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 1-8 of 8 results.

A226141 Sum of the squared parts of the partitions of n into exactly two parts.

Original entry on oeis.org

0, 2, 5, 18, 30, 64, 91, 156, 204, 310, 385, 542, 650, 868, 1015, 1304, 1496, 1866, 2109, 2570, 2870, 3432, 3795, 4468, 4900, 5694, 6201, 7126, 7714, 8780, 9455, 10672, 11440, 12818, 13685, 15234, 16206, 17936, 19019, 20940, 22140, 24262, 25585, 27918, 29370, 31924, 33511
Offset: 1

Views

Author

Wesley Ivan Hurt, May 27 2013

Keywords

Examples

			a(5) = 30; 5 has exactly 2 partitions into two parts, (4,1) and (3,2). Squaring the parts and adding, we get: 1^2 + 2^2 + 3^2 + 4^2 = 30.

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), this sequence (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).
Cf. A000330, A308422.

Programs

  • Magma
    [n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8 : n in [1..80]]; // Wesley Ivan Hurt, Jun 22 2024
  • Maple
    a:=n->sum(i^2 + (n-i)^2, i=1..floor(n/2)); seq((a(k), k=1..40);
  • Mathematica
    Array[Sum[i^2 + (# - i)^2, {i, Floor[#/2]}] &, 39] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,2,5,18,30,64,91},50] (* Harvey P. Dale, Jul 23 2019 *)

Formula

a(n) = Sum_{i=1..floor(n/2)} (i^2 + (n-i)^2).
a(n) = n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8. - Giovanni Resta, May 29 2013
G.f.: x^2*(2+3*x+7*x^2+3*x^3+x^4) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Jun 07 2013
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Wesley Ivan Hurt, Jun 22 2024
a(n) = A000330(n) - A308422(n). - Wesley Ivan Hurt, Jul 16 2025

A294270 Sum of the cubes of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 9, 44, 100, 252, 441, 848, 1296, 2150, 3025, 4572, 6084, 8624, 11025, 14912, 18496, 24138, 29241, 37100, 44100, 54692, 64009, 77904, 90000, 107822, 123201, 145628, 164836, 192600, 216225, 250112, 278784, 319634, 354025, 402732, 443556, 501068, 549081
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Comments

a(n) is a square when n is odd. In fact: a(2*k+1) = (2*k^2 + k)^2; a(2*k) = k^2*(4*k^2 - 3*k + 1), where (2*k)^2 > 4*k^2 - 3*k + 1 > (2*k - 1)^2 for k>0. - Bruno Berselli, Nov 20 2017

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), this sequence (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Programs

  • Magma
    [0] cat &cat[[k^2*(4*k^2-3*k+1),k^2*(2*k+1)^2]: k in [1..20]]; // Bruno Berselli, Nov 22 2017
  • Mathematica
    Table[Sum[i^3 + (n - i)^3, {i, Floor[n/2]}], {n, 80}]
  • PARI
    concat(0, Vec(x^2*(2 + 7*x + 27*x^2 + 28*x^3 + 24*x^4 + 7*x^5 + x^6) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^3 + (n-i)^3.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 7*x + 27*x^2 + 28*x^3 + 24*x^4 + 7*x^5 + x^6) / ((1 - x)^5*(1 + x)^4).
a(n) = n^2*(4*n^2 - 7*n + 4 + n*(-1)^n)/16.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>9. (End)

A294271 Sum of the fourth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 17, 114, 354, 1060, 2275, 4932, 8772, 15958, 25333, 41270, 60710, 91672, 127687, 182408, 243848, 333930, 432345, 572666, 722666, 931788, 1151403, 1451980, 1763020, 2182206, 2610621, 3180478, 3756718, 4514624, 5273999, 6263056, 7246096, 8515538, 9768353
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), this sequence (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Programs

  • Magma
    [(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4))/480 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025
  • Mathematica
    Table[Sum[i^4 + (n - i)^4, {i, Floor[n/2]}], {n, 60}]
Table[Total[Flatten[IntegerPartitions[n,{2}]]^4],{n,40}] (* Harvey P. Dale, Mar 01 2019 *)
  • PARI
    concat(0, Vec(x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ Colin Barker, Nov 20 2017
  • PARI
    a(n) = sum(i=1, n\2, i^4 + (n-i)^4); \\ Michel Marcus, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^4 + (n-i)^4.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 15*x + 87*x^2 + 165*x^3 + 241*x^4 + 165*x^5 + 77*x^6 + 15*x^7 + x^8) / ((1 - x)^6*(1 + x)^5).
a(n) = (1/480)*(n*(-16 + 160*n^2 + 15*(-15 + (-1)^n)*n^3 + 96*n^4)).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>11.
(End)

