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.

Previous Showing 11-20 of 22 results. Next

A204467 Number of 3-element subsets that can be chosen from {1,2,...,6*n+3} having element sum 9*n+6.

Original entry on oeis.org

1, 8, 25, 50, 85, 128, 181, 242, 313, 392, 481, 578, 685, 800, 925, 1058, 1201, 1352, 1513, 1682, 1861, 2048, 2245, 2450, 2665, 2888, 3121, 3362, 3613, 3872, 4141, 4418, 4705, 5000, 5305, 5618, 5941, 6272, 6613, 6962, 7321, 7688, 8065, 8450, 8845, 9248, 9661
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 9*n+6 into 3 distinct parts <= 6*n+3.

Examples

			a(1) = 8 because there are 8 3-element subsets that can be chosen from {1,2,...,9} having element sum 15: {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}.

Links

Crossrefs

Bisection of column k=3 of A204459.
Cf. A104185.

Programs

  • Maple
    a:= n-> 1 +floor((3+9/2*n)*n):
seq(a(n), n=0..50);
  • Mathematica
    Table[(6n(3n+2)+(-1)^n+3)/4,{n,0,50}] (* or *) LinearRecurrence[{2,0,-2,1},{1,8,25,50},50] (* Harvey P. Dale, May 25 2015 *)

Formula

a(n) = 1+floor((3+9/2*n)*n).
G.f.: -(2*x+1)*(x^2+4*x+1)/((x+1)*(x-1)^3).
a(n) = (6*n*(3*n+2)+(-1)^n+3)/4. - Bruno Berselli, Jan 17 2012
a(0)=1, a(1)=8, a(2)=25, a(3)=50, a(n)=2*a(n-1)-2*a(n-3)+a(n-4). - Harvey P. Dale, May 25 2015

A204468 Number of 4-element subsets that can be chosen from {1,2,...,4*n} having element sum 8*n+2.

Original entry on oeis.org

0, 1, 8, 33, 86, 177, 318, 519, 790, 1143, 1588, 2135, 2796, 3581, 4500, 5565, 6786, 8173, 9738, 11491, 13442, 15603, 17984, 20595, 23448, 26553, 29920, 33561, 37486, 41705, 46230, 51071, 56238, 61743, 67596, 73807, 80388, 87349, 94700, 102453, 110618, 119205
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 8*n+2 into 4 distinct parts <= 4*n.

Examples

			a(2) = 8 because there are 8 4-element subsets that can be chosen from {1,2,...,8} having element sum 18: {1,2,7,8}, {1,3,6,8}, {1,4,5,8}, {1,4,6,7}, {2,3,5,8}, {2,3,6,7}, {2,4,5,7}, {3,4,5,6}.

Links

Crossrefs

Column k=4 of A204459.

Programs

  • Maple
    a:= n-> ((9+(16*n-18)*n)*n +[0, 2, -2][irem(n, 3)+1])/9:
seq(a(n), n=0..50);
  • Mathematica
    LinearRecurrence[{3,-3,2,-3,3,-1},{0,1,8,33,86,177},50] (* or *) CoefficientList[Series[(x (1+5 x+12 x^2+9 x^3+5 x^4))/((-1+x)^4 (1+x+x^2)),{x,0,50}],x] (* Harvey P. Dale, Feb 25 2021 *)

Formula

G.f.: x*(5*x^4+9*x^3+12*x^2+5*x+1)/((x^2+x+1)*(x-1)^4).
a(n) = (1/9)*(2*A102283(n) + n*(16*n^2-18*n+9)). - Bruno Berselli, Jan 19 2012

A204469 Number of 5-element subsets that can be chosen from {1,2,...,10*n+5} having element sum 25*n+15.

Original entry on oeis.org

1, 141, 1394, 5910, 17053, 39361, 78602, 141702, 236833, 373309, 561704, 813722, 1142341, 1561651, 2087034, 2734970, 3523243, 4470721, 5597592, 6925112, 8475873, 10273519, 12343044, 14710482, 17403231, 20449711, 23879724, 27724080, 32014983, 36785631, 42070632
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 25*n+15 into 5 distinct parts <= 10*n+5.

Examples

			a(0) = 1 because there is 1 5-element subset that can be chosen from {1,2,3,4,5} having element sum 15: {1,2,3,4,5}.

Links

Crossrefs

Bisection of column k=5 of A204459.

