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 1-9 of 9 results.

A152725 a(n) = n*(n+1)*(n^4 + 2*n^3 - 2*n^2 - 3*n + 3)/2.

Original entry on oeis.org

0, 1, 63, 666, 3430, 12195, 34461, 83188, 178956, 352485, 647515, 1124046, 1861938, 2964871, 4564665, 6825960, 9951256, 14186313, 19825911, 27219970, 36780030, 48986091, 64393813, 83642076, 107460900, 136679725, 172236051, 215184438
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Links

Crossrefs

Cf. A062392, A062393 (for 5th powers), A011934, A152726 (for 7th powers).

Programs

  • Magma
    [n*(n+1)*(n^4+2*n^3-2*n^2-3*n+3)/2: n in [0..50]]; // G. C. Greubel, Sep 01 2018
  • Mathematica
    k=0;lst={k};Do[k=n^6-k;AppendTo[lst,k],{n,1,5!}];lst
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,1,63,666,3430,12195,34461}, 50] (* G. C. Greubel, Sep 01 2018 *)
CoefficientList[Series[-((x (1+56 x+246 x^2+56 x^3+x^4))/(-1+x)^7),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2024 *)
  • PARI
    a(n)=n*(n+1)*(n^4+2*n^3-2*n^2-3*n+3)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = n^6 - (n-1)^6 + (n-2)^6 - ... + ((-1)^n)*0^6.
G.f.: x*(1 + 56*x + 246*x^2 + 56*x^3 + x^4) / (1-x)^7. - R. J. Mathar, Jul 08 2013
a(n) = A050492(A000217(n)). - Kelvin Voskuijl, Jun 18 2025
E.g.f.: exp(x)*x*(2 + 61*x + 160*x^2 + 95*x^3 + 18*x^4 + x^5)/2. - Stefano Spezia, Jun 19 2025

Extensions

Offset set to 0 by R. J. Mathar, Aug 15 2010

A255177 Second differences of seventh powers (A001015).

Original entry on oeis.org

1, 126, 1932, 12138, 47544, 140070, 341796, 730002, 1412208, 2531214, 4270140, 6857466, 10572072, 15748278, 22780884, 32130210, 44327136, 59978142, 79770348, 104476554, 134960280, 172180806, 217198212, 271178418
Offset: 0

Views

Author

Luciano Ancora, Feb 21 2015

Keywords

Examples

			Second differences:  1, 126, 1932, 12138,  47544, ... (this sequence)
First differences:   1, 127, 2060, 14324,  63801, ... (A152726)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ... (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ... (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ... (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ... (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ... (A254646)
Fifth partial sums:  1, 133, 2842, 29274, 197400, ... (A254684)

Links

Crossrefs

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254684, A255178, A255179.

Programs

  • Magma
    [1] cat [14*(-1+n)*(9-22*n+23*n^2-12*n^3+3*n^4): n in [2..30]]; // Vincenzo Librandi, Mar 12 2015
  • Mathematica
    Join[{1}, Table[14 n (3 n^4 + 5 n^2 + 1), {n, 1, 30}], {n, 0, 24}] (* or *)
CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(1 - x)^6, {x, 0, 22}], x]
  • Python
    def A255177(n): return 14*n*(n**2*(3*n**2 + 5) + 1) if n else 1 # Chai Wah Wu, Oct 07 2024

Formula

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^6.
a(n) = 14*n*(3*n^4 + 5*n^2 + 1) for n>0, a(0)=1.
a(n) = A022523(n)-A022523(n-1). - R. J. Mathar, Jul 16 2015

Extensions

Edited by Bruno Berselli, Mar 19 2015

A254684 Fifth partial sums of seventh powers (A001015).

Original entry on oeis.org

1, 133, 2842, 29274, 197400, 1001952, 4137966, 14597934, 45454773, 127861825, 330540028, 795609724, 1801339176, 3867558072, 7926516900, 15591322404, 29566276257, 54259095093, 96674782246, 167695627750, 283882296880
Offset: 1

Views

Author

Luciano Ancora, Feb 12 2015

Keywords

Examples

			First differences:   1, 127, 2060, 14324,  63801, ...  (A152726)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (A254646)
Fifth partial sums:  1, 133, 2842, 29274, 197400, ...  (this sequence)

Links

Crossrefs

Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254681, A254682, A254683.

