405339, 639595, 639596, 971372, 971373, 1428349, 1428350, 2043006, 2043007, 2853007, 2853008, 3901584, 3901585
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Maple
    a:= n-> (Matrix([[275,274,18,17, 1,0,0,-1,-17, -18,-274]]). Matrix(11, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,5,-5,-10,10,10, -10,-5,5,1,-1][i] else 0 fi)^n)[1,6]: seq(a(n), n=1..33); # Alois P. Heinz, Aug 06 2008
  • Mathematica
    a = {}; r = 0; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
nxt[{n_,a_}]:={n+1,If[OddQ[n+1],a+1,a+(n+1)^4]}; Transpose[ NestList[ nxt,{1,1},40]][[2]] (* Harvey P. Dale, Dec 24 2012 *)

Formula

O.g.f.: x*(x^8+16*x^7-4*x^6+176*x^5+6*x^4+176*x^3-4*x^2+16*x+1)/((-1+x)^6*(1+x)^5) - R. J. Mathar, May 17 2008

A140145 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^3 if n is even.

Original entry on oeis.org

1, 9, 12, 76, 81, 297, 304, 816, 825, 1825, 1836, 3564, 3577, 6321, 6336, 10432, 10449, 16281, 16300, 24300, 24321, 34969, 34992, 48816, 48841, 66417, 66444, 88396, 88425, 115425, 115456, 148224, 148257, 187561, 187596, 234252, 234289, 289161
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 1; s = 3; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)

Formula

a(n)=a(n-1)+4a(n-2)-4a(n-3)-6a(n-4)+6a(n-5)+4a(n-6)-4a(n-7)-a(n-8)+a(n-9). G.f.: -x*(1+8*x-x^2+32*x^3-x^4+8*x^5+x^6)/((1+x)^4*(x-1)^5). [From R. J. Mathar, Feb 22 2009]

A140146 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^4 if n is even.

Original entry on oeis.org

1, 17, 20, 276, 281, 1577, 1584, 5680, 5689, 15689, 15700, 36436, 36449, 74865, 74880, 140416, 140433, 245409, 245428, 405428, 405449, 639705, 639728, 971504, 971529, 1428505, 1428532, 2043188, 2043217, 2853217, 2853248, 3901824, 3901857
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 1; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[OddQ[n+1],a+n+1,a+(n+1)^4]}; Transpose[NestList[nxt,{1,1},40]][[2]] (* Harvey P. Dale, Mar 19 2013 *)

Formula

G.f.: -x*(x^2+1)*(x^6-16*x^5-3*x^4-160*x^3+3*x^2-16*x-1)/((1+x)^5*(x-1)^6). [From R. J. Mathar, Feb 22 2009]

A140147 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^5 if n is even.

Original entry on oeis.org

1, 33, 36, 1060, 1065, 8841, 8848, 41616, 41625, 141625, 141636, 390468, 390481, 928305, 928320, 1976896, 1976913, 3866481, 3866500, 7066500, 7066521, 12220153, 12220176, 20182800, 20182825, 32064201, 32064228, 49274596, 49274625
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 1; s = 5; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:=If[OddQ[n+1],{n+1,a+n+1},{n+1,a+(n+1)^5}]; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* Harvey P. Dale, Jun 27 2012 *)

Formula

G.f.: -x*(1+32*x-3*x^2+832*x^3+2*x^4+2112*x^5+2*x^6+832*x^7-3*x^8+32*x^9+x^10)/ ((1+x)^6*(x-1)^7). [From R. J. Mathar, Feb 22 2009]

A140148 a(1)=1, a(n)=a(n-1)+n^2 if n odd, a(n)=a(n-1)+ n^0 if n is even.

Original entry on oeis.org

1, 2, 11, 12, 37, 38, 87, 88, 169, 170, 291, 292, 461, 462, 687, 688, 977, 978, 1339, 1340, 1781, 1782, 2311, 2312, 2937, 2938, 3667, 3668, 4509, 4510, 5471, 5472, 6561, 6562, 7787, 7788, 9157, 9158, 10679, 10680, 12361, 12362, 14211, 14212, 16237
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 2; s = 0; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^2,a+1]}; NestList[nxt,{1,1},50][[All,2]] (* Harvey P. Dale, Sep 05 2021 *)

Formula

a(n)=a(n-1)+3a(n-2)-3a(n-3)-3a(n-4)+3a(n-5)+a(n-6)-a(n-7). G.f.: x*(1+x+6*x^2-2*x^3+x^4+x^5)/((1+x)^3*(x-1)^4). [From R. J. Mathar, Feb 22 2009]

A140149 a(1)=1, a(n)=a(n-1)+n^2 if n odd, a(n)=a(n-1)+ n^3 if n is even.

Original entry on oeis.org

1, 9, 18, 82, 107, 323, 372, 884, 965, 1965, 2086, 3814, 3983, 6727, 6952, 11048, 11337, 17169, 17530, 25530, 25971, 36619, 37148, 50972, 51597, 69173, 69902, 91854, 92695, 119695, 120656, 153424, 154513, 193817, 195042, 241698, 243067
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 2; s = 3; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^2,a+(n+1)^3]}; NestList[nxt,{1,1},40][[;;,2]] (* or *) LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,9,18,82,107,323,372,884,965},40] (* Harvey P. Dale, May 27 2024 *)

Formula

From R. J. Mathar, Feb 22 2009: (Start)
G.f.: x*(-1-8*x-5*x^2-32*x^3+5*x^4-8*x^5+x^6)/((1+x)^4*(x-1)^5).
a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9). (End)

A140151 a(1)=1, a(n)=a(n-1)+n^2 if n odd, a(n)=a(n-1)+ n^5 if n is even.

