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.

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A188124 Number of strictly increasing arrangements of 5 nonzero numbers in -(n+3)..(n+3) with sum zero.

Original entry on oeis.org

0, 4, 16, 42, 90, 172, 296, 482, 740, 1092, 1554, 2154, 2906, 3846, 4992, 6382, 8038, 10004, 12302, 14984, 18074, 21626, 25670, 30266, 35442, 41266, 47770, 55024, 63064, 71966, 81766, 92548, 104350, 117258, 131316, 146616, 163200, 181168, 200566
Offset: 0

Views

Author

R. H. Hardin, Mar 21 2011

Keywords

Comments

Row 5 of A188122.

Examples

			4*x + 16*x^2 + 42*x^3 + 90*x^4 + 172*x^5 + 296*x^6 + 482*x^7 + 740*x^8 + ...
Some solutions for n=6
.-7...-7...-6...-7...-8...-8...-4...-9...-7...-5...-6...-4...-6...-9...-7...-5
.-5...-5...-4...-6...-6...-2...-3...-5...-5...-4...-3...-3...-3...-5...-4...-3
..1....2....2....2....1...-1...-2....1...-4...-2...-2...-2....1....2...-2....1
..5....3....3....5....4....4....4....5....7....4....4....1....2....5....6....3
..6....7....5....6....9....7....5....8....9....7....7....8....6....7....7....4

Links

Programs

  • PARI
    {a(n) = local(v, c, m); m = n+3; forvec( v = vector( 5, i, [-m, m]), if( 0==prod( k=1, 5, v[k]), next); if( 0==sum( k=1, 5, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

Formula

Empirical: a(n)=2*a(n-1)-a(n-3)-2*a(n-5)+2*a(n-6)+a(n-8)-2*a(n-10)+a(n-11) = 269/1728 +235*n^2/144 +161*n/96 +23*n^4/288 +83*n^3/144 +(-1)^n*(1/64-3*n/32) -2*(-1)^n*A130815(n+2)/27 +A057077(n+1)/8.
Empirical: G.f. -2*x*(2+4*x+5*x^2+5*x^3+4*x^4+x^5+2*x^6) / ( (x^2+1)*(1+x+x^2)*(1+x)^2*(x-1)^5 ). - R. J. Mathar, Mar 21 2011

A188125 Number of strictly increasing arrangements of 6 nonzero numbers in -(n+4)..(n+4) with sum zero.

Original entry on oeis.org

4, 16, 52, 137, 308, 624, 1154, 1999, 3278, 5144, 7772, 11387, 16230, 22602, 30830, 41303, 54440, 70734, 90706, 114963, 144146, 178984, 220244, 268797, 325548, 391514, 467756, 555449, 655816, 770208, 900020, 1046787, 1212094, 1397668
Offset: 0

Views

Author

R. H. Hardin, Mar 21 2011

Keywords

Comments

Row 6 of A188122.

Examples

			4 + 16*x + 52*x^2 + 137*x^3 + 308*x^4 + 624*x^5 + 1154*x^6 + 1999*x^7 + 3278*x^8 + ...
Some solutions for n=6
-10...-8...-7...-8...-8...-9...-9...-9...-9...-7..-10...-9...-7..-10...-9...-9
.-8...-6...-5...-5...-6...-3...-7...-3...-2...-5...-6...-5...-5...-6...-4...-5
.-1....1...-1...-1...-1...-2...-2....1...-1...-2...-2...-1...-1...-2...-2...-4
..4....3....1....1....2....3....3....2....1....1....2....1....3....3....1....3
..7....4....2....3....5....4....5....4....2....4....6....6....4....5....5....6
..8....6...10...10....8....7...10....5....9....9...10....8....6...10....9....9

Links

Crossrefs

Cf. A188122

Programs

  • PARI
    {a(n) = local(v, c, m); m = n+4; forvec( v = vector( 6, i, [-m, m]), if( 0==prod( k=1, 6, v[k]), next); if( 0==sum( k=1, 6, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

Formula

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+2*a(n-8)-a(n-11)-a(n-13)+2*a(n-15)-a(n-16)
= 168587/43200 +187*n/32 +3593*n^3/2160 +619*n^2/144 +457*n^4/1440 +11*n^5/450 -(-1)^n/64-3*n*(-1)^n/32 +4*(-1)^n*A119910(n+1)/27 -2*A117444(n+2)/25 +A057077(n)/8.
Empirical: G.f. -x*(-16 -20*x -33*x^2 -50*x^3 -60*x^4 -59*x^5 -51*x^6 -41*x^7 -18*x^8 -3*x^9 -x^10 +x^11 +4*x^12 -2*x^13 -7*x^14 +4*x^15) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x)^2 *(x-1)^6 ). - R. J. Mathar, Mar 21 2011

A188126 Number of strictly increasing arrangements of 7 nonzero numbers in -(n+5)..(n+5) with sum zero.