Programs

  • Maple
    a:= n-> (Matrix(11, (i, j)-> `if`(i=j-1, 1, `if`(i=11, [1, -2, 0, 1, 0, 2, -2, 0, -1, 0, 2][j], 0)))^n. <<1, 141, 1394, 5910, 17053, 39361, 78602, 141702, 236833, 373309, 561704>>)[1, 1]: seq(a(n), n=0..50);

Formula

G.f.: -(12*x^10 +390*x^9 +1821*x^8 +4057*x^7 +6070*x^6 +6651*x^5 +5374*x^4 +3123*x^3 +1112*x^2 +139*x+1) / ((x^2+x+1)*(x^2+1)*(x+1)^2*(x-1)^5).

A204470 Number of 6-element subsets that can be chosen from {1,2,...,6*n} having element sum 18*n+3.

Original entry on oeis.org

0, 1, 58, 676, 3486, 11963, 32134, 73294, 148718, 276373, 479632, 787986, 1237756, 1872809, 2745266, 3916220, 5456444, 7447107, 9980486, 13160678, 17104314, 21941271, 27815384, 34885162, 43324496, 53323377, 65088604, 78844500, 94833624, 113317483, 134577246
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 18*n+3 into 6 distinct parts <= 6*n.

Examples

			a(2) = 58 because there are 58 6-element subsets that can be chosen from {1,2,...,12} having element sum 39: {1,2,3,10,11,12}, {1,2,4,9,11,12}, ..., {3,5,6,7,8,10}, {4,5,6,7,8,9}.

Links

Crossrefs

Column k=6 of A204459.

Programs

  • Maple
    a:= n-> (Matrix(11, (i, j)-> `if`(i=j-1, 1, `if`(i=11, [-1, 4, -5, 0, 5, -3, -3, 5, 0, -5, 4][j], 0)))^n. <<0, 1, 58, 676, 3486, 11963, 32134, 73294, 148718, 276373, 479632>>)[1, 1]: seq(a(n), n=0..50);

Formula

G.f.: x*(32*x^9 +352*x^8 +979*x^7 +1370*x^6 +1425*x^5 +1394*x^4 +1072*x^3 +449*x^2 +54*x+1) / ((x+1)*(x^4+x^3+x^2+x+1)*(x-1)^6).

A204471 Number of 7-element subsets that can be chosen from {1,2,...,14*n+7} having element sum 49*n+28.

Original entry on oeis.org

1, 3370, 108108, 957332, 4721127, 16627422, 47043624, 114106128, 246902225, 489197948, 903720482, 1576984058, 2624673317, 4197566692, 6488021194, 9736993054, 14241624013, 20363359008, 28536634496, 39278092476, 53196371385, 71002416300, 93520372350, 121698990952
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 49*n+28 into 7 distinct parts <= 14*n+7.

Examples

			a(0) = 1 because there is 1 7-element subset that can be chosen from {1,2,...,7} having element sum 28: {1,2,3,4,5,6,7}.

Links

Crossrefs

Bisection of column k=7 of A204459.

Programs

  • Maple
    a:= n-> (Matrix(22, (i, j)-> `if`(i=j-1, 1, `if`(i=22, [1, -2, 0, 1, 0, 1, -1, 2, -2, -1, 0, 0, 0, 1, 2, -2, 1, -1, 0, -1, 0, 2][j], 0)))^n. <<1, 3370, 108108, 957332, 4721127, 16627422, 47043624, 114106128, 246902225, 489197948, 903720482, 1576984058, 2624673317, 4197566692, 6488021194, 9736993054, 14241624013, 20363359008, 28536634496, 39278092476, 53196371385, 71002416300>>)[1, 1]: seq(a(n), n=0..50);

Formula

G.f.: -(94*x^21 +18950*x^20 +265472*x^19 +1391863*x^18 +4387222*x^17 +10120300*x^16 +18809933*x^15 +29668549*x^14 +40847915*x^13 +49820911*x^12 +54281003*x^11 +53032087*x^10 +46410392*x^9 +36173353*x^8 +24844747*x^7 +14749481*x^6 +7293277*x^5 +2809833*x^4 +741117*x^3 +101368*x^2 +3368*x+1) / ((x^2-x+1)*(x^4+x^3+x^2+x+1)*(x^2+1)*(x^2+x+1)^2*(x+1)^3*(x-1)^7).

A204472 Number of 8-element subsets that can be chosen from {1,2,...,8*n} having element sum 32*n+4.

Original entry on oeis.org

0, 1, 526, 17575, 178870, 1016737, 4083008, 13011585, 35154340, 83916031, 181913856, 365087337, 687884214, 1229647953, 2102332580, 3459670513, 5507918992, 8518310823, 12841335118, 18922973607, 27323018256, 38735595881, 54012025302, 74186132807, 100502151596
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 32*n+4 into 8 distinct parts <= 8*n.

