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.

User: Xavier Acloque

Xavier Acloque's wiki page.

Xavier Acloque has authored 8 sequences.

A088892 Polynexus numbers of order 14.

Original entry on oeis.org

0, 1, 291, 16096, 356232, 4411517, 36621423, 227095448, 1128128568, 4708376529, 17078744419, 55199550120, 161993768080, 438011626365, 1103841220991, 2616890599056, 5880356075792, 12602902382337, 25897027973187, 51245013077968, 98017089897528, 181801058663389
Offset: 1

Views

Author

Xavier Acloque, Oct 21 2003

Keywords

Links

Crossrefs

Cf. A079547, A083200, A088889, A088890, A088891, A088893, A088894.

Programs

  • Mathematica
    Table[((n^14 - (n - 1)^14) - (n^2 - (n - 1)^2))/16380, {n, 20}] (* Bruno Berselli, Feb 08 2012 *)
LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{0,1,291,16096,356232,4411517,36621423,227095448,1128128568,4708376529,17078744419,55199550120,161993768080,438011626365},20] (* Harvey P. Dale, Mar 04 2024 *)

Formula

a(n) = ((n^14-(n-1)^14)-(n^2-(n-1)^2))/16380 = ((n^14-(n-1)^14)-(n^2-(n-1)^2))/(2^14-2^2).
G.f.: x^2*(1+x)*(1+276*x+11837*x^2+145168*x^3+638914*x^4+1068728*x^5+638914*x^6+145168*x^7+11837*x^8+276*x^9+x^10)/(1-x)^14. - Bruno Berselli, Feb 08 2012

Extensions

First term added according to the formula from Bruno Berselli, Feb 08 2012

A088891 Polynexus numbers of order 9.

Original entry on oeis.org

0, 1, 38, 481, 3355, 16120, 60071, 186238, 502386, 1215435, 2694340, 5559191, 10803013, 19953466, 35282365, 60071660, 98945236, 158276613, 246683346, 375619645, 560079455, 819422956, 1178340163, 1667966026, 2327162150, 3203980975, 4357328976, 5858846163
Offset: 1

Views

Author

Xavier Acloque, Oct 21 2003

Keywords

Links

Crossrefs

Cf. A079547, A083200, A088889, A088890, A088892, A088893, A088894.

Programs

  • Mathematica
    Table[((n^9-(n-1)^9)-(n^3-(n-1)^3))/504,{n,30}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{0,1,38,481,3355,16120,60071,186238,502386},30] (* Harvey P. Dale, Jan 18 2012 *)

Formula

a(n) = ((n^9-(n-1)^9)-(n^3-(n-1)^3))/504 = ((n^9-(n-1)^9)-(n^3-(n-1)^3))/(2^9-2^3).
a(1)=1, a(2)=38, a(3)=481, a(4)=3355, a(5)=16120, a(6)=60071, a(7)=186238, a(8)=502386, a(9)=1215435, a(n)=9*a(n-1)-36*a(n-2)+ 84*a(n-3)- 126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Jan 18 2012
G.f.: x^2*(1+29*x+175*x^2+310*x^3+175*x^4+29*x^5+x^6)/(1-x)^9. - Bruno Berselli, Feb 10 2012

Extensions

Offset changed and first term added (according to the formula) from Bruno Berselli, Feb 08 2012

A088889 Polynexus numbers of order 8.

Original entry on oeis.org

0, 1, 26, 245, 1353, 5368, 17017, 45878, 109446, 237291, 476476, 898403, 1607255, 2750202, 4529539, 7216924, 11169884, 16850757, 24848238, 35901697, 50928437, 71054060, 97646109, 132351154, 177135490, 234329615, 306676656, 397384911, 510184675, 649389518
Offset: 1

Views

Author

Xavier Acloque, Oct 21 2003

Keywords

Links

Crossrefs

Cf. A079547, A083200, A088890, A088891, A088892, A088893, A088894.

Programs

  • Mathematica
    Table[((n^8 - (n - 1)^8) - (n^4 - (n - 1)^4))/240, {n, 26}] (* Bruno Berselli, Feb 10 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,26,245,1353,5368,17017,45878},30] (* Harvey P. Dale, Oct 31 2024 *)

Formula

a(n) = ((n^8-(n-1)^8)-(n^4-(n-1)^4))/240 = ((n^8-(n-1)^8)-(n^4-(n-1)^4))/(2^8-2^4).
G.f.: x^2*(1+x)*(1+17*x+48*x^2+17*x^3+x^4)/(1-x)^8. - Bruno Berselli, Feb 10 2012

Extensions

First term added according to the formula from Bruno Berselli, Feb 10 2012

A088890 Polynexus numbers of order 8.