Original entry on oeis.org

42, 152, 426, 1032, 2216, 4376, 8044, 13994, 23210, 37030, 57086, 85506, 124816, 178186, 249308, 342708, 463550, 618042, 813186, 1057238, 1359422, 1730468, 2182232, 2728362, 3383832, 4165678, 5092482, 6185216, 7466594, 8962070
Offset: 1

Views

Author

R. H. Hardin, Mar 21 2011

Keywords

Examples

			Some solutions for n=6
-10..-10...-6...-7...-6..-11...-8..-10...-8..-11..-10...-9..-11..-11...-9...-9
.-9...-4...-3...-6...-5...-9...-7...-7...-7...-4...-7...-8...-9...-8...-6...-7
.-4...-2...-2...-4...-4...-3...-4...-6...-1...-3...-3...-3...-4...-4...-5...-4
..4....2...-1....1...-1...-1...-3...-1....1...-2...-1...-1....1....3...-4...-2
..5....3....1....3....3....4....5....6....3....1....1....5....2....4....7....4
..6....4....2....6....4....9....6....8....4....8....9....6...10....6....8....8
..8....7....9....7....9...11...11...10....8...11...11...10...11...10....9...10

Links

Crossrefs

Row 7 of A188122.

Formula

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+a(n-6)-2*a(n-7)+2*a(n-8)+a(n-9)-a(n-13)-2*a(n-14)+2*a(n-15)-a(n-16)+a(n-17)+a(n-19)-2*a(n-21)+a(n-22) =
208637*n/12960 +413*(-1)^n/1152 +6403*n^3/1296 +355951*n^2/28800 +11*(-1)^n*n^2/384 +13*(-1)^n*n/96 +28669*n^4/25920 +709*n^5/5400 +841*n^6/129600 +6124649/777600 + (157*A049347(n)+74*A049347(n-1))/486 + 5*A128214(n+3)/81 +2*b(n)/25 + A057079(n+2)/18 -(-1)^(floor((n+1)/2))*A000034(n+1)/8 where b(n) is the 5-periodic sequence (-3,-1,-1,2,3,...) with offset 0.
Empirical: G.f. -2*x *(21 +34*x +61*x^2 +111*x^3 +152*x^4 +206*x^5 +217*x^6 +240*x^7 +212*x^8 +172*x^9 +120*x^10 +77*x^11 +36*x^12 +9*x^13 +11*x^14 -x^15 +4*x^16 +4*x^18 -8*x^20 +4*x^21) / ( (x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x+x^2)^2 *(1+x)^3 *(x-1)^7 ). - R. J. Mathar, Mar 21 2011

A188127 Number of strictly increasing arrangements of 8 nonzero numbers in -(n+6)..(n+6) with sum zero.

Original entry on oeis.org

137, 484, 1398, 3528, 7970, 16547, 32035, 58595, 102113, 170844, 275878, 432018, 658432, 979785, 1427065, 2039067, 2863403, 3958322, 5393994, 7254686, 9640296, 12669003, 16479033, 21231771, 27113883, 34340884, 43159574, 53852210, 66739242
Offset: 1

Views

Author

R. H. Hardin Mar 21 2011

Keywords

Comments

Row 8 of A188122

Examples

			Some solutions for n=6
-12..-11..-12...-8..-10..-12...-9..-10...-6..-12...-8..-10..-12..-10..-12..-11
.-9..-10...-7...-7...-7..-11...-8...-8...-5...-9...-7...-6...-4...-8...-9...-7
.-8...-2...-6...-4...-4...-3...-7...-7...-2...-3...-3...-3...-3...-5...-7...-5
.-2....1...-2...-3...-3...-2...-6...-5...-1...-2...-1...-1....1...-3....2...-4
..5....2...-1....3...-2....1....1....4....1....1....1....1....2....1....4...-2
..7....3....5....4....5....4....6....7....2....4....3....2....3....5....5....8
..9....5...11....7...10...11...11....8....5...10....4....5....6....8....7....9
.10...12...12....8...11...12...12...11....6...11...11...12....7...12...10...12

Links

Formula

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+a(n-6)-a(n-7)+a(n-9)+a(n-10)+a(n-12)-2*a(n-13)-2*a(n-16)+a(n-17)+a(n-19)+a(n-20)-a(n-22)+a(n-23)-a(n-24)-a(n-26)+2*a(n-28)-a(n-29).
Empirical: G.f. -x*(-137 -210*x -3284*x^12 -869*x^3 -1398*x^4 -2142*x^5 -2816*x^6 -3546*x^7 -4084*x^8 -4269*x^10 -3951*x^11 -2644*x^13 -1892*x^14 -1202*x^15 -768*x^16 -389*x^17 -202*x^18 -87*x^19 -5*x^20 +20*x^21 -26*x^22 +32*x^23 +5*x^24 +29*x^25 -9*x^26 -57*x^27 +31*x^28 -4356*x^9 -430*x^2) / ( (x^2-x+1) *(x^5-1) *(x^7-1) *(x^2+1) *(1+x+x^2)^2 *(1+x)^3 *(x-1)^6 ). - R. J. Mathar, Mar 21 2011
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