A294273 Sum of the sixth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 65, 858, 4890, 21244, 67171, 188916, 446964, 994030, 1978405, 3796622, 6735950, 11680408, 19092295, 30745064, 47260136, 71929146, 105409929, 153455810, 216455810, 303993492, 415601835, 566623708, 754740700, 1003708134, 1307797101, 1702747126
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), this sequence (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Programs

  • Magma
    [(n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7) : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025
  • Mathematica
    Table[Sum[i^6 + (n - i)^6, {i, Floor[n/2]}], {n, 50}]
  • PARI
    concat(0, Vec(x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^6 + (n-i)^6.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7).
a(n) = (n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 21*a(n-4) + 21*a(n-5) + 35*a(n-6) - 35*a(n-7) - 35*a(n-8) + 35*a(n-9) + 21*a(n-10) - 21*a(n-11) - 7*a(n-12) + 7*a(n-13) + a(n-14) - a(n-15) for n>15.
(End)

A294274 Sum of the seventh powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 129, 2444, 18700, 99012, 376761, 1216688, 3297456, 8158550, 18080425, 37847532, 73399404, 136971464, 241561425, 414517952, 680856256, 1095977898, 1703414961, 2607286700, 3877286700, 5697862412, 8172733129, 11613390384, 16164030000, 22330294142
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), this sequence (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Programs

  • Magma
    [n^2*(64 - 224*n^2 + 448*n^4 - 381*n^5 + 96*n^6 + 3*n^5*(-1)^n)/768 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025
  • Mathematica
    Table[Sum[i^7 + (n - i)^7, {i, Floor[n/2]}], {n, 40}]
  • PARI
    concat(0, Vec(x^2*(2 + 127*x + 2299*x^2 + 15240*x^3 + 61848*x^4 + 151257*x^5 + 262139*x^6 + 306832*x^7 + 260914*x^8 + 151257*x^9 + 60777*x^10 + 15240*x^11 + 2180*x^12 + 127*x^13+ x^14) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^7 + (n-i)^7.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 127*x + 2299*x^2 + 15240*x^3 + 61848*x^4 + 151257*x^5 + 262139*x^6 + 306832*x^7 + 260914*x^8 + 151257*x^9 + 60777*x^10 + 15240*x^11 + 2180*x^12 + 127*x^13+ x^14) / ((1 - x)^9*(1 + x)^8).
a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - 28*a(n-4) + 28*a(n-5) + 56*a(n-6) - 56*a(n-7) - 70*a(n-8) + 70*a(n-9) + 56*a(n-10) - 56*a(n-11) - 28*a(n-12) + 28*a(n-13) + 8*a(n-14) - 8*a(n-15) - a(n-16) + a(n-17) for n>17.
(End)
a(n) = n^2*(64 - 224*n^2 + 448*n^4 - 381*n^5 + 96*n^6 + 3*n^5*(-1)^n)/768. - Wesley Ivan Hurt, Jul 12 2025

A294275 Sum of the eighth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 257, 7074, 72354, 469540, 2142595, 7972932, 24684612, 68121958, 167731333, 383769830, 812071910, 1633567432, 3103591687, 5683259528, 9961449608, 16980253770, 27957167625, 45040730666, 70540730666, 108577948908, 163239463563, 241980430540, 351625763020
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), this sequence (k=8), A294276 (k=9), A294279 (k=10).

Programs

  • Magma
    [-n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025
  • Mathematica
    Table[Sum[i^8 + (n - i)^8, {i, Floor[n/2]}], {n, 40}]
  • PARI
    concat(0, Vec( x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^8 + (n-i)^8.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9).
a(n) = a(n-1) + 9*a(n-2) - 9*a(n-3) - 36*a(n-4) + 36*a(n-5) + 84*a(n-6) - 84*a(n-7) - 126*a(n-8) + 126*a(n-9) + 126*a(n-10) - 126*a(n-11) - 84*a(n-12) + 84*a(n-13) + 36*a(n-14) - 36*a(n-15) - 9*a(n-16) + 9*a(n-17) + a(n-18) - a(n-19) for n>19.
(End)
a(n) = -n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040. - Wesley Ivan Hurt, Jul 12 2025

A294276 Sum of the ninth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 513, 20708, 282340, 2255148, 12313161, 52928912, 186884496, 576258110, 1574304985, 3942330372, 9092033028, 19736886008, 40357579185, 78935156288, 147520415296, 266495712282, 464467582161, 788155279940, 1299155279940, 2095793274212, 3300704544313
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), this sequence (k=9), A294279 (k=10).

