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.

A294300 Sum of the fifth powers of the parts in the partitions of n into two distinct parts.

Original entry on oeis.org

0, 0, 33, 244, 1300, 4182, 12201, 27984, 61776, 117700, 220825, 374100, 630708, 985194, 1539825, 2266432, 3347776, 4708584, 6657201, 9033300, 12333300, 16256350, 21571033, 27758544, 35970000, 45364332, 57617001, 71428084, 89176276, 108928050, 133987425
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 27 2017

Keywords

Links

Crossrefs

Cf. A294286, A294287, A294288.

Programs

  • Mathematica
    Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2] - Mod[n + 1, 2]}], {n, 40}]
Table[Total[Flatten[Select[IntegerPartitions[n,{2}],#[[1]]!=#[[2]]&]]^5],{n,40}] (* Harvey P. Dale, Sep 04 2024 *)
  • PARI
    concat(vector(2), Vec(x^3*(33 + 211*x + 858*x^2 + 1616*x^3 + 2178*x^4 + 1656*x^5 + 858*x^6 + 236*x^7 + 33*x^8 + x^9) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ Colin Barker, Nov 21 2017
  • PARI
    a(n) = sum(i=1, (n-1)\2, i^5 + (n-i)^5); \\ Michel Marcus, Nov 22 2017

Formula

a(n) = Sum_{i=1..floor(n/2)-((n+1) mod 2)} i^5 + (n-i)^5.
G.f.: -x^3*(33 +211*x +858*x^2 +1616*x^3 +2178*x^4 +1656*x^5 +858*x^6 +236*x^7 +33*x^8 +x^9) /(1+x)^6 /(x-1)^7. - R. J. Mathar, Nov 07 2017
From Colin Barker, Nov 21 2017: (Start)
a(n) = (1/192)*(n^2*(-16 + 80*n^2 - 3*(33 + (-1)^n)*n^3 + 32*n^4)).
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>13.
(End)

A294301 Sum of the sixth powers of the parts in the partitions of n into two distinct parts.

Original entry on oeis.org

0, 0, 65, 730, 4890, 19786, 67171, 180724, 446964, 962780, 1978405, 3703310, 6735950, 11445110, 19092295, 30220776, 47260136, 70866264, 105409929, 151455810, 216455810, 300450370, 415601835, 560651740, 754740700, 994054516, 1307797101, 1687688054, 2177107894
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 27 2017

Keywords

Links

Crossrefs

Cf. A294286, A294287, A294288, A294300.

Programs

  • Mathematica
    Table[Sum[i^6 + (n - i)^6, {i, Floor[(n-1)/2]}], {n, 40}]
  • PARI
    a(n) = sum(i=1, (n-1)\2, i^6 + (n-i)^6); \\ Michel Marcus, Nov 08 2017
  • PARI
    concat(vector(2), Vec(x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^6 + (n-i)^6.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7).
a(n) = (n/42 - n^3/6 + n^5/2 - 1/128*(65 + (-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)

A294302 Sum of the seventh powers of the parts in the partitions of n into two distinct parts.

Original entry on oeis.org

0, 0, 129, 2188, 18700, 94638, 376761, 1183920, 3297456, 8002300, 18080425, 37287660, 73399404, 135324378, 241561425, 410323648, 680856256, 1086411960, 1703414961, 2587286700, 3877286700, 5658888070, 8172733129, 11541726768, 16164030000, 22204797108
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 27 2017

Keywords

Links

Crossrefs

Cf. A294286, A294287, A294288, A294300, A294301.

Programs

  • Mathematica
    Table[Sum[i^7 + (n - i)^7, {i, Floor[(n-1)/2]}], {n, 40}]
CoefficientList[Series[x^3(129+2059x+15480x^2+59466x^3+153639x^4+257307x^5+ 311664x^6+ 258532x^7+153639x^8+60537x^9+15480x^10+2178x^11+129x^12+x^13)/ ((1-x)^9 (1+x)^8),{x,0,60}],x] (* or *) LinearRecurrence[{1,8,-8,-28,28,56,-56,-70,70,56,-56,-28,28,8,-8,-1,1},{0,0,0,129,2188,18700,94638,376761,1183920,3297456,8002300,18080425,37287660,73399404,135324378,241561425,410323648},60] (* Harvey P. Dale, Aug 05 2021 *)
  • PARI
    a(n) = sum(i=1, (n-1)\2, i^7 + (n-i)^7); \\ Michel Marcus, Nov 08 2017
  • PARI
    concat(vector(2), Vec(x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^7 + (n-i)^7.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8).
a(n) = (1/768)*(n^2*(64 - 224*n^2 + 448*n^4 - 3*(129 + (-1)^n)*n^5 + 96*n^6)).
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)

A294303 Sum of the eighth powers of the parts in the partitions of n into two distinct parts.

