A075671 -id:A075671 - cp's OEIS frontend

A075664 Sum of next n cubes.

0, 1, 35, 405, 2584, 11375, 38961, 111475, 278720, 627669, 1300375, 2516921, 4604040, 8030035, 13446629, 21738375, 34080256, 52004105, 77474475, 112974589, 161603000, 227181591, 314375545, 428825915, 577295424, 767828125, 1009923551, 1314725985, 1695229480, 2166499259

Offset: 0

Zak Seidov, Sep 24 2002

			a(1) = 1^3 = 1; a(2) = 2^3 + 3^3 = 35; a(3) = 4^3 + 5^3 + 6^3 = 64 + 125 + 216 = 405.
From _Philippe Deléham_, Mar 09 2014: (Start)
a(1) = 1*2*3/8 = 1;
a(2) = 8*5*7/8 = 35;
a(3) = 27*10*12/8 = 405;
a(4) = 64*17*19/8 = 2584;
a(5) = 125*26*28/8 = 11375; etc. (End)

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000578 (cubes).

Cf. A006003, A072474 (for squares), A075665 - A075671 (4th to 10th powers), A069876 (n-th powers).

[(n^7+4*n^5+3*n^3)/8: n in [1..30]]; // Vincenzo Librandi, Mar 11 2014

A075664:=n->(n^7 + 4n^5 + 3n^3)/8; seq(A075664(n), n=1..30); # Wesley Ivan Hurt, Mar 10 2014

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=3; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
CoefficientList[Series[(1 + 27 x + 153 x^2 + 268 x^3 + 153 x^4 + 27 x^5 + x^6)/(1 - x)^8, {x, 0, 40}], x](* Vincenzo Librandi, Mar 11 2014 *)
With[{nn=30},Total/@TakeList[Range[(nn(nn+1))/2]^3,Range[nn]]] (* Requires Mathematica version 11 or later *) (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,35,405,2584,11375,38961,111475,278720},30] (* Harvey P. Dale, Jun 05 2021 *)

a(n)=(n^7+4*n^5+3*n^3)/8 \\ Charles R Greathouse IV, Oct 07 2015

def A075664(n): return n*(m:=n**2)*(m*(m+4)+3)>>3 # Chai Wah Wu, Feb 09 2025

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^3. [Corrected by Stefano Spezia, Jun 22 2024]
a(n) = (n^7 + 4n^5 + 3n^3)/8. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(1+27*x+153*x^2+268*x^3+153*x^4+27*x^5+x^6)/(1-x)^8. - Colin Barker, May 25 2012
a(n) = n^3*(n^2 + 1)*(n^2 + 3)/8 = A000578(n)*A002522(n)*A117950(n)/8. - Philippe Deléham, Mar 09 2014
E.g.f.: exp(x)*x*(8 + 132*x + 404*x^2 + 390*x^3 + 144*x^4 + 21*x^5 + x^6)/8. - Stefano Spezia, Jun 22 2024

Formula from Charles R Greathouse IV, Sep 17 2009
More terms from Vincenzo Librandi, Mar 11 2014
a(0) added by Chai Wah Wu, Feb 09 2025

A069876 a(n) = (k-n+1)^n + (k-n+2)^n + ... + (k-1)^n + k^n, where k = n(n+1)/2.

Original entry on oeis.org

1, 13, 405, 23058, 2078375, 271739011, 48574262275, 11373936899396, 3377498614484589, 1240006139651007925, 551449374186192949841, 292093390490112799117190, 181694111127303339553250275, 131144830297438122797495823519, 108709456000518111261404495694375

Offset: 1

Amarnath Murthy, Apr 25 2002

nonn

Sum of next n n-th powers.

			a(1) = 1^1 = 1; a(2) = 2^2 + 3^2 = 13; a(3) = 4^3 + 5^3 + 6^3 = 405; a(4) = 7^4 + 8^4 + 9^4 + 10^4 = 23058.

Cf. A072474 (for squares), A075664 - A075671 (3rd to 10th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; Table[Sum[i^n, {i, i1, i2}], {n, 20}]

def A069876(n): return sum(((n*(n+1)>>1)-i)**n for i in range(n)) # Chai Wah Wu, Feb 10 2025

More terms from Larry Reeves (larryr(AT)acm.org) and Zak Seidov, Sep 24 2002

A075665 Sum of next n 4th powers.

Original entry on oeis.org

1, 97, 2177, 23058, 152979, 738835, 2839571, 9191876, 26037717, 66301333, 154762069, 336050870, 686502375, 1331121351, 2467171687, 4396168328, 7566347369, 12628007049, 20504452585, 32481640666, 50320004987, 76392352443, 113852150523, 166836980044, 240712403645

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^4 = 1; a(2) = 2^4 + 3^4 = 97; a(3) = 4^4 + 5^4 + 6^4 = 2177; a(4) = 7^4 + 8^4 + 9^4 + 10^4 = 23058.