Programs

  • Magma
    [-n^2*(768-2560*n^2+3584*n^4-3840*n^6+2555*n^7-512*n^8-5*n^7*(-1)^n)/5120 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025
  • Mathematica
    Table[Sum[i^9 + (n - i)^9, {i, Floor[n/2]}], {n, 40}]
  • PARI
    concat(0, Vec(x^2*(2 + 511*x + 20175*x^2 + 256522*x^3 + 1770948*x^4 + 7464688*x^5 + 21796206*x^6 + 45087574*x^7 + 69569484*x^8 + 79813090*x^9 + 69501528*x^10 + 45087574*x^11 + 21722580*x^12 + 7464688*x^13 + 1756842*x^14 + 256522*x^15 + 19674*x^16 + 511*x^17+ x^18) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ Colin Barker, Nov 21 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} i^9 + (n-i)^9.
From Colin Barker, Nov 21 2017: (Start)
G.f.: x^2*(2 + 511*x + 20175*x^2 + 256522*x^3 + 1770948*x^4 + 7464688*x^5 + 21796206*x^6 + 45087574*x^7 + 69569484*x^8 + 79813090*x^9 + 69501528*x^10 + 45087574*x^11 + 21722580*x^12 + 7464688*x^13 + 1756842*x^14 + 256522*x^15 + 19674*x^16 + 511*x^17+ x^18) / ((1 - x)^11*(1 + x)^10).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.
(End)
a(n) = -n^2*(768-2560*n^2+3584*n^4-3840*n^6+2555*n^7-512*n^8-5*n^7*(-1)^n)/5120. - Wesley Ivan Hurt, Jul 12 2025

A294279 Sum of the tenth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 1025, 61098, 1108650, 10933324, 71340451, 354864276, 1427557524, 4924107550, 14914341925, 40912232702, 102769130750, 240910097848, 529882277575, 1107606410024, 2206044295976, 4225524980826, 7792505423049, 13933571680850, 24163571680850, 40869390083652
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

Links

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), this sequence (k=10).

Programs

  • Magma
    [n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584 : n in [1..50]]; // Wesley Ivan Hurt, Jul 13 2025
  • Maple
    f:= proc(n)
if n::even then (1/66)*n*(6*n^10-(16863/512)*n^9+55*n^8-66*n^6+66*n^4-33*n^2+5)
  else (1/66*(n-1))*n*(2*n-1)*(n^2-n-1)*(3*n^6-9*n^5+2*n^4+11*n^3+3*n^2-10*n-5)
fi end proc:
map(f, [$1..50]); # Robert Israel, Oct 27 2017
  • Mathematica
    Table[Sum[i^10 + (n - i)^10, {i, Floor[n/2]}], {n, 30}]

Formula

a(n) = Sum_{i=1..floor(n/2)} i^10 + (n-i)^10.
From Robert Israel, Oct 27 2017: (Start)
a(2*k) = (6144*k^10-16863*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.
a(2*k+1) = (6144*k^10+16896*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.
G.f.: x^2*(x^20+1023*x^19+59039*x^18+1036299*x^17+9117154*x^16+48940320*x^15
+178348744*x^14+465661416*x^13+907378474*x^12+1340492142*x^11+1528402822*x^10
+1340492142*x^9+908233636*x^8+465661416*x^7+178756096*x^6+48940320*x^5
+9163981*x^4+1036299*x^3+60051*x^2+1023*x+2)/((x^2-1)^11*(x-1)). (End)
a(n) = n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584. - Wesley Ivan Hurt, Jul 13 2025
a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 55*a(n-4) + 55*a(n-5) + 165*a(n-6) - 165*a(n-7) - 330*a(n-8) + 330*a(n-9) + 462*a(n-10) - 462*a(n-11) - 462*a(n-12) + 462*a(n-13) + 330*a(n-14) - 330*a(n-15) - 165*a(n-16) + 165*a(n-17) + 55*a(n-18) - 55*a(n-19) - 11*a(n-20) + 11*a(n-21) + a(n-22) - a(n-23). - Wesley Ivan Hurt, Jul 13 2025
Showing 1-8 of 8 results.

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