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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 11-15 of 15 results.

A220132 Number of tilings of an n X 7 rectangle using integer-sided rectangular tiles of area n.

Original entry on oeis.org

1, 1, 21, 9, 65, 4, 88, 2, 65, 9, 26, 1, 180, 1, 22, 12, 65, 1, 88, 1, 70, 10, 21, 1, 180, 4, 21, 9, 66, 1, 93, 1, 65, 9, 21, 5, 180, 1, 21, 9, 70, 1, 89, 1, 65, 12, 21, 1, 180, 2, 26, 9, 65, 1, 88, 4, 66, 9, 21, 1, 185, 1, 21, 10, 65, 4, 88, 1, 65, 9, 27, 1
Offset: 0

Views

Author

Alois P. Heinz, Dec 06 2012

Keywords

Comments

1 followed by period 420: (1, 21, ..., 186) repeated; offset 0.

Examples

			a(5) = 4, because there are 4 tilings of a 5 X 7 rectangle using integer-sided rectangular tiles of area 5:
._._._._._._._.  ._________._._.
| | | | | | | |  |_________| | |
| | | | | | | |  |_________| | |
| | | | | | | |  |_________| | |
| | | | | | | |  |_________| | |
|_|_|_|_|_|_|_|  |_________|_|_|
._._________._.  ._._._________.
| |_________| |  | | |_________|
| |_________| |  | | |_________|
| |_________| |  | | |_________|
| |_________| |  | | |_________|
|_|_________|_|  |_|_|_________|

Links

Crossrefs

Row n=7 of A220122.

Programs

  • Maple
    gf:= -(185*x^26 +186*x^25 +392*x^24 +402*x^23 +673*x^22 +687*x^21 +1046*x^20 +877*x^19 +1300*x^18 +1119*x^17 +1374*x^16 +1128*x^15 +1353*x^14 +1010*x^13 +1169*x^12 +760*x^11 +822*x^10 +567*x^9 +564*x^8 +325*x^7 +310*x^6 +135*x^5 +121*x^4 +34*x^3 +24*x^2 +2*x +1) /
(x^26 +x^25 +2*x^24 +2*x^23 +3*x^22 +3*x^21 +4*x^20 +3*x^19 +4*x^18 +3*x^17 +3*x^16 +2*x^15 +x^14 -x^12 -2*x^11 -3*x^10 -3*x^9 -4*x^8 -3*x^7 -4*x^6 -3*x^5 -3*x^4 -2*x^3 -2*x^2 -x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);

Formula

G.f.: see Maple program.

A220133 Number of tilings of an n X 8 rectangle using integer-sided rectangular tiles of area n.

Original entry on oeis.org

1, 1, 34, 13, 143, 5, 209, 3, 250, 13, 44, 1, 472, 1, 36, 19, 250, 1, 209, 1, 153, 15, 34, 1, 681, 5, 34, 13, 145, 1, 221, 1, 250, 13, 34, 7, 472, 1, 34, 13, 260, 1, 211, 1, 143, 19, 34, 1, 681, 3, 44, 13, 143, 1, 209, 5, 252, 13, 34, 1, 484, 1, 34, 15, 250, 5
Offset: 0

Views

Author

Alois P. Heinz, Dec 06 2012

Keywords

Comments

1 followed by period 840: (1, 34, ..., 695) repeated; offset 0.

Examples

			a(5) = 5, because there are 5 tilings of a 5 X 8 rectangle using integer-sided rectangular tiles of area 5:
._._._._._._._._.   ._________._._._.   ._._________._._.
| | | | | | | | |   |_________| | | |   | |_________| | |
| | | | | | | | |   |_________| | | |   | |_________| | |
| | | | | | | | |   |_________| | | |   | |_________| | |
| | | | | | | | |   |_________| | | |   | |_________| | |
|_|_|_|_|_|_|_|_|   |_________|_|_|_|   |_|_________|_|_|
._._._________._.   ._._._._________.
| | |_________| |   | | | |_________|
| | |_________| |   | | | |_________|
| | |_________| |   | | | |_________|
| | |_________| |   | | | |_________|
|_|_|_________|_|   |_|_|_|_________|

Links

Crossrefs

Row n=8 of A220122.