Original entry on oeis.org

0, 0, 257, 6562, 72354, 456418, 2142595, 7841860, 24684612, 67340708, 167731333, 380410598, 812071910, 1622037830, 3103591687, 5649705096, 9961449608, 16894160328, 27957167625, 44840730666, 70540730666, 108149231146, 163239463563, 241120467148, 351625763020
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 27 2017

Keywords

Links

Crossrefs

Cf. A294286, A294287, A294288, A294300, A294301, A294302.

Programs

  • Mathematica
    Table[Sum[i^8 + (n - i)^8, {i, Floor[(n-1)/2]}], {n, 30}]
  • PARI
    first(n) = {my(res = vector(n, i, 1/9*i^9 - 1/2*i^8 + 2/3*i^7 - 7/15*i^5 + 2/9*i^3 - 1/30*i)); forstep(i = 2, #res, 2, res[i] -= i^8/256); res} \\ David A. Corneth, Nov 05 2017
  • PARI
    concat(vector(2), Vec(x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^8 + (n-i)^8.
From David A. Corneth, Nov 05 2017: (Start)
For odd n, a(n) = n^9 / 9 - n^8/2 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30
For even n, a(n) = n^9 / 9 - 129*n^8/256 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30.
(End)
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((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)

A294304 Sum of the ninth powers of the parts of the partitions of n into two distinct parts.

Original entry on oeis.org

0, 0, 513, 19684, 282340, 2215782, 12313161, 52404624, 186884496, 572351860, 1574304985, 3922174980, 9092033028, 19656178794, 40357579185, 78666720832, 147520415296, 265720871304, 464467582161, 786155279940, 1299155279940, 2091077378830, 3300704544313
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 27 2017

Keywords

Links

Crossrefs

Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303.

Programs

  • Mathematica
    Table[Sum[i^9 + (n - i)^9, {i, Floor[(n-1)/2]}], {n, 30}]
  • PARI
    a(n) = sum(i=1, (n-1)\2, i^9 + (n-i)^9); \\ Michel Marcus, Nov 08 2017
  • PARI
    concat(vector(2), Vec(x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ Colin Barker, Nov 20 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^9 + (n-i)^9.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((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)

A294473 Sum of the areas of the squares on the sides of the distinct rectangles that can be made with positive integer sides such that L + W = n, W < L.

Original entry on oeis.org

0, 0, 10, 20, 60, 92, 182, 248, 408, 520, 770, 940, 1300, 1540, 2030, 2352, 2992, 3408, 4218, 4740, 5740, 6380, 7590, 8360, 9800, 10712, 12402, 13468, 15428, 16660, 18910, 20320, 22880, 24480, 27370, 29172, 32412, 34428, 38038, 40280, 44280, 46760, 51170
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 31 2017

Keywords

Examples

			a(14) = 1540; the rectangles are 1 X 13, 2 X 12, 3 X 11, 4 X 10, 5 X 9, 6 X 8 (7 X 7 is not considered since W < L). The sum of the areas of the squares on the sides of the rectangles is 2*1^2+2*13^2 + 2*2^2+2*12^2 + 2*3^2+2*11^2 + 2*4^2+2*10^2 + 2*5^2+2*9^2 + 2*6^2+2*8^2 = 340 + 296 + 260 + 232 + 212 + 200 = 1540.

Links

Crossrefs

Cf. A294286, A050187, A294457.

