A072474 -id:A072474 - 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

A050410 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2.

Original entry on oeis.org

0, 1, 13, 50, 126, 255, 451, 728, 1100, 1581, 2185, 2926, 3818, 4875, 6111, 7540, 9176, 11033, 13125, 15466, 18070, 20951, 24123, 27600, 31396, 35525, 40001, 44838, 50050, 55651, 61655, 68076, 74928, 82225, 89981, 98210, 106926, 116143

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

Starting with offset 1 = binomial transform of [1, 12, 25, 14, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009

			1^2 + 1;
2^2 + 3^2 = 13;
3^2 + 4^2 + 5^2 = 50; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A072474, A240137.

List([0..40], n-> n*(7*n-1)*(2*n-1)/6); # G. C. Greubel, Oct 30 2019

[n*(7*n-1)*(2*n-1)/6: n in [0..40]]; // Vincenzo Librandi, Apr 27 2012

seq(n*(7*n-1)*(2*n-1)/6, n=0..36); # Zerinvary Lajos, Dec 01 2006

Table[Sum[k^2,{k,n,2n-1}],{n,0,40}] (* or *) LinearRecurrence[{4,-6,4, -1}, {0,1,13,50},40] (* Harvey P. Dale, Feb 29 2012 *)

for(n=1,100,print1(sum(i=0,n-1,(n+i)^2),","))

vector(40, n, (n-1)*(7*n-8)*(2*n-3)/6) \\ G. C. Greubel, Oct 30 2019

[n*(7*n-1)*(2*n-1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019

a(n) = n*(7*n-1)*(2*n-1)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=13, a(3)=50. - Harvey P. Dale, Feb 29 2012
G.f.: x*(1 + 9*x + 4*x^2)/(1-x)^4. - Colin Barker, Mar 23 2012
E.g.f.: x*(6 + 33*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

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

A072475 Sum of next n composite numbers.

Original entry on oeis.org

4, 14, 31, 63, 112, 176, 264, 385, 529, 712, 932, 1184, 1503, 1833, 2234, 2689, 3207, 3779, 4408, 5117, 5913, 6747, 7657, 8667, 9766, 10938, 12240, 13612, 15071, 16578, 18266, 20081, 22007, 23989, 26100, 28334, 30695, 33221, 35811, 38569, 41474

Offset: 1

Amarnath Murthy, Jun 20 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A072474, A007468, A006003.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Sum[ Composite[i], {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 42}]
With[{terms=50},cnos=With[{c=(terms(terms+1)(terms+2))/6}, Complement[ Range[5,c], Prime[Range[PrimePi[c]]]]];Join[{4}, Total/@Table[Take[ cnos,{n (n+1)/2,(n+1) (n+2)/2-1}],{n,terms-1}]]] (* Harvey P. Dale, Oct 10 2011 *)

More terms from Jim Nastos and Robert G. Wilson v, Jun 21 2002

A372583 a(n) = (3n^5 + 5n^3)/8.

Original entry on oeis.org

1, 17, 108, 424, 1250, 3051, 6517, 12608, 22599, 38125, 61226, 94392, 140608, 203399, 286875, 395776, 535517, 712233, 932824, 1205000, 1537326, 1939267, 2421233, 2994624, 3671875, 4466501, 5393142, 6467608, 7706924, 9129375, 10754551, 12603392, 14698233

Offset: 1

Kelvin Voskuijl, May 05 2024

Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.

			The first ten pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, and 145.  Taking them in groups, respectively, of 1, 2, 3, and 4, i.e., (1), (5, 12), (22, 35, 51), and (70, 92, 117, 145), and summing each group separately gives 1, 17, 108, 424.

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. A260513 (for triangular numbers), A072474 (for squares).

Cf. A000326 (pentagonal numbers), A002411 (their partial sums).

A372583[n_] := (3*n^5 + 5*n^3)/8; Array[A372583, 50] (* Paolo Xausa, May 25 2024 *)

From Stefano Spezia, May 06 2024: (Start)
G.f.: x*(1 + 11*x + 21*x^2 + 11*x^3 + x^4)/(1 - x)^6.
E.g.f.: exp(x)*x*(8 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/8. (End)

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)

A380353 a(n) = (n^2 - n + 2) * (5n^2 - 5n + 2) / 4.

Original entry on oeis.org

1, 12, 64, 217, 561, 1216, 2332, 4089, 6697, 10396, 15456, 22177, 30889, 41952, 55756, 72721, 93297, 117964, 147232, 181641, 221761, 268192, 321564, 382537, 451801, 530076, 618112, 716689, 826617, 948736, 1083916, 1233057, 1397089, 1576972, 1773696, 1988281, 2221777

Offset: 1

Kelvin Voskuijl, Jan 22 2025

First differences of A072474 (sum of next n squares).

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A072474 (partial sums), A051624, A000124.

Cf. A005448 (first difference of sum of next n natural numbers).

Table[((n^2 - n + 2)*(5*n^2 - 5*n + 2))/4, {n, 1, 40}]

a(n) = (n^2 - n + 2) * (5*n^2 - 5*n + 2) / 4

a(n) = A051624(A000124(n-1)).
G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^5. - Jinyuan Wang, Jan 23 2025
E.g.f.: exp(x)*(4 + 22*x^2 + 20*x^3 + 5*x^4)/4 - 1. - Stefano Spezia, Jan 28 2025

cp's OEIS Frontend

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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A050410 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2.

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A075671 Sum of next n 10th powers.

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A072475 Sum of next n composite numbers.

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A372583 a(n) = (3*n^5 + 5*n^3)/8.

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A075665 Sum of next n 4th powers.

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A372583 a(n) = (3n^5 + 5n^3)/8.

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

A380353 a(n) = (n^2 - n + 2) * (5n^2 - 5n + 2) / 4.