Programs

  • Maple
    gf:= -(694*x^46 +x^45 +728*x^44 +708*x^43 +872*x^42 +1441*x^41 +1789*x^40 +928*x^39 +2784*x^38 +1967*x^37 +2307*x^36 +3029*x^35 +3122*x^34 +2593*x^33 +4196*x^32 +2514*x^31 +3854*x^30 +3978*x^29 +3762*x^28
+3055*x^27 +4448*x^26 +2969*x^25 +4154*x^24 +3352*x^23 +3461*x^22 +2969*x^21 +3755*x^20 +2362*x^19 +3069*x^18 +2592*x^17 +2468*x^16 +1821*x^15 +2117*x^14 +1207*x^13 +1736*x^12 +950*x^11 +921*x^10 +581*x^9 +705*x^8 +235*x^7 +403*x^6 +55*x^5 +179*x^4 +15*x^3 +35*x^2 +x +1) /
(x^46 +x^44 +x^43 +x^42 +2*x^41 +2*x^40 +x^39 +3*x^38 +2*x^37 +2*x^36 +3*x^35 +2*x^34 +2*x^33 +3*x^32 +x^31 +2*x^30 +2*x^29 +x^28 +x^27 +x^26
+x^24 -x^22 -x^20 -x^19 -x^18 -2*x^17 -2*x^16 -x^15 -3*x^14 -2*x^13 -2*x^12 -3*x^11 -2*x^10 -2*x^9 -3*x^8 -x^7 -2*x^6 -2*x^5 -x^4 -x^3 -x^2 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..80);

Formula

G.f.: see Maple program.

A220134 Number of tilings of an n X 9 rectangle using integer-sided rectangular tiles of area n.

Original entry on oeis.org

1, 1, 55, 19, 281, 6, 473, 4, 495, 37, 75, 1, 1091, 1, 60, 30, 495, 1, 609, 1, 309, 22, 55, 1, 1509, 6, 55, 37, 286, 1, 499, 1, 495, 19, 55, 9, 1259, 1, 55, 19, 523, 1, 478, 1, 281, 48, 55, 1, 1509, 4, 75, 19, 281, 1, 609, 6, 500, 19, 55, 1, 1125, 1, 55, 40
Offset: 0

Views

Author

Alois P. Heinz, Dec 06 2012

Keywords

Comments

1 followed by period 2520: (1, 55, ..., 1716) repeated; offset 0.

Examples

			a(7) = 4, because there are 4 tilings of a 7 X 9 rectangle using integer-sided rectangular tiles of area 7:
._._._._._._._._._.   ._____________._._.
| | | | | | | | | |   |_____________| | |
| | | | | | | | | |   |_____________| | |
| | | | | | | | | |   |_____________| | |
| | | | | | | | | |   |_____________| | |
| | | | | | | | | |   |_____________| | |
| | | | | | | | | |   |_____________| | |
|_|_|_|_|_|_|_|_|_|   |_____________|_|_|
._._____________._.   ._._._____________.
| |_____________| |   | | |_____________|
| |_____________| |   | | |_____________|
| |_____________| |   | | |_____________|
| |_____________| |   | | |_____________|
| |_____________| |   | | |_____________|
| |_____________| |   | | |_____________|
|_|_____________|_|   |_|_|_____________|

Links

Crossrefs

Row n=9 of A220122.

Programs

  • Maple
    gf:= -(1715*x^84 -1714*x^83 +1769*x^82 -35*x^81 +2032*x^80 -255*x^79 +2518*x^78 -443*x^77 +4730*x^76 -427*x^75 +4827*x^74 +13*x^73 +9665*x^72 -2709*x^71 +8592*x^70 +1768*x^69 +11758*x^68 -521*x^67 +13222*x^66 -578*x^65 +17124*x^64 +1707*x^63 +15652*x^62 +761*x^61 +24022*x^60
-1869*x^59 +20608*x^58 +3671*x^57 +25352*x^56 +340*x^55 +25138*x^54 +265*x^53 +28854*x^52 +3728*x^51 +25940*x^50 -281*x^49 +33322*x^48 +443*x^47 +26340*x^46 +4094*x^45 +30372*x^44 +578*x^43 +28000*x^42 +578*x^41 +28658*x^40 +4094*x^39 +24626*x^38 -1271*x^37 +28180*x^36
+1433*x^35 +20798*x^34 +2014*x^33 +21998*x^32 +265*x^31 +18282*x^30 +340*x^29 +16782*x^28 +1957*x^27 +13752*x^26 -1869*x^25 +13738*x^24 +761*x^23 +8796*x^22 -7*x^21 +8554*x^20 -578*x^19 +6366*x^18 -521*x^17 +4902*x^16 +54*x^15 +3450*x^14 -995*x^13 +2809*x^12 +13*x^11 +1399*x^10 -427*x^9 +1302*x^8 -443*x^7 +804*x^6 -255*x^5 +318*x^4-35*x^3 +55*x^2+1)/
(x^84 -x^83 +x^82 +x^80 +x^78 +2*x^76 +2*x^74 +4*x^72 -x^71 +3*x^70 +x^69 +4*x^68 +4*x^66 +5*x^64 +x^63 +4*x^62 +6*x^60 +4*x^58 +x^57 +5*x^56 +4*x^54 +4*x^52 +x^51 +3*x^50 -x^49 +3*x^48 +x^47 +x^46 +x^44 -x^40 -x^38
-x^37 -3*x^36 +x^35 -3*x^34 -x^33 -4*x^32 -4*x^30 -5*x^28 -x^27 -4*x^26 -6*x^24 -4*x^22 -x^21 -5*x^20 -4*x^18 -4*x^16 -x^15 -3*x^14 +x^13 -4*x^12 -2*x^10 -2*x^8 -x^6 -x^4 -x^2 +x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);

Formula

G.f.: see Maple program.