Original entry on oeis.org

0, 1, 25, 234, 1290, 5115, 16211, 43700, 104244, 226005, 453805, 855646, 1530750, 2619279, 4313895, 6873320, 10638056, 16048425, 23665089, 34192210, 48503410, 67670691, 92996475, 126048924, 168700700, 223171325, 292073301, 378462150, 485890534, 618466615
Offset: 1

Views

Author

Xavier Acloque, Oct 21 2003

Keywords

Links

Crossrefs

Cf. A079547, A083200, A088889, A088891, A088892, A088893, A088894.

Programs

  • Mathematica
    Table[((n^8 - (n - 1)^8) - (n^2 - (n - 1)^2))/252, {n, 26}] (* Bruno Berselli, Feb 10 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,25,234,1290,5115,16211,43700},30] (* Harvey P. Dale, Nov 24 2019 *)

Formula

a(n) = ((n^8-(n-1)^8)-(n^2-(n-1)^2))/252 = ((n^8-(n-1)^8)-(n^2-(n-1)^2))/(2^8-2^2).
G.f.: x^2*(1+x)*(1+16*x+46*x^2+16*x^3+x^4)/(1-x)^8. - Bruno Berselli, Feb 10 2012

Extensions

First term added according to the formula from Bruno Berselli, Feb 10 2012

A088894 Polynexus numbers of order 16.

Original entry on oeis.org

0, 1, 656, 64895, 2263323, 40728358, 464160877, 3788798018, 23985732786, 124343403711, 548683300726, 2120490369833, 7334132330405, 23085114905772, 67009386720139, 181293280189204, 461148470725844, 1110780513139917, 2548868780082828, 5600100525524707
Offset: 1

Views

Author

Xavier Acloque, Oct 21 2003

Keywords

Links

Crossrefs

Cf. A079547, A083200, A088889, A088890, A088891, A088892, A088893.

Programs

  • Mathematica
    Table[((n^16 - (n - 1)^16) - (n^4 - (n - 1)^4))/65520, {n, 20}] (* Bruno Berselli, Feb 08 2012 *)

Formula

a(n) = ((n^16-(n-1)^16)-(n^4-(n-1)^4))/65520 = ((n^16-(n-1)^16)-(n^4-(n-1)^4))/(2^16-2^4).
G.f.: x^2*(1+x)*(1+639*x+53880*x^2+1249283*x^3+10687767*x^4+38266494*x^5+59151072*x^6+38266494*x^7+10687767*x^8+1249283*x^9+53880*x^10+639*x^11+x^12)/(1-x)^16. - Bruno Berselli, Feb 08 2012

Extensions

First term added according to the formula from Bruno Berselli, Feb 08 2012

A088893 Polynexus numbers of order 15.

Original entry on oeis.org

0, 1, 437, 32338, 898774, 13420861, 130567049, 929084572, 5210829060, 24240197433, 96985597357, 342789175982, 1092151142842, 3186269918917, 8618003504977, 21826239750488, 52182586901800, 118565859736497, 257462955231909, 536839834252906, 1079191653936254
Offset: 1

Views

Author

Xavier Acloque, Oct 21 2003

Keywords

Links

Crossrefs

Cf. A079547, A083200, A088889, A088890, A088891, A088892, A088894.

Programs

  • Mathematica
    Table[((n^15 - (n - 1)^15) - (n^3 - (n - 1)^3))/32760, {n, 20}] (* Bruno Berselli, Feb 08 2012 *)

Formula

a(n) = ((n^15-(n-1)^15)-(n^3-(n-1)^3))/32760 = ((n^15-(n-1)^15)-(n^3-(n-1)^3))/(2^15-2^3).
G.f.: x^2*(1+422*x+25888*x^2+459134*x^3+3137271*x^4+9505116*x^5+13661136*x^6+9505116*x^7+3137271*x^8+459134*x^9+25888*x^10+422*x^11+x^12)/(1-x)^15. - Bruno Berselli, Feb 08 2012

Extensions

First term added according to the formula from Bruno Berselli, Feb 08 2012

A083200 Polynexus numbers of order 7.