Programs

  • Mathematica
    Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 3 + 5 n + n^2) (- 2 + 5 n + n^2) (5 + 5 n + n^2)/95040, {n,21}] (* or *)
CoefficientList[Series[(- 1 - 120 x - 1191 x^2 - 2416 x^3 - 1191 x^4 - 120 x^5 - x^6)/(-1 + x)^13, {x,0,20}], x]
  • PARI
    a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-3+5*n+n^2)*(-2+5*n+n^2)*(5+5*n+n^2)/95040 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (- x - 120*x^2 - 1191*x^3 - 2416*x^4 - 1191*x^5 - 120*x^6 - x^7)/(- 1 + x)^13.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(-3 + 5*n + n^2)*(-2 + 5*n + n^2)*(5 + 5*n + n^2)/95040.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^7.

A152729 a(n) = (n-2)^4 - a(n-1) - a(n-2), with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 15, 65, 176, 384, 736, 1281, 2079, 3201, 4720, 6720, 9296, 12545, 16575, 21505, 27456, 34560, 42960, 52801, 64239, 77441, 92576, 109824, 129376, 151425, 176175, 203841, 234640, 268800, 306560, 348161, 393855, 443905, 498576, 558144
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

a(n+2) - a(n-1) = n^4 - (n-1)^4 = A005917(n) for all n in Z. - Michael Somos, Sep 02 2018

Examples

			0 + 0 + 1 = 1^4; 0 + 1 + 15 = 2^4; 1 + 15 + 65 = 3^4; ...
G.f. = x^3 + 15*x^4 + 65*x^5 + 176*x^6 + 384*x^7 + 736*x^8 + 1281*x^9 + ... - _Michael Somos_, Sep 02 2018

Links

Crossrefs

Cf. A005917, A152728, A152725, A152726, A000212.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(x+1)*(x^2+10*x+1)/((1-x)^5*(x^2+x+1)))); // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^4-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,4!}];lst
LinearRecurrence[{4,-6,5,-5,6,-4,1}, {0,0,1,15,65,176,384}, 50] (* G. C. Greubel, Sep 01 2018 *)
a[ n_] := With[ {m = Max[n, 2 - n]}, SeriesCoefficient[ x^3 (1 + x) (1 + 10 x + x^2) / ((1 - x)^5 (1 + x + x^2)), {x , 0, m}]]; (* Michael Somos, Sep 02 2018 *)
  • PARI
    concat([0,0], Vec(-x^3*(x+1)*(x^2+10*x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Oct 28 2014
  • PARI
    {a(n) = my(m = max(n, 2 - n)); polcoeff( x^3 * (1 + x) * (1 + 10*x + x^2) / ((1 - x)^5 * (1 + x + x^2)) + x * O(x^m), m)}; /* Michael Somos, Sep 02 2018 */

Formula

G.f.: -x^3*(x+1)*(x^2+10*x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Oct 28 2014
a(n) = a(2 - n) for all n in Z. - Michael Somos, Sep 02 2018

Extensions

Definition adapted to offset by Georg Fischer, Jun 18 2021

A152730 a(n) + a(n+1) + a(n+2) = n^5, with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 31, 211, 782, 2132, 4862, 9813, 18093, 31143, 50764, 79144, 118924, 173225, 245675, 340475, 462426, 616956, 810186, 1048957, 1340857, 1694287, 2118488, 2623568, 3220568, 3921489, 4739319, 5688099, 6782950, 8040100, 9476950
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Examples

			0 + 0 + 1 = 1^5; 0 + 1 + 31 = 2^5; 1 + 31 + 211 = 3^5; ...