A220135 Number of tilings of an n X 10 rectangle using integer-sided rectangular tiles of area n.

Original entry on oeis.org

1, 1, 89, 28, 590, 8, 1002, 5, 1209, 64, 254, 1, 2861, 1, 99, 47, 1209, 1, 1274, 1, 1045, 34, 89, 1, 4146, 8, 89, 64, 600, 1, 1527, 1, 1209, 28, 89, 12, 3197, 1, 89, 28, 1968, 1, 1014, 1, 590, 83, 89, 1, 4146, 5, 254, 28, 590, 1, 1274, 8, 1219, 28, 89, 1, 3904
Offset: 0

Views

Author

Alois P. Heinz, Dec 06 2012

Keywords

Comments

1 followed by period 2520: (1, 89, ..., 5841) repeated; offset 0.

Examples

			a(7) = 5, because there are 5 tilings of a 7 X 10 rectangle using integer-sided rectangular tiles of area 7:
._._._._._._._._._._.   ._____________._._._.   ._._____________._._.
| | | | | | | | | | |   |_____________| | | |   | |_____________| | |
| | | | | | | | | | |   |_____________| | | |   | |_____________| | |
| | | | | | | | | | |   |_____________| | | |   | |_____________| | |
| | | | | | | | | | |   |_____________| | | |   | |_____________| | |
| | | | | | | | | | |   |_____________| | | |   | |_____________| | |
| | | | | | | | | | |   |_____________| | | |   | |_____________| | |
|_|_|_|_|_|_|_|_|_|_|   |_____________|_|_|_|   |_|_____________|_|_|
._._._____________._.   ._._._._____________.
| | |_____________| |   | | | |_____________|
| | |_____________| |   | | | |_____________|
| | |_____________| |   | | | |_____________|
| | |_____________| |   | | | |_____________|
| | |_____________| |   | | | |_____________|
| | |_____________| |   | | | |_____________|
|_|_|_____________|_|   |_|_|_|_____________|

Links

Crossrefs

Row n=10 of A220122.