Original entry on oeis.org

1, 33, 42, 1066, 1091, 8867, 8916, 41684, 41765, 141765, 141886, 390718, 390887, 928711, 928936, 1977512, 1977801, 3867369, 3867730, 7067730, 7068171, 12221803, 12222332, 20184956, 20185581, 32066957, 32067686, 49278054, 49278895
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 2; s = 5; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^2,a+(n+1)^5]}; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* Harvey P. Dale, Aug 20 2015 *)

Formula

G.f.: x*(-1-32*x-3*x^2-832*x^3+14*x^4-2112*x^5-14*x^6-832*x^7+3*x^8-32*x^9+x^10 )/((1+x)^6*(x-1)^7). [From R. J. Mathar, Feb 22 2009]

A140153 a(1)=1, a(n) = a(n-1) + n^3 if n odd, a(n) = a(n-1) + n^1 if n is even.

Original entry on oeis.org

1, 3, 30, 34, 159, 165, 508, 516, 1245, 1255, 2586, 2598, 4795, 4809, 8184, 8200, 13113, 13131, 19990, 20010, 29271, 29293, 41460, 41484, 57109, 57135, 76818, 76846, 101235, 101265, 131056, 131088, 167025, 167059, 209934, 209970, 260623
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • GAP
    a:=[1];; for n in [2..40] do a[n]:=a[n-1]+((1-(-1)^n)/2)*n^3+((1+(-1)^n)/2)*n; od; a; # Muniru A Asiru, Jul 12 2018
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5))); // G. C. Greubel, Jul 12 2018
  • Maple
    a:=proc(n) option remember: if n=1 then 1 elif modp(n,2)<>0 then procname(n-1)+n^3 else procname(n-1)+n; fi: end; seq(a(n),n=1..40); # Muniru A Asiru, Jul 12 2018
  • Mathematica
    a = {}; r = 3; s = 1; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
CoefficientList[Series[x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5), {x,0,30}], x] (* G. C. Greubel, Jul 12 2018 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^3,a+n+1]}; NestList[nxt,{1,1},40][[All,2]] (* or *) LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,3,30,34,159,165,508,516,1245},40] (* Harvey P. Dale, Aug 26 2021 *)
  • PARI
    x='x+O('x^30); Vec(x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5)) \\ G. C. Greubel, Jul 12 2018

Formula

a(n) = a(n-1) + {[1-(-1)^n]/2}*n^3 + {[1+(-1)^n]/2}*n, with a(1)=1.
From R. J. Mathar, Feb 22 2009: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
G.f.: x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5). (End)

A140154 a(1)=1, a(n) = a(n-1) + n^3 if n odd, a(n) = a(n-1) + n^2 if n is even.

Original entry on oeis.org

1, 5, 32, 48, 173, 209, 552, 616, 1345, 1445, 2776, 2920, 5117, 5313, 8688, 8944, 13857, 14181, 21040, 21440, 30701, 31185, 43352, 43928, 59553, 60229, 79912, 80696, 105085, 105985, 135776, 136800, 172737, 173893, 216768, 218064, 268717
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • GAP
    a:=[1];; for n in [2..40] do a[n]:=a[n-1]+((1-(-1)^n)/2)*n^3+((1+(-1)^n)/2)*n^2; od; a; # Muniru A Asiru, Jul 12 2018
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(x^2+1)*(x^4-4*x^3+22*x^2+4*x+1)/((1+x)^4*(1-x)^5))); // G. C. Greubel, Jul 12 2018
  • Maple
    a:=proc(n) option remember: if n=1 then 1 elif modp(n,2)<>0 then procname(n-1)+n^3 else procname(n-1)+n^2; fi: end; seq(a(n),n=1..40); # Muniru A Asiru, Jul 12 2018
  • Mathematica
    a = {}; r = 3; s = 2; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
CoefficientList[Series[x*(x^2+1)*(x^4-4*x^3+22*x^2+4*x+1)/((1+x)^4*(1-x)^5), {x,0,30}], x] (* G. C. Greubel, Jul 12 2018 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^3,a+(n+1)^2]}; NestList[nxt,{1,1},40][[All,2]] (* Harvey P. Dale, Aug 05 2019 *)
  • PARI
    x='x+O('x^30); Vec(x*(x^2+1)*(x^4-4*x^3+22*x^2+4*x+1)/((1+x)^4*(1-x)^5)) \\ G. C. Greubel, Jul 12 2018

Formula

a(n) = a(n-1) + {[1-(-1)^n]/2}*n^3 + {[1+(-1)^n]/2}*n^2, with a(1)=1.
From R. J. Mathar, Feb 22 2009: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
G.f.: x*(x^2+1)*(x^4-4*x^3+22*x^2+4*x+1)/((1+x)^4*(1-x)^5). (End)

A140155 a(1)=1, a(n)=a(n-1)+n^3 if n odd, a(n)=a(n-1)+ n^4 if n is even.

Original entry on oeis.org

1, 17, 44, 300, 425, 1721, 2064, 6160, 6889, 16889, 18220, 38956, 41153, 79569, 82944, 148480, 153393, 258369, 265228, 425228, 434489, 668745, 680912, 1012688, 1028313, 1485289, 1504972, 2119628, 2144017, 2954017, 2983808, 4032384
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 3; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^3,a+(n+1)^4]}; NestList[nxt,{1,1},40][[;;,2]] (* Harvey P. Dale, Oct 22 2023 *)

Formula

G.f.: -x*(x^2+1)*(x^6-16*x^5+21*x^4-160*x^3-21*x^2-16*x-1)/((1+x)^5*(x-1)^6). [From R. J. Mathar, Feb 22 2009]
Previous Showing 11-20 of 31 results. Next

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