Examples

			a(2) = 526 because there are 526 8-element subsets that can be chosen from {1,2,...,16} having element sum 68: {1,2,3,4,13,14,15,16}, {1,2,3,5,12,14,15,16}, ..., {4,6,7,8,9,10,11,13}, {5,6,7,8,9,10,11,12}.

Links

Crossrefs

Column k=8 of A204459.

Programs

  • Maple
    a:= n-> (Matrix(22, (i, j)-> `if`(i=j-1, 1, `if`(i=22, [-1, 4, -6, 6, -9, 13, -13, 13, -16, 19, -19, 18, -19, 19, -16, 13, -13, 13, -9, 6, -6, 4][j], 0)))^n. <<0, 1, 526, 17575, 178870, 1016737, 4083008, 13011585, 35154340, 83916031, 181913856, 365087337, 687884214, 1229647953, 2102332580, 3459670513, 5507918992, 8518310823, 12841335118, 18922973607, 27323018256, 38735595881>>)[1, 1]: seq(a(n), n=0..50);

Formula

G.f.: x*(289*x^20 +11190*x^19 +91493*x^18 +352388*x^17 +898356*x^16 +1737191*x^15 +2761013*x^14 +3796426*x^13 +4655081*x^12 +5159765*x^11 +5190716*x^10 +4740985*x^9 +3917109*x^8 +2893806*x^7 +1858105*x^6 +988551*x^5 +403560*x^4 +111720*x^3 +15477*x^2 +522*x+1) / ((x^4+x^3+x^2+x+1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x^2+x+1)^2*(x-1)^8).

A204473 Number of 9-element subsets that can be chosen from {1,2,...,18*n+9} having element sum 81*n+45.

Original entry on oeis.org

1, 94257, 9853759, 182262067, 1537408202, 8262875230, 33131331832, 108130342498, 302954110225, 754561227653, 1711557426281, 3597716377411, 7099506906934, 13283048544410, 23746473266386, 40814224081470, 67780377968751, 109208632376183, 171297152624647
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 81*n+45 into 9 distinct parts <= 18*n+9.

Examples

			a(0) = 1 because there is 1 9-element subset that can be chosen from {1,2,...,9} having element sum 45: {1,2,3,4,5,6,7,8,9}.

Links

Crossrefs

Bisection of column k=9 of A204459.

Programs

  • Maple
    a:= n-> (Matrix(31, (i, j)-> `if`(i=j-1, 1, `if`(i=31, [1, -3, 1, 5, -6, 1, 5, -8, 2, 9, -11, 0, 11, -11, 1, 11, -11, -1, 11, -11, 0, 11, -9, -2, 8, -5, -1, 6, -5, -1, 3][j], 0)))^n.
<<1, 94257, 9853759, 182262067, 1537408202, 8262875230, 33131331832, 108130342498, 302954110225, 754561227653, 1711557426281, 3597716377411, 7099506906934, 13283048544410, 23746473266386, 40814224081470, 67780377968751,
109208632376183, 171297152624647, 262317027786495, 393133642429336, 577820819931050, 834378155041412, 1185562499279422, 1659845127394359, 2292506656080729, 3126882354024441, 4215771021760211, 5623021191591966, 7425308933006994, 9714122125363850>>)[1, 1]: seq(a(n), n=0..50);

Formula

G.f.: -(910*x^30 +1040804*x^29 +38021156*x^28 +382272336*x^27 +1924406509*x^26 +6310497232*x^25 +15598550757*x^24 +31691324994*x^23 +55644068089*x^22 +87101380417*x^21 +124235349095*x^20 +163834246902*x^19 +201423850605*x^18 +231972434360*x^17 +250948109605*x^16 +255267409282*x^15 +244185313288*x^14 +219577712922*x^13 +185287461384*x^12 +146192435862*x^11 +107203950569*x^10 +72278724244*x^9 +44015118309*x^8 +23603015876*x^7 +10732396387*x^6 +3881615945*x^5 +1000947039*x^4 +152795052*x^3 +9570989*x^2 +94254*x+1) / ((x^4+x^3+x^2+x+1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x^4+1)*(x^2+1)^2*(x+1)^4*(x-1)^9).

A204474 Number of 10-element subsets that can be chosen from {1,2,...,10*n} having element sum 50*n+5.