Programs

  • Maple
    gf:= -(-5840*x^136 +5839*x^135 -5928*x^134 +60*x^133 +5189*x^132 -5285*x^131 -1496*x^130 +928*x^129 -7484*x^128 +557*x^127 -494*x^126 -836*x^125 -14180*x^124 +13384*x^123 -15627*x^122 -5927*x^121 +10767*x^120 -12422*x^119 -11498*x^118 +8324*x^117 -24921*x^116 +5813*x^115 -7409*x^114 -3505*x^113 -22788*x^112 +13672*x^111 -27634*x^110 -12862*x^109 +11206*x^108 -17207*x^107 -26452*x^106
+17129*x^105 -50277*x^104 +11512*x^103 -17938*x^102 -12787*x^101 -23042*x^100 +7805*x^99 -45002*x^98 -10518*x^97 -2969*x^96 -17338*x^95 -39604*x^94 +21144*x^93 -68673*x^92 +12881*x^91 -28074*x^90 -22885*x^89 -22229*x^88 +3198*x^87 -63456*x^86 +9*x^85 -20501*x^84 -17035*x^83 -44066*x^82 +17258*x^81 -76763*x^80 +11371*x^79 -35515*x^78 -26786*x^77 -19392*x^76 +967*x^75 -73127*x^74 +8938*x^73 -34070*x^72 -17281*x^71
-40042*x^70 +11259*x^69 -71956*x^68 +11259*x^67 -40042*x^66 -23120*x^65 -16553*x^64 -2740*x^63 -67288*x^62 +12645*x^61 -36909*x^60 -15108*x^59 -29676*x^58 +5532*x^57 -59246*x^56 +11419*x^55 -38227*x^54 -17035*x^53 -8823*x^52 -5830*x^51 -51778*x^50 +14876*x^49 -33907*x^48 -11207*x^47 -16396*x^46 +1203*x^45 -39478*x^44 +9466*x^43 -27926*x^42 -11499*x^41 -2969*x^40 -4679*x^39 -33324*x^38 +13644*x^37 -23042*x^36 -6948*x^35
-6260*x^34 -166*x^33 -21082*x^32 +5451*x^31 -14774*x^30 -5529*x^29 -472*x^28 -1184*x^27 -15956*x^26 +7833*x^25 -11110*x^24 -3505*x^23 -1570*x^22 -26*x^21 -7404*x^20 +2485*x^19 -5659*x^18 -744*x^17 -911*x^16 -88*x^15 -3949*x^14 +1706*x^13 -2502*x^12 -836*x^11 -494*x^10 +557*x^9 -1645*x^8 +928*x^7 -1496*x^6 +554*x^5 -650*x^4 +60*x^3 -89*x^2 -1) /
(-x^136 +x^135 -x^134 +x^132 -x^131 -x^128 -2*x^124 +2*x^123 -2*x^122 -x^121 +2*x^120 -2*x^119 -x^118 +x^117 -3*x^116 +x^115 -x^114 -2*x^112 +x^111 -2*x^110 -2*x^109 +2*x^108 -2*x^107 -2*x^106 +2*x^105 -5*x^104 +2*x^103 -2*x^102 -x^101 -x^99 -2*x^98 -x^97 -x^95 -2*x^94 +2*x^93
-5*x^92 +2*x^91 -2*x^90 -2*x^89 +2*x^88 -2*x^87 -2*x^86 +x^85 -2*x^84 -x^82 +x^81 -3*x^80 +x^79 -x^78 -2*x^77 +3*x^76 -2*x^75 -x^74 +2*x^73 -3*x^72 +x^71 -x^65 +3*x^64 -2*x^63 +x^62 +2*x^61 -3*x^60 +2*x^59 +x^58 -x^57 +3*x^56 -x^55 +x^54 +2*x^52 -x^51 +2*x^50 +2*x^49 -2*x^48 +2*x^47 +2*x^46
-2*x^45 +5*x^44 -2*x^43 +2*x^42 +x^41 +x^39 +2*x^38 +x^37 +x^35 +2*x^34 -2*x^33 +5*x^32 -2*x^31 +2*x^30 +2*x^29 -2*x^28 +2*x^27 +2*x^26 -x^25 +2*x^24 +x^22 -x^21 +3*x^20 -x^19 +x^18 +2*x^17 -2*x^16 +x^15 +2*x^14 -2*x^13 +2*x^12 +x^8 +x^5 -x^4 +x^2 -x +1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);

Formula

G.f.: see Maple program.

A182106 Number of tilings of an n X n square with rectangles with integer sides and area n.

Original entry on oeis.org

1, 1, 2, 2, 9, 2, 46, 2, 250, 37, 254, 2, 31052, 2, 1480, 896, 306174, 2, 2097506, 2, 6025516, 6638, 59930, 2, 22119057652, 1141, 400776, 1028162, 1205138020, 2, 188380348290, 2
Offset: 0

Views

Author

Felix A. Pahl, Apr 12 2012

Keywords

Links

Crossrefs

Diagonal of A220122. - Alois P. Heinz, Dec 07 2012

Programs

  • Java
    public class Question130758 {
    final static int maxn = 23;
    static int n;
    static int [] divisors = new int [maxn];
    static int ndivisors;
    static boolean [] [] grid;
    static int count;
    public static void main (String [] args) {
        for (n = 0;n <= maxn;n++) {
            ndivisors = 0;
            for (int divisor = 1;divisor <= n;divisor++)
                if (n % divisor == 0)
                    divisors [ndivisors++] = divisor;
            grid = new boolean [n] [n];
            count = 0;
            recurse (0,0,0);
            System.out.print (count + ",");
        }
        System.out.println ();
    }
    static void recurse (int x,int y,int depth) {
        if (depth == n) {
            count++;
            return;
        }
        while (grid [x] [y])
            if (++x == n) {
                x = 0;
                y++;
            }
        outer:
        for (int k = 0;k < ndivisors;k++) {
            int w = divisors [k];
            int h = n / w;
            if (x + w > n || y + h > n)
                continue;
            for (int i = 0;i < w;i++)
                for (int j = 0;j < h;j++)
                    if (grid [x + i] [y + j])
                        continue outer;
            for (int i = 0;i < w;i++)
                for (int j = 0;j < h;j++)
                    grid [x + i] [y + j] = true;
            recurse (x,y,depth + 1);
            for (int i = 0;i < w;i++)
                for (int j = 0;j < h;j++)
                    grid [x + i] [y + j] = false;
        }
    }
}

Extensions

a(24)-a(31) from Lars Blomberg, Oct 09 2023
Previous Showing 11-15 of 15 results.

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