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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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]

A140156 a(1)=1, a(n) = a(n-1) + n^3 if n odd, a(n) = a(n-1) + n^5 if n is even.

Original entry on oeis.org

1, 33, 60, 1084, 1209, 8985, 9328, 42096, 42825, 142825, 144156, 392988, 395185, 933009, 936384, 1984960, 1989873, 3879441, 3886300, 7086300, 7095561, 12249193, 12261360, 20223984, 20239609, 32120985, 32140668, 49351036, 49375425
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(1/48)*(9*(-1 +(-1)^n) + 4*(1 -12(-1)^n)*n^2 + 12*(1 -(-1)^n)*n^3 + (16 + 30*(-1)^n)*n^4 + 12*(1 +(-1)^n)*n^5 + 4*n^6): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 3; 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)^3,a+(n+1)^5]}; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* or *) LinearRecurrence[ {1,6,-6,-15, 15,20,-20,-15,15,6,-6,-1,1},{1,33,60,1084,1209,8985,9328, 42096, 42825, 142825,144156, 392988,395185},40] (* Harvey P. Dale, Aug 27 2013 *)
Table[(1/48)*(9*(-1 +(-1)^n) + 4*(1 -12(-1)^n)*n^2 + 12*(1 -(-1)^n)*n^3 + (16 + 30*(-1)^n)*n^4 + 12*(1 +(-1)^n)*n^5 + 4*n^6), {n, 1, 50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1, 50, print1((1/48)*(9*(-1 +(-1)^n) + 4*(1 -12(-1)^n)*n^2 + 12*(1 -(-1)^n)*n^3 + (16 + 30*(-1)^n)*n^4 + 12*(1 +(-1)^n)*n^5 + 4*n^6), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(1 + 32*x + 21*x^2 + 832*x^3 - 22*x^4 + 2112*x^5 - 22*x^6 + 832*x^7 + 21*x^8 + 32*x^9 + x^10)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

A140157 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^0 if n is even.

Original entry on oeis.org

1, 2, 83, 84, 709, 710, 3111, 3112, 9673, 9674, 24315, 24316, 52877, 52878, 103503, 103504, 187025, 187026, 317347, 317348, 511829, 511830, 791671, 791672, 1182297, 1182298, 1713739, 1713740, 2421021, 2421022, 3344543, 3344544, 4530465
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; 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 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 2, 83, 84, 709, 710, 3111, 3112, 9673, 9674, 24315}, 50] (* or *) Table[(1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1,50, print1((1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(1 + x + 76*x^2 - 4*x^3 + 230*x^4 + 6*x^5 + 76*x^6 - 4*x^7 + x^8 + x^9)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140158 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^1 if n is even.

Original entry on oeis.org

1, 3, 84, 88, 713, 719, 3120, 3128, 9689, 9699, 24340, 24352, 52913, 52927, 103552, 103568, 187089, 187107, 317428, 317448, 511929, 511951, 791792, 791816, 1182441, 1182467, 1713908, 1713936, 2421217, 2421247, 3344768, 3344800, 4530721
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; 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 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 3, 84, 88, 713, 719, 3120, 3128, 9689, 9699, 24340}, 50] (* or *) Table[(1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^4,a+n+1]}; NestList[nxt,{1,1},40][[;;,2]] (* Harvey P. Dale, Dec 28 2024 *)
  • PARI
    for(n=1,50, print1((1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(1 + 2*x + 76*x^2 - 6*x^3 + 230*x^4 + 6*x^5 + 76*x^6 - 2*x^7 + x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009
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