A152725 -id:A152725 - cp's OEIS frontend

A152726 a(n) = n^7 - (n-1)^7 + (n-2)^7 - ... + ((-1)^n)*0^7.

0, 1, 127, 2060, 14324, 63801, 216135, 607408, 1489744, 3293225, 6706775, 12780396, 23051412, 39697105, 65716399, 105142976, 163292480, 247046193, 365173839, 528697900, 751302100, 1049786441, 1444571447, 1960254000

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Cf. A152725 (6th powers).

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+120*x+1191*x^2+2416*x^3+1191*x^4 +120*x^5+x^6)/( (1+x)*(x-1)^8))); // G. C. Greubel, Sep 01 2018

k=0;lst={k};Do[k=n^7-k;AppendTo[lst,k],{n,1,5!}];lst
LinearRecurrence[{7, -20, 28, -14, -14, 28, -20, 7, -1}, {0, 1, 127, 2060, 14324, 63801, 216135, 607408, 1489744}, 50] (* G. C. Greubel, Sep 01 2018 *)
Table[Total[(Times@@@Partition[Riffle[Range[n,1,-1],{1,-1},{2,-1,2}],2])^7],{n,0,30}] (* Harvey P. Dale, Mar 14 2023 *)

x='x+O('x^50); concat([0], Vec(x*(1+120*x+1191*x^2 +2416*x^3 +1191*x^4+120*x^5+x^6)/((1+x)*(x-1)^8))) \\ G. C. Greubel, Sep 01 2018

G.f.: x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/((1+x)*(x-1)^8). - R. J. Mathar, Jul 08 2013

a(n) = (17*(-1)^n + 84*n^2 - 17 + 28*n^6 + 8*n^7 - 70*n^4)/16. - R. J. Mathar, Jul 08 2013

Offset corrected by R. J. Mathar, Jul 08 2013

A062393 a(n) = n^5 - (n-1)^5 + (n-2)^5 - ... +(-1)^n*0^5.

Original entry on oeis.org

0, 1, 31, 212, 812, 2313, 5463, 11344, 21424, 37625, 62375, 98676, 150156, 221137, 316687, 442688, 605888, 813969, 1075599, 1400500, 1799500, 2284601, 2869031, 3567312, 4395312, 5370313, 6511063, 7837844, 9372524, 11138625, 13161375

Offset: 0

Henry Bottomley, Jun 21 2001

The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus a(k) = |2^(-6)(P(5,1)-(-1)^k P(5,2k+1))|. - Peter Luschny, Jul 12 2009

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Cf. A000539, A000584. A062392 for 4th powers, A152725 for 6th powers.

a := n -> (1-(-1)^n+n^2*(n^2*(2*n+5)-5))/4; # Peter Luschny, Jul 12 2009

k=0;lst={k};Do[k=n^5-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[Total[(Times@@@Partition[Riffle[Range[n,0,-1],{1,-1},{2,-1,2}],2])^5],{n,0,30}] (* or *) LinearRecurrence[ {5,-9,5,5,-9,5,-1},{0,1,31,212,812,2313,5463},40] (* Harvey P. Dale, Feb 01 2013 *)

{ a=0; for (n=0, 1000, write("b062393.txt", n, " ", a=n^5 - a) ) } \\ Harry J. Smith, Aug 07 2009

a(n) = (2*n^5 + 5*n^4 - 5*n^2 + 1 - (-1)^n)/4 = n^5 - a(n-1).
G.f.: x*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1)/((x-1)^6*(x+1)). - Colin Barker, Sep 19 2012
a(0)=0, a(1)=1, a(2)=31, a(3)=212, a(4)=812, a(5)=2313, a(6)=5463, a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7). - Harvey P. Dale, Feb 01 2013

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

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

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

			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

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

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

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

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 *)

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

{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 */

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

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

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

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

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-10,11,-10,5,-1).

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

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

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 *)

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

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

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

nonn

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

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (6,-15,21,-21,21,-21,15,-6,1).

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

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

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 *)

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

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

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

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

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

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 *)

concat([0,0],Vec(2/(1-2*x)/(1+x+x^2)+O(x^99))) \\ Charles R Greathouse IV, Feb 24 2012

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

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

nonn

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 2, 3).

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

[n le 2 select 0 else 3^(n-2) -Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 31 2014

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 *)

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

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

A187620 a(n) = n^6 - a(n-1), a(0)=1.

Original entry on oeis.org

1, 0, 64, 665, 3431, 12194, 34462, 83187, 178957, 352484, 647516, 1124045, 1861939, 2964870, 4564666, 6825959, 9951257, 14186312, 19825912, 27219969, 36780031, 48986090, 64393814, 83642075, 107460901

Offset: 0

Vincenzo Librandi, Mar 12 2011

a(n)+a(n-1) is a perfect 6th power, hence a perfect square and a perfect cube.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A152725.

[(-1)^n + n*(3-5*n^2+3*n^4+n^5)/2: n in [0..30]]; // Vincenzo Librandi, Oct 04 2013

CoefficientList[Series[(- 1 - 78 x^2 - 267 x^3 - 337 x^4 - 36 x^5 - 8 x^6 + x^7 + 6 x)/((1 + x) (x - 1)^7), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 04 2013 *)
LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{1,0,64,665,3431,12194,34462,83187},30] (* Harvey P. Dale, Apr 30 2020 *)

From R. J. Mathar, Mar 15 2011: (Start)
a(n) = (-1)^n + n*(3-5*n^2+3*n^4+n^5)/2.
a(n) = (-1)^n + A152725(n).
G.f.: ( -1-78*x^2-267*x^3-337*x^4-36*x^5-8*x^6+x^7+6*x ) / ( (1+x)*(x-1)^7 ). (End)

A261032 a(n) = (-1)^n(n^8 + 4n^7 - 14n^5 + 28n^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

Ilya Gutkovskiy, Nov 18 2015

Alternating sum of eighth powers (A001016).

For n>0, a(n) is divisible by A000217(n).

			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.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-9,-36,-84,-126,-126,-84,-36,-9,-1 ).

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

[(-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

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

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}]

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

[(-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

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

cp's OEIS Frontend

A152726 a(n) = n^7 - (n-1)^7 + (n-2)^7 - ... + ((-1)^n)*0^7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A062393 a(n) = n^5 - (n-1)^5 + (n-2)^5 - ... +(-1)^n*0^5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

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