A075664 -id:A075664 - cp's OEIS frontend

A072474 Sum of next n squares.

1, 13, 77, 294, 855, 2071, 4403, 8492, 15189, 25585, 41041, 63218, 94107, 136059, 191815, 264536, 357833, 475797, 623029, 804670, 1026431, 1294623, 1616187, 1998724, 2450525, 2980601, 3598713, 4315402, 5142019, 6090755, 7174671, 8407728, 9804817, 11381789, 13155485

Offset: 1

Amarnath Murthy, Jun 20 2002

			a(1) = 1^2 = 1;
a(2) = 2^2 + 3^2 = 13;
a(3) = 4^2 + 5^2 + 6^2 = 77.

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000290, A000330, A072475, A050410.

Cf. A006003 (for natural numbers), A260513 (for triangular numbers), A372583 (for pentagonal numbers), A372751 (for hexagonal numbers), A075664 (for cubes).

[n*(3*n^2+1)*(n^2+2)/12: n in [1..35]]; // Vincenzo Librandi, Dec 31 2024

Table[Sum[ i^2, {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 35}]

PARI
```
a(n) = n*(3*n^2+1)*(n^2+2)/12
```

a(n) = k*(k+1)*(2*k+1)/6 - r*(r+1)*(2*r+1)/6, where k = n*(n+1)/2 and r = n*(n-1)/2.
a(n) = A000330(n*(n+1)/2) - A000330(n*(n-1)/2).
a(n) = (n/12)*(3*n^2 + 1)*(n^2 + 2). - Benoit Cloitre, Jun 26 2002
G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^6. - Colin Barker, Mar 23 2012
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. - Jinyuan Wang, May 25 2020
E.g.f.: exp(x)*x*(12 + 66*x + 82*x^2 + 30*x^3 + 3*x^4)/12. - Stefano Spezia, May 14 2024

Edited by Robert G. Wilson v, Jun 21 2002

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

A075671 Sum of next n 10th powers.

Original entry on oeis.org

1, 60073, 71280377, 14843001474, 1091618326275, 39736919990851, 870012241054523, 12967387960026452, 143075291905145949, 1240006139651007925, 8817026830146599701, 53151169903167142598, 278615540073819826527, 1295610629596485350799, 5430916505417064431575

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^10 = 1; a(2) = 2^10 + 3^10 = 60073; a(3) = 4^10 + 5^10 + 6^10 = 71280377; a(4) = 7^10 + 8^10 + 9^10 + 10^10 = 14843001474.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (22, -231, 1540, -7315, 26334, -74613, 170544, -319770, 497420, -646646, 705432, -646646, 497420, -319770, 170544, -74613, 26334, -7315, 1540, -231, 22, -1).

Cf. A008454 (10th powers).

Cf. A072474 (for squares), A075664 - A075670 (3rd to 9th powers), A069876 (n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=10; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^10,Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 18 2018 *)

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^10.
a(n) = (33n^21 + 825n^19 + 6336n^17 + 18546n^15 + 14289n^13 - 14187n^11 - 418n^9 + 20592n^7 - 10560n^5 - 4224n^3 + 2560n)/33792. - Charles R Greathouse IV, Sep 17 2009
G.f.: (x^20 +60051*x^19 +69959002*x^18 +13288708503*x^17 +781445555829*x^16 +19040717780376*x^15 +225625446425352*x^14 +1431958892640624*x^13 +5170348336132746*x^12 +11021721646301518*x^11 +14154518527431996*x^10 +11021721646301518*x^9 +5170348336132746*x^8 +1431958892640624*x^7 +225625446425352*x^6 +19040717780376*x^5 +781445555829*x^4 +13288708503*x^3 +69959002*x^2 +60051*x +1) / (x -1)^22. - Colin Barker, Dec 19 2012

More terms from Colin Barker, Dec 19 2012

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

A372751 a(n) = (3n^5 + 4n^3 - n)/6.

Original entry on oeis.org

1, 21, 139, 554, 1645, 4031, 8631, 16724, 30009, 50665, 81411, 125566, 187109, 270739, 381935, 527016, 713201, 948669, 1242619, 1605330, 2048221, 2583911, 3226279, 3990524, 4893225, 5952401, 7187571, 8619814, 10271829, 12167995, 14334431, 16799056, 19591649

Offset: 1

Kelvin Voskuijl, May 12 2024

Sums of hexagonal numbers (A000384) in successive groups of length 1,2,3,etc, so 1, 6+15, 28+45+66, 91+120+153+190, etc.

			The hexagonal numbers and their groups summed begin
  1, 6, 15, 28, 45, 66, 91, 120, 153, 190
  \/ \---/  \--------/  \---------------/
  1,   21,     139,            554

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000384 (hexagonal numbers), A002412 (their partial sums).

Cf. A260513 (for triangular numbers), A072474 (for squares), A372583 (for pentagonal numbers), A075664 (cubes).

A372751[n_] := (3*n^5 + 4*n^3 - n)/6; Array[A372751, 50] (* Paolo Xausa, Jun 19 2024 *)

From Stefano Spezia, May 12 2024: (Start)
G.f.: x*(1 + 15*x + 28*x^2 + 15*x^3 + x^4)/(1 - x)^6.
E.g.f.: exp(x)*x*(6 + 57*x + 79*x^2 + 30*x^3 + 3*x^4)/6. (End)

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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A072474 Sum of next n squares.

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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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A075671 Sum of next n 10th powers.

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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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A372751 a(n) = (3*n^5 + 4*n^3 - n)/6.

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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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A372751 a(n) = (3n^5 + 4n^3 - n)/6.