Links

Crossrefs

Cf. A152728, A152729, A152725, A152726, A000212.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(x^4+26*x^3+66*x^2+26*x+1)/((x-1)^6*(x^2+x+1)))); // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^5-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{5,-10,11,-10,11,-10,5,-1}, {0,0,1,31,211,782,2132, 4862}, 50] (* G. C. Greubel, Sep 01 2018 *)
CoefficientList[Series[x^2*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1) / ((x - 1)^6*(x^2 + x + 1)),{x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)
  • PARI
    concat([0,0], Vec(x^3*(x^4+26*x^3+66*x^2+26*x+1)/((x-1)^6*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Oct 28 2014

Formula

G.f.: x^3*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1) / ((x-1)^6*(x^2 + x + 1)). - Colin Barker, Oct 28 2014

A152731 a(n) + a(n+1) + a(n+2) = n^6, a(1)=a(2)=0.

Original entry on oeis.org

0, 0, 1, 63, 665, 3368, 11592, 31696, 74361, 156087, 300993, 542920, 927648, 1515416, 2383745, 3630375, 5376505, 7770336, 10990728, 15251160, 20803993, 27944847, 37017281, 48417776, 62600832, 80084368, 101455425, 127375983
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

0 + 0 + 1 = 1^6; 0 + 1 + 63 = 2^6; ...

Links

Crossrefs

Cf. A152728, A152729, A152730, A152725, A152726, A000212.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(1+x)*(x^4+56*x^3+246*x^2+56*x+1)/((1-x)^7*(1 +x+ x^2)))); // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^6-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{6, -15, 21, -21, 21, -21, 15, -6, 1}, {0, 0, 1, 63, 665, 3368, 11592, 31696, 74361}, 5000]
CoefficientList[Series[x^2*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)),{x, 0, 5000}], x] (* Stefano Spezia, Sep 02 2018 *)
  • PARI
    x='x+O('x^30); concat([0,0], Vec(x^3*(1+x)*(x^4+56*x^3 +246*x^2 +56*x+1)/((1-x)^7*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018

Formula

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = -26*n/3 + 20*n^3/3 - 5*n^2 + 7/3 - 2*n^5 + n^6/3 + 5*n^4/3 - 7*A131713(n)/3.
G.f.: x^3*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)). (End)

A152732 a(n) + a(n+1) + a(n+2) = 2^n.

Original entry on oeis.org

0, 0, 2, 2, 4, 10, 18, 36, 74, 146, 292, 586, 1170, 2340, 4682, 9362, 18724, 37450, 74898, 149796, 299594, 599186, 1198372, 2396746, 4793490, 9586980, 19173962, 38347922, 76695844, 153391690, 306783378, 613566756, 1227133514, 2454267026, 4908534052
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

0 + 0 + 2 = 2^1; 0 + 2 + 2 = 2^2; 2 + 2 + 4 = 2^3; 2 + 4 + 10 = 2^4; ...
With a(0)=1, a(n) is the number of length n strings in the language over alphabet {0,1} generated by the regular expression: ((0+1)(0*(11)*)*10)*. - Geoffrey Critzer, Jan 25 2014

Links

Crossrefs

Cf. A152728, A152729, A152730, A152731, A152725, A152726, A000212.

Programs

  • Magma
    I:=[0,0,2]; [n le 3 select I[n] else Self(n-1) +Self(n-2) +2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=2^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{1, 1, 2}, {0, 0, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
  • PARI
    concat([0,0],Vec(2/(1-2*x)/(1+x+x^2)+O(x^99))) \\ Charles R Greathouse IV, Feb 24 2012

Formula

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 2*A077947(n-3).
G.f.: 2*x^3/((1-2*x)*(1+x+x^2)). (End)
a(n) = (1/21)*(3*2^n + 18*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)). - Zak Seidov, Dec 12 2008

A152733 a(n) + a(n+1) + a(n+2) = 3^n.