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000583 (4th powers).

Cf. A006003 (for natural numbers), A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=4; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
Table[Total[Range[(n(n+1))/2+1,((n+1)(n+2))/2]^4],{n,0,20}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,97,2177,23058,152979,738835,2839571,9191876,26037717,66301333},30] (* Harvey P. Dale, Dec 18 2015 *)

a(n) = Sum_{i=n*(n-1)/2+1..n*(n-1)/2+n} i^4.
a(n) = (15*n^9 + 90*n^7 + 123*n^5 + 20*n^3 - 8*n)/240. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(1+87*x+1252*x^2+5533*x^3+8934*x^4+5533*x^5+1252*x^6+87*x^7+x^8)/ (1-x)^10. - Colin Barker, May 25 2012

A075670 Sum of next n 9th powers.

Original entry on oeis.org

1, 20195, 12292965, 1561991824, 77226633575, 2014634387961, 33098483802475, 383318212734080, 3377498614484589, 23898971839102975, 141290020118952881, 719054471032657200, 3223613105991831475, 12964037775857022869, 47453810583528962775, 159982264435790734336

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^9 = 1; a(2) = 2^9 + 3^9 = 20195; a(3) = 4^9 + 5^9 + 6^9 = 12292965; a(4) = 7^9 + 8^9 + 9^9 + 10^9 = 1561991824.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Cf. A001017 (9th powers).

Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

[(5*n^19 + 105*n^17 + 666*n^15 + 1530*n^13 + 689*n^11 - 995*n^9 + 304*n^7 + 640*n^5 - 384*n^3)/2560 : n in [1..20]]; // Vincenzo Librandi, Oct 06 2011

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=9; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
Total[#^9]&/@(Range[First[#]+1,Last[#]]&/@Partition[Accumulate[Range[ 0,15]],2,1]) (* Harvey P. Dale, Oct 05 2011 *)
With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^9,Range[nn]]] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^9.
a(n) = (5n^19 + 105n^17 + 666n^15 + 1530n^13 + 689n^11 - 995n^9 + 304n^7 + 640n^5 - 384n^3)/2560. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^18 +20175*x^17 +11889255*x^16 +1319968434*x^15 +48299442990*x^14 +752964012192*x^13 +5757432094050*x^12 +23468751060270*x^11 +53583908362248*x^10 +70362713036770*x^9 +53583908362248*x^8 +23468751060270*x^7 +5757432094050*x^6+752964012192*x^5 +48299442990*x^4 +1319968434*x^3 +11889255*x^2 +20175*x +1)/(x -1)^20. - Colin Barker, Sep 06 2012

A075666 Sum of next n 5th powers.

Original entry on oeis.org

1, 275, 11925, 208624, 2078375, 14118201, 72758875, 304553600, 1084203549, 3390961375, 9540835601, 24582546000, 58801331875, 131987718149, 280410672375, 567799960576, 1102105900025, 2060382328875, 3724847929549, 6534040766000, 11154010982751, 18575718271825

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^5 = 1; a(2) = 2^5 + 3^5 = 275; a(3) = 4^5 + 5^5 + 6^5 = 11925; a(4) = 7^5 + 8^5 + 9^5 + 10^5 = 208624.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000584 (5th powers).

Cf. A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=5; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
nn=30;With[{p5=Range[((nn+1)(nn+2))/2]^5},Join[{1},Table[Total[Take[p5,{(n(n+1))/2+1,((n+1)(n+2))/2}]],{n,nn}]]] (* Harvey P. Dale, Mar 09 2014 *)
Module[{nn=25,p5},p5=Range[(nn(nn+1))/2]^5;Total/@TakeList[p5,Range[nn]]] (* Harvey P. Dale, Oct 13 2023 *)

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^5.
a(n) = (3n^11 + 25n^9 + 53n^7 + 23n^5 - 8n^3)/96. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^10 +263*x^9 +8691*x^8 +83454*x^7 +301932*x^6 +458718*x^5 +301932*x^4 +83454*x^3 +8691*x^2 +263*x+1) / (x-1)^12. - Colin Barker, Jul 22 2012

A075667 Sum of next n 6th powers.