Original entry on oeis.org

0, 1, 17, 118, 514, 1681, 4529, 10612, 22380, 43473, 79057, 136202, 224302, 355537, 545377, 813128, 1182520, 1682337, 2347089, 3217726, 4342394, 5777233, 7587217, 9847036, 12642020, 16069105, 20237841, 25271442, 31307878, 38501009, 47021761, 57059344, 68822512
Offset: 1

Views

Author

Xavier Acloque, Jun 01 2003

Keywords

Links

Crossrefs

Cf. A079547, A088889-A088894 (similar sequences).

Programs

  • Mathematica
    Table[((n^7 - (n - 1)^7) - (n^3 - (n - 1)^3))/120, {n, 31}] (* Bruno Berselli, Feb 13 2012 *)

Formula

a(n) = ((n^7-(n-1)^7)-(n^3-(n-1)^3))/120.
G.f.: x^2*(1+10*x+20*x^2+10*x^3+x^4)/(1-x)^7. - Bruno Berselli, Feb 13 2012

Extensions

More terms from David Wasserman, Oct 26 2004
Offset changed (according to the formula) from Bruno Berselli, Feb 13 2012

A079547 a(n) = ((n^6 - (n-1)^6) - (n^2 - (n-1)^2))/60.

Original entry on oeis.org

0, 1, 11, 56, 192, 517, 1183, 2408, 4488, 7809, 12859, 20240, 30680, 45045, 64351, 89776, 122672, 164577, 217227, 282568, 362768, 460229, 577599, 717784, 883960, 1079585, 1308411
Offset: 1

Views

Author

Xavier Acloque, Jan 22 2003

Keywords

Comments

Polynexus numbers of order 6.
A polynexus (subtractive) function is composed of two or more subtracted nexus numbers divided by an integer x. The general form of the formula is a(n)=((n^p-(n-1)^p)-(n^q-(n-1)^q))/x, where n, p, q and x are integers.
Already known: ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24, giving A006322 for n>1; ((n^4-(n-1)^4) - (n^2-(n-1)^2))/12, giving A000330; ((n^3-(n-1)^3) - (n^1-(n-1)^1))/6, giving A000217; ((n^2-(n-1)^2) - (n^1-(n-1)^1))/2, giving n; ((n^2-(n-1)^2) - (n^0-(n-1)^0))/1, giving 2*n-1. In those examples, x is equal to 1,2,6,12,24, and 3 is also possible.
Also number of monotone n-weightings of complete bipartite digraph K(3,2) if offset were 0; cf. A085464-A085465. - Goran Kilibarda, Vladeta Jovovic, Jul 01 2003
Partial sums of A037270. - J. M. Bergot, Jun 07 2012

Links

Crossrefs

Cf. A006322, A000330, A000217, A047969, A003215, A083200, A088889-A088894.

Programs

  • GAP
    List([1..30], n-> n*(6*n^4-15*n^3+20*n^2-15*n+4)/60) # G. C. Greubel, Jun 19 2019
  • Magma
    [n*(6*n^4-15*n^3+20*n^2-15*n+4)/60: n in [1..30]]; // G. C. Greubel, Jun 19 2019
  • Mathematica
    Table[((n^6 -(n-1)^6) - (n^2 -(n-1)^2))/60, {n, 30}] (* Bruno Berselli, Feb 13 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,11,56,192,517},30] (* Harvey P. Dale, Feb 21 2023 *)
  • PARI
    a(n) = n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 \\ Charles R Greathouse IV, Jan 16 2013
  • Sage
    [n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 for n in (1..30)] # G. C. Greubel, Jun 19 2019

Formula

a(n+1) = Sum_{i=1..n} (i^2 + i^4)/2 = n*(2*n+1)*(n+1)*(3*n^2+3*n+4)/60. - Vladeta Jovovic, Mar 17 2006
G.f.: x^2*(x+1)*(1+4*x+x^2)/(1-x)^6. - Bruno Berselli, Feb 13 2012
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^3. - Enrique Pérez Herrero, Jan 16 2013
E.g.f.: x^2*(30 + 80*x + 45*x^2 + 6*x^3)*exp(x)/60. - G. C. Greubel, Jun 19 2019

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