Original entry on oeis.org

0, 0, 3, 6, 18, 57, 168, 504, 1515, 4542, 13626, 40881, 122640, 367920, 1103763, 3311286, 9933858, 29801577, 89404728, 268214184, 804642555, 2413927662, 7241782986, 21725348961, 65176046880, 195528140640, 586584421923, 1759753265766, 5279259797298
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Examples

			0 + 0 + 3 = 3^1; 0 + 3 + 6 = 3^2; 3 + 6 + 18 = 3^3; ...

Links

Crossrefs

Cf. A152728, A152729, A152730, A152731, A152732, A152725, A152726, A000212.

Programs

  • Magma
    [n le 2 select 0 else 3^(n-2) -Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 31 2014
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=3^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
Rest[CoefficientList[Series[3x^3/((1-3x)(1+x+x^2)),{x,0,30}],x]] (* Harvey P. Dale, Aug 31 2014 *)
  • PARI
    x='x+O('x^30); concat([0,0], Vec(3*x^3/((1-3*x)*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018

Formula

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 3*A077834(n-3).
G.f.: 3*x^3/((1-3*x)*(1+x+x^2)). (End)
a(n) = (1/13)*(3^n + 12*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)), n=1,2,... - Zak Seidov, Dec 12 2008

A261032 a(n) = (-1)^n*(n^8 + 4*n^7 - 14*n^5 + 28*n^3 - 17*n)/2.

Original entry on oeis.org

0, -1, 255, -6306, 59230, -331395, 1348221, -4416580, 12360636, -30686085, 69313915, -145044966, 284936730, -530793991, 944995065, -1617895560, 2677071736, -4298685705, 6721274871, -10262288170, 15337711830, -22485147531, 32390726005, -45920259276, 64155054900, -88432835725
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2015

Keywords

Comments

Alternating sum of eighth powers (A001016).
For n>0, a(n) is divisible by A000217(n).

Examples

			a(0) = 0^8 = 0,
a(1) = 0^8 -1^8 = -1,
a(2) = 0^8 -1^8 + 2^8 = 255,
a(3) = 0^8 -1^8 + 2^8 - 3^8 = -6306,
a(4) = 0^8 -1^8 + 2^8 - 3^8 + 4^8 = 59230,
a(5) = 0^8 -1^8 + 2^8 - 3^8 + 4^8 - 5^8 = -331395, etc.

Links

Crossrefs

Cf. A000217, A001016, A000542, A089594, A232599, A062392, A062393, A152725, A152726.

Programs

  • Magma
    [(-1)^n*(n^8+4*n^7-14*n^5+28*n^3-17*n)/2: n in [0..30]]; // Vincenzo Librandi, Nov 20 2015
  • Maple
    seq((-1)^n*(n^8 + 4*n^7 - 14*n^5 + 28*n^3 - 17*n)/2, n = 0 .. 100); # Robert Israel, Nov 18 2015
  • Mathematica
    Table[(1/2) (-1)^n n (n + 1) (n^6 + 3 n^5 - 3 n^4 - 11 n^3 + 11 n^2 + 17 n - 17), {n, 0, 25}]
  • PARI
    vector(100, n, n--; (-1)^n*(n^8+4*n^7-14*n^5+28*n^3-17*n)/2) \\ Altug Alkan, Nov 18 2015
  • Sage
    [(-1)^n*(n^8 +4*n^7 -14*n^5 +28*n^3 -17*n)/2 for n in (0..40)] # G. C. Greubel, Apr 02 2021

Formula

G.f.: -x*(1 - 246*x + 4047*x^2 - 11572*x^3 + 4047*x^4 - 246*x^5 + x^6)/(1 + x)^9.
a(n) = Sum_{k = 0..n} (-1)^k*k^8.
a(n) = (-1)^n*n*(n + 1)*(n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 11*n^2 + 17*n - 17)/2.
Sum_{n>0} 1/a(n) = -0.9962225712723456482...
Sum_{j=0..9} binomial(9,j)*a(n-j) = 0. - Robert Israel, Nov 18 2015
E.g.f.: (x/2)*(-2 +253*x -1848*x^2 +2961*x^3 -1596*x^4 +350*x^5 -32*x^6 +x^7)*exp(-x). - G. C. Greubel, Apr 02 2021
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