Original entry on oeis.org

1, 793, 66377, 1911234, 28504515, 271739011, 1874885963, 10136389172, 45311985069, 173957200405, 589679082421, 1802148522758, 5045944649967, 13108508706879, 31915866810295, 73427944186856, 160710828298553, 336507487921137, 677266380588289, 1315464522556810

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^6 = 1; a(2) = 2^6 + 3^6 = 793; a(3) = 4^6 + 5^6 + 6^6 = 66377; a(4) = 7^6 + 8^6 + 9^6 + 10^6 = 1911234.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A001014 (6th powers).

Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=6; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^6,Range[nn]]] (* or *) LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,793,66377,1911234,28504515,271739011,1874885963,10136389172,45311985069,173957200405,589679082421,1802148522758,5045944649967,13108508706879},20] (* Harvey P. Dale, Mar 29 2022 *)

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^6.
a(n) = (21n^13 + 231n^11 + 693n^9 + 549n^7 - 126n^5 - 56n^3 + 32n)/1344. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^12 +779*x^11 +55366*x^10 +1053755*x^9 +7499895*x^8 +23228658*x^7 +33620292*x^6 +23228658*x^5 +7499895*x^4 +1053755*x^3 +55366*x^2 +779*x +1)/(x-1)^14. - Colin Barker, Jul 22 2012

A075668 Sum of next n 7th powers.

Original entry on oeis.org

1, 2315, 374445, 17703664, 394340375, 5265954441, 48574262275, 338837482880, 1900477947429, 8950536157375, 36536761179281, 132397570996560, 433806511149115, 1303971065324669, 3637715990646375, 9507513902672896, 23461050872397545, 55013865421504275

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^7 = 1; a(2) = 2^7 + 3^7 = 2315; a(3) = 4^7 + 5^7 + 6^7 = 374445; a(4) = 7^7 + 8^7 + 9^7 + 10^7 = 17703664.

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A001015 (7th powers).

Cf. A006003 (for natural numbers), A072474 (for squares), A075664 - A075671 (for 3rd to 10th powers), A069876 (n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=7; Table[Sum[i^s, {i, i1, i2}], {n, 20}]

a(n) = Sum_{i=n*(n-1)/2+1..n*(n-1)/2+n} i^7.
a(n) = (3*n^15 + 42*n^13 + 168*n^11 + 206*n^9 - 11*n^7 - 56*n^5 + 32*n^3)/384. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^14 +2299*x^13 +337525*x^12 +11989784*x^11 +154720571*x^10 +875467853*x^9 +2397170367*x^8 +3336829200*x^7 +2397170367*x^6 +875467853*x^5 +154720571*x^4 +11989784*x^3 +337525*x^2 +2299*x +1)/(x-1)^16. - Colin Barker, Jul 22 2012

A075669 Sum of next n 8th powers.

Original entry on oeis.org

1, 6817, 2135777, 165588738, 5498750979, 102697107715, 1264908663011, 11373936899396, 79985007371877, 461856872635333, 2269365182729029, 9747136491367430, 37362375267437415, 129917413702762791, 415196000174767687, 1232554282743058568, 3428668198703973449

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^8 = 1; a(2) = 2^8 + 3^8 = 6817; a(3) = 4^8 + 5^8 + 6^8 = 2135777; a(4) = 7^8 + 8^8 + 9^8 + 10^8 = 165588738.

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (18,-153,816,-3060,8568,-18564,31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Cf. A001016 (8th powers).

Cf. A006003 (for natural numbers), A072474 (for squares), A075664 - A075671 (for 3rd to 10th powers), A069876 (for n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=8; Table[Sum[i^s, {i, i1, i2}], {n, 20}]

a(n) = Sum_{i=n*(n-1)/2+1..n*(n-1)/2+n} i^8.
a(n) = (45*n^17 + 780*n^15 + 3990*n^13 + 6900*n^11 + 1205*n^9 - 3240*n^7 + 1584*n^5 + 640*n^3 - 384*n)/11520. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^16 +6799*x^15 +2013224*x^14 +128186937*x^13 +2839367964*x^12 +27332724427*x^11 +129026301848*x^10 +319786366637*x^9 +431174080326*x^8 +319786366637*x^7 +129026301848*x^6 +27332724427*x^5 +2839367964*x^4 +128186937*x^3 +2013224*x^2 +6799*x +1)/(x -1)^18. - Colin Barker, Sep 06 2012

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A075664 Sum of next n cubes.

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A069876 a(n) = (k-n+1)^n + (k-n+2)^n + ... + (k-1)^n + k^n, where k = n(n+1)/2.

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A075665 Sum of next n 4th powers.

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A075670 Sum of next n 9th powers.

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A075666 Sum of next n 5th powers.

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A075667 Sum of next n 6th powers.

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A075668 Sum of next n 7th powers.

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A075669 Sum of next n 8th powers.

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