Programs

  • Magma
    [n*(2*n-3-(-1)^n)*(8*n-3+(-1)^n)/24: n in [1..60]]; // Wesley Ivan Hurt, Dec 02 2023
  • Mathematica
    Table[2 Sum[i^2 + (n - i)^2, {i, Floor[(n-1)/2]}], {n, 40}]
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,0,10,20,60,92,182},50] (* Harvey P. Dale, Jul 05 2025 *)
  • PARI
    concat(vector(2), Vec(2*x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3) + O(x^60))) \\ Colin Barker, Oct 31 2017
  • PARI
    a(n) = 2*sum(i=1, (n-1)\2, i^2 + (n-i)^2); \\ Michel Marcus, Nov 08 2017

Formula

a(n) = 2 * Sum_{i=1..floor((n-1)/2)} i^2 + (n-i)^2.
a(n) = 2 * A294286(n).
From Colin Barker, Oct 31 2017: (Start)
G.f.: 2*x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3).
a(n) = n*(8*n^2 - 18*n + 4) / 12 for n even.
a(n) = n*(8*n^2 - 12*n + 4) / 12 for n odd.
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) for n > 7. (End)
a(n) = n*(2*n-3-(-1)^n)*(8*n-3+(-1)^n)/24. - Wesley Ivan Hurt, Dec 02 2023
E.g.f.: x*(x*(3 + 4*x)*cosh(x) - (3 - 6*x - 4*x^2)*sinh(x))/6. - Stefano Spezia, May 19 2025

A294305 Sum of the tenth powers of the parts in the partitions of n into two distinct parts.

Original entry on oeis.org

0, 0, 1025, 59050, 1108650, 10815226, 71340451, 352767124, 1427557524, 4904576300, 14914341925, 40791300350, 102769130750, 240345147350, 529882277575, 1105458926376, 2206044295976, 4218551412024, 7792505423049, 13913571680850, 24163571680850, 40817515234450
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 27 2017

Keywords

Links

Crossrefs

Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303, A294304.

Programs

  • Mathematica
    Table[Sum[i^10 + (n - i)^10, {i, Floor[(n-1)/2]}], {n, 30}]
  • PARI
    a(n) = sum(i=1, (n-1)\2, i^10 + (n-i)^10); \\ Michel Marcus, Nov 05 2017
  • PARI
    concat(vector(2), Vec(x^3*(1025 + 58025*x + 1038325*x^2 + 9068301*x^3 + 49036000*x^4 + 177845712*x^5 + 466571800*x^6 + 905612928*x^7 + 1343112850*x^8 + 1525782114*x^9 + 1343112850*x^10 + 906468090*x^11 + 466571800*x^12 + 178253064*x^13 + 49036000*x^14 + 9115128*x^15 + 1038325*x^16 + 59037*x^17 + 1025*x^18 + x^19) / ((1 - x)^12*(1 + x)^11) + O(x^40))) \\ Colin Barker, Nov 21 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^10 + (n-i)^10.
From Colin Barker, Nov 21 2017: (Start)
G.f.: x^3*(1025 + 58025*x + 1038325*x^2 + 9068301*x^3 + 49036000*x^4 + 177845712*x^5 + 466571800*x^6 + 905612928*x^7 + 1343112850*x^8 + 1525782114*x^9 + 1343112850*x^10 + 906468090*x^11 + 466571800*x^12 + 178253064*x^13 + 49036000*x^14 + 9115128*x^15 + 1038325*x^16 + 59037*x^17 + 1025*x^18 + x^19) / ((1 - x)^12*(1 + x)^11).
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) for n>23.
(End)

A308025 a(n) = n*(2*n - 3 - (-1)^n)*(5*n - 2 + (-1)^n)/16.

Original entry on oeis.org

0, 0, 9, 19, 55, 87, 168, 234, 378, 490, 715, 885, 1209, 1449, 1890, 2212, 2788, 3204, 3933, 4455, 5355, 5995, 7084, 7854, 9150, 10062, 11583, 12649, 14413, 15645, 17670, 19080, 21384, 22984, 25585, 27387, 30303, 32319, 35568, 37810, 41410, 43890, 47859
Offset: 1

Views

Author

Wesley Ivan Hurt, May 09 2019

Keywords

Comments

Consider the rectangular prisms with dimensions s X t X t, where n = s + t and s < t. Then a(n) is the sum of the areas of the squares that rest on a given space diagonal in each of the rectangular prisms.
Sum of the squares of the smaller parts and twice the sum of the squares of the larger parts in the partitions of n into two distinct parts.

Links

Crossrefs

Cf. A294286.

Programs

  • Mathematica
    Table[n*(2 n - 3 - (-1)^n)*(5 n - 2 + (-1)^n)/16, {n, 60}]
  • PARI
    concat([0,0], Vec(x^3*(9 + 10*x + 9*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, May 17 2019

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^2 + 2*(n-i)^2.
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).
G.f.: x^3*(9 + 10*x + 9*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3). - Colin Barker, May 17 2019
Showing 1-8 of 8 results.

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