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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A135095 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^2 if n is even.

Original entry on oeis.org

1, 5, 248, 264, 3389, 3425, 20232, 20296, 79345, 79445, 240496, 240640, 611933, 612129, 1371504, 1371760, 2791617, 2791941, 5268040, 5268440, 9352541, 9353025, 15789368, 15789944, 25555569, 25556245, 39905152, 39905936, 60417085
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(1/24)*(3 + 2*n + 5*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 2*n^6 - 3*(-1)^n*(1 + n* (-2 - 7*n + 5*n^3 + 2*n^4))): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 5; 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*)
Table[(1/24)*(3 + 2*n + 5*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 2*n^6 - 3*(-1)^n*(1 + n* (-2 - 7*n + 5*n^3 + 2*n^4))), {n,1,50}] (* G. C. Greubel, Sep 23 2016 *)
  • PARI
    for(n=1,50, print1((1/24)*(3 + 2*n + 5*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 2*n^6 - 3*(-1)^n*(1 + n* (-2 - 7*n + 5*n^3 + 2*n^4))), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(x^8 - 4*x^7 + 236*x^6 + 12*x^5 + 1446*x^4 - 12*x^3 + 236*x^2 + 4*x + 1)*(x^2 + 1)/( (1+x)^6 * (x-1)^7 ). - R. J. Mathar, Feb 22 2009
E.g.f.: (1/24)*( (-3 - 6*x - 17*x^2 + 240*x^3 - 75*x^4 + 6*x^5)*exp(x) + (3 + 24*x + 204*x^2 + 364*x^3 + 195*x^4 + 36*x^5 + 2*x^6)*exp(x) ). - G. C. Greubel, Sep 23 2016

A135099 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^3 if n is even.

Original entry on oeis.org

1, 9, 252, 316, 3441, 3657, 20464, 20976, 80025, 81025, 242076, 243804, 615097, 617841, 1377216, 1381312, 2801169, 2807001, 5283100, 5291100, 9375201, 9385849, 15822192, 15836016, 25601641, 25619217, 39968124, 39990076, 60501225
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*n^2*(n+1)^2*(n^2 +n+1) -6*(-1)^n*n^2*(n + 2)*(2*n^2 +n-4)): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 5; 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
Table[(1/48)*(9*(1 - (-1)^n) + 4*n^2*(n + 1)^2*(n^2 + n + 1) - 6*(-1)^n*n^2*(n + 2)*(2*n^2 + n - 4)), {n, 1, 50}] (* G. C. Greubel, Sep 23 2016 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^5,a+(n+1)^3]}; NestList[nxt,{1,1},30][[All,2]] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,9,252,316,3441,3657,20464,20976,80025,81025,242076,243804,615097},30] (* Harvey P. Dale, Oct 02 2022 *)
  • PARI
    for(n=1,50, print1((1/48)*(9*(1-(-1)^n) +4*n^2*(n+1)^2*(n^2 +n +1) -6*(-1)^n*n^2*(n+2)*(2*n^2 +n-4)), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(1 + 8*x + 237*x^2 + 16*x^3 + 1682*x^4 - 48*x^5 + 1682*x^6 + 16*x^7 + 237*x^8 + 8*x^9 + x^ 10)/((1+x)^6 * (x-1)^7). - R. J. Mathar, Feb 22 2009
E.g.f.: (1/48)*( (-9 - 18*x - 306*x^2 + 468*x^3 - 150*x^4 + 12*x^5)*exp(-x) + (9 + 48*x + 456*x^2 + 768*x^3 + 396*x^4 + 72*x^5 + 4*x^6)*exp(x) ). - G. C. Greubel, Sep 23 2016

A135214 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^4 if n is even.

Original entry on oeis.org

1, 17, 260, 516, 3641, 4937, 21744, 25840, 84889, 94889, 255940, 276676, 647969, 686385, 1445760, 1511296, 2931153, 3036129, 5512228, 5672228, 9756329, 9990585, 16426928, 16758704, 26524329, 26981305, 41330212, 41944868, 62456017
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

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

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6))) // G. C. Greubel, Jul 04 2018
  • Mathematica
    a = {}; r = 5; 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
LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {1, 17, 260, 516, 3641, 4937, 21744, 25840, 84889, 94889, 255940, 276676, 647969}, 50] (* G. C. Greubel, Oct 04 2016 *)
  • PARI
    x='x+O('x^50); Vec(x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6)) \\ G. C. Greubel, Jul 04 2018

Formula

From R. J. Mathar, May 17 2008: (Start)
O.g.f.: x*(1 + x^2)*(x^8 - 16*x^7 + 236*x^6 - 144*x^5 + 1446*x^4 + 144*x^3 + 236*x^2 + 16*x + 1)/((1-x)^7 *(1+x)^6).
a(2*n-1) = n*(-8 + 80*n^2 + 48*n^4 + 80*n^5 + 35*n - 220*n^3)/15.
a(2*n) = n*(-8 + 80*n^2 + 48*n^4 + 80*n^5 + 35*n + 20*n^3)/15 . (End)

A135301 a(1)=1, a(n)=a(n-1)+n^0 if n odd, a(n)=a(n-1)+ n^2 if n is even.

Original entry on oeis.org

1, 5, 6, 22, 23, 59, 60, 124, 125, 225, 226, 370, 371, 567, 568, 824, 825, 1149, 1150, 1550, 1551, 2035, 2036, 2612, 2613, 3289, 3290, 4074, 4075, 4975, 4976, 6000, 6001, 7157, 7158, 8454, 8455, 9899, 9900, 11500, 11501, 13265, 13266, 15202, 15203
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 = 0; 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*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+1,a+(n+1)^2]}; Transpose[NestList[nxt,{1,1},50]][[2]] (* or *) LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,5,6,22,23,59,60},50] (* Harvey P. Dale, Jul 16 2014 *)

Formula

O.g.f.: x*(x^4+4*x^3-2*x^2+4*x+1)/((-1+x)^4*(1+x)^3) . a(2n-1) = 4*n^3/3-2*n^2+5*n/3, a(2n) = 4*n^3/3+2*n^2+5*n/3. - R. J. Mathar, May 17 2008
a(1)=1, a(2)=5, a(3)=6, a(4)=22, a(5)=23, a(6)=59, a(7)=60, a(n)=a(n-1)+ 3*a(n-2)- 3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a (n-7). - Harvey P. Dale, Jul 16 2014
a(n) = ( (2*n+1)*(n^2+n+3)+3*(n^2+n-1)*(-1)^n )/12. - Luce ETIENNE, Jul 26 2014

A135332 a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^3 if n is even.

Original entry on oeis.org

1, 9, 10, 74, 75, 291, 292, 804, 805, 1805, 1806, 3534, 3535, 6279, 6280, 10376, 10377, 16209, 16210, 24210, 24211, 34859, 34860, 48684, 48685, 66261, 66262, 88214, 88215, 115215, 115216, 147984, 147985, 187289, 187290, 233946, 233947, 288819
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,9,10,74,75,291,292,804,805},40] (* Harvey P. Dale, Nov 28 2014 *)

Formula

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 + 8*x - 3*x^2 + 32*x^3 + 3*x^4 +8*x^5 -x^6)/((1+x)^4*(1-x)^5). (End)

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]
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