Original entry on oeis.org

0, 1, 5448, 517971, 10388788, 97809616, 587267282, 2615047418, 9408671330, 28851490163, 78132541528, 191563698893, 432971108530, 914435915008, 1823570327812, 3461969170632, 6297974185728, 11037428544793, 18716682230144, 30822735206253, 49446074607790
Offset: 0

Views

Author

Alois P. Heinz, Jan 16 2012

Keywords

Comments

a(n) is the number of partitions of 50*n+5 into 10 distinct parts <= 10*n.

Examples

			a(2) = 5448 because there are 5448 10-element subsets that can be chosen from {1,2,...,20} having element sum 105: {1,2,3,4,5,16,17,18,19,20},{1,2,3,4,6,15,17,18,19,20}, ..., {6,7,8,9,10,11,12,13,14,15}.

Links

Crossrefs

Column k=10 of A204459.

Programs

  • Maple
    a:= n-> (Matrix(32, (i, j)-> `if`(i=j-1, 1, `if`(i=32, [-1, 4, -5, 2, -2, 2, 5, -7, 2, -2, -1, 9, -6, -1, 1, -7, 14, -7, 1, -1, -6, 9, -1, -2, 2, -7, 5, 2, -2, 2, -5, 4][j], 0)))^n.
<<0, 1, 5448, 517971, 10388788, 97809616, 587267282, 2615047418, 9408671330, 28851490163, 78132541528, 191563698893, 432971108530, 914435915008, 1823570327812, 3461969170632, 6297974185728, 11037428544793, 18716682230144,
30822735206253, 49446074607790, 77472474506552, 118820782200294, 178734513401148, 264135920159882, 384052079868415, 550123481245484, 777206553792057, 1084082600833512, 1494286652957642, 2037070858360564, 2748518169368740>>)[1, 1]: seq(a(n), n=0..50);

Formula

G.f.: (2934*x^31 +348266*x^30 +6597024*x^29 +49734090*x^28 +217728951*x^27 +667463008*x^26 +1585348242*x^25 +3103463874*x^24 +5228487221*x^23 +7832917054*x^22 +10700328447*x^21 +13574603633*x^20 +16172892190*x^19 +18185529447*x^18 +19316679453*x^17 +19372433191*x^16 +18342644114*x^15 +16405735056*x^14 +13851791136*x^13 +10991882803*x^12 +8111849599*x^11 +5470428985*x^10 +3289797304*x^9 +1708913237*x^8 +735273835*x^7 +246947710*x^6 +58833425*x^5 +8344142*x^4 +496184*x^3 +5444*x^2+x) / ((x^2+1)*(x^6+x^3+1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x+1)^2*(x^2+x+1)^3*(x-1)^10).

A185062 Number of n-element subsets that can be chosen from {1,2,...,2*n^3} having element sum n^4.

Original entry on oeis.org

1, 1, 7, 351, 56217, 18878418, 11163952389, 10292468330630, 13703363417260677, 24932632800863823135, 59509756600504616529186, 180533923700628895521591343, 678854993880375551144618682344, 3100113915888360851262910882014885
Offset: 0

Views

Author

Alois P. Heinz, Jan 22 2012

Keywords

Comments

a(n) is the number of partitions of n^4 into n distinct parts <= 2*n^3.

Examples

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.

Crossrefs

Column k=3 of A185282.
Cf. A202261, A186730, A204459.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n^4, 2*n^3, n):
seq(a(n), n=0..5);
  • Mathematica
    $RecursionLimit = 10000;
b[n_, i_, t_] := b[n, i, t] =
     If[i < t || n < t(t+1)/2 || n > t(2i - t + 1)/2, 0,
     If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]];
a[n_] := b[n^4, 2 n^3, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

A204462 Number of 2*n-element subsets that can be chosen from {1,2,...,12*n} having element sum n*(12*n+1).

Original entry on oeis.org

1, 6, 318, 32134, 4083008, 587267282, 91403537276, 15027205920330, 2572042542065646, 454018964549333284, 82122490665668040962, 15150820045467016057500, 2841258381788564812646472, 540201085284535788002286246, 103917818379993516623446237348
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2012

Keywords

Comments

a(n) is the number of partitions of n*(12*n+1) into 2*n distinct parts <=12*n.

Examples

			a(1) = 6 because there are 6 2-element subsets that can be chosen from {1,2,...,12} having element sum 13: {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}.

Crossrefs

Bisection of row n=6 of A204459.

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n*(12*n+1), 12*n, 2*n):
seq(a(n), n=0..12);
  • Mathematica
    b[n_, i_, t_] /; it(2i-t+1)/2 = 0; b[0, , ] = 1;
b[n_, i_, t_] := b[n, i, t] = b[n, i-1, t] + If[nJean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
Previous Showing 11-20 of 22 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.