A083200 -id:A083200 - cp's OEIS frontend

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

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

Xavier Acloque, Jan 22 2003

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

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

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

List([1..30], n-> n*(6*n^4-15*n^3+20*n^2-15*n+4)/60) # G. C. Greubel, Jun 19 2019

[n*(6*n^4-15*n^3+20*n^2-15*n+4)/60: n in [1..30]]; // G. C. Greubel, Jun 19 2019

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 *)

a(n) = n*(6*n^4-15*n^3+20*n^2-15*n+4)/60 \\ Charles R Greathouse IV, Jan 16 2013

[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

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

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

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [dead link, domain owned by a domain grabber].
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

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

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 *)

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

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

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
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. A079547, A083200, A088889, A088890, A088891, A088892, A088893.

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

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

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

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

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

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 *)

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

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

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

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

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 *)

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

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

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

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

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

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 *)

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

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

Xavier Acloque, Oct 21 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

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

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

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

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

A114239 a(n) = (n+1)(n+2)^3*(n+3)(n^2 + 4n + 5)/120.

Original entry on oeis.org

1, 18, 136, 650, 2331, 6860, 17472, 39852, 83325, 162382, 298584, 522886, 878423, 1423800, 2236928, 3419448, 5101785, 7448874, 10666600, 15008994, 20786227, 28373444, 38220480, 50862500, 66931605, 87169446, 112440888, 143748766

Offset: 0

Emeric Deutsch, Nov 18 2005

Kekulé numbers for certain benzenoids.

First differences of A107891. Partial sums of A083200. - Peter Bala, Sep 21 2007

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 167, Table 10.5/I/6).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A047819, A083200, A107891.

a:=n->(n+1)*(n+2)^3*(n+3)*(n^2+4*n+5)/120: seq(a(n),n=0..33);

a(n)=n-=2;(n^7-n^3)/120 \\ Charles R Greathouse IV, Feb 09 2012

a(n-2) = (n^7-n^3)/(2^7-2^3). - David Radcliffe, Dec 27 2008

G.f.: (1+10*x+20*x^2+10*x^3+x^4)/(1-x)^8. - Colin Barker, Feb 09 2012

cp's OEIS Frontend

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

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A088889 Polynexus numbers of order 8.

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A088894 Polynexus numbers of order 16.

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A088890 Polynexus numbers of order 8.

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A088891 Polynexus numbers of order 9.

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A088892 Polynexus numbers of order 14.

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A088893 Polynexus numbers of order 15.

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A114239 a(n) = (n+1)(n+2)^3*(n+3)(n^2 + 4n + 5)/120.

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