A233427 -id:A233427 - cp's OEIS frontend

A054318 a(n)-th star number (A003154) is a square.

1, 5, 45, 441, 4361, 43165, 427285, 4229681, 41869521, 414465525, 4102785725, 40613391721, 402031131481, 3979697923085, 39394948099365, 389969783070561, 3860302882606241, 38213059042991845, 378270287547312205

Offset: 1

Ignacio Larrosa Cañestro, Feb 27 2000

A two-way infinite sequence which is palindromic.
Also indices of centered hexagonal numbers (A003215) which are also centered square numbers (A001844). - Colin Barker, Jan 02 2015
Also positive integers y in the solutions to 4*x^2 - 6*y^2 - 4*x + 6*y = 0. - Colin Barker, Jan 02 2015

			a(2) = 5 because the 5th Star number (A003154) 121=11^2 is the 2nd that is a square.

Colin Barker, Table of n, a(n) for n = 1..1000
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

A031138 is 3*a(n)-2. Cf. A003154, A006061, A182432, A211955.
Quintisection of column k=2 of A233427.
Cf. A001844, A003215, A253475.

a:=[1,5,45];; for n in [4..30] do a[n]:=11*a[n-1]-11*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jul 23 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1-6*x+x^2)/((1-x)*(1-10*x+x^2)) )); // G. C. Greubel, Jul 23 2019

CoefficientList[Series[x(1-6x+x^2)/((1-x)(1-10x+x^2)), {x,0,30}], x] (* Michael De Vlieger, Aug 11 2016 *)
LinearRecurrence[{11,-11,1},{1,5,45},30] (* Harvey P. Dale, Nov 05 2016 *)

a(n)=if(n<1,a(1-n),1/2+subst(poltchebi(n)+poltchebi(n-1),x,5)/12)

Vec(x*(1-6*x+x^2)/((1-x)*(1-10*x+x^2)) + O(x^30)) \\ Colin Barker, Jan 02 2015

(x*(1-6*x+x^2)/((1-x)*(1-10*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019

a(n) = 11*(a(n-1) - a(n-2)) + a(n-3).
a(n) = 1/2 + (3 - sqrt(6))/12*(5 + 2*sqrt(6))^n + (3 + sqrt(6))/12*(5 - 2*sqrt(6))^n.
From Michael Somos, Mar 18 2003: (Start)
G.f.: x*(1-6*x+x^2)/((1-x)*(1-10*x+x^2)).
12*a(n)*a(n-1) + 4 = (a(n) + a(n-1) + 2)^2.
a(n) = a(1-n) = 10*a(n-1) - a(n-2) - 4.
a(n) = 12*a(n-1)^2/(a(n-1) + a(n-2)) - a(n-1).
a(n) = (a(n-1) + 4)*a(n-1)/a(n-2). (End)
From Peter Bala, May 01 2012: (Start)
a(n+1) = 1 + (1/2)*Sum_{k = 1..n} 8^k*binomial(n+k,2*k).
a(n+1) = R(n,4), where R(n,x) is the n-th row polynomial of A211955.
a(n+1) = (1/u)*T(n,u)*T(n+1,u) with u = sqrt(3) and T(n,x) the Chebyshev polynomial of the first kind.
Sum {k>=0} 1/a(k) = sqrt(3/2). (End)
A003154(a(n)) = A006061(n). - Zak Seidov, Oct 22 2012
a(n) = (4*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - Seiichi Manyama, Aug 11 2016
2*a(n) = 1+A072256(n). - R. J. Mathar, Feb 07 2022

More terms from James Sellers, Mar 01 2000

A247125 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.

Original entry on oeis.org

1, 0, 2, 1, 16, 10, 59, 60, 330, 397, 1520, 2218, 7875, 12820, 39250, 70045, 202168, 384866, 1038051, 2073580, 5385754, 11156701, 28015232, 59580154, 146333795, 317517636, 766142242, 1686735709, 4019319048, 8946988370, 21116854115, 47386013020, 111065223914

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

			a(4) = 16:
._______.     ._______.     ._______.
| ._____|     | ._____|     | ._| ._|
|_| |_. |     |_| |_. |     | | | | |
|_. ._| |     |_. ._| |     | | | | |
| |_|___|     | |_| | |     |_| |_| |
|_______| (2) |_____|_| (4) |___|___| (4)
._______.     ._______.
| ._____|     | ._____|
|_| ._. |     |_|_. | |
| |_| |_|     | ._| | |
|_____| |     | |___| |
|_______| (2) |___|___| (4) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,1,12,6,2).

Cf. A174249, A233427, A234312, A247124, A247268.

a:= n-> (<<0|1|0|0|0|0>, <0|0|1|0|0|0>, <0|0|0|1|0|0>,
          <0|0|0|0|1|0>, <0|0|0|0|0|1>, <2|6|12|1|2|0>>^n)[6,6]:
seq(a(n), n=0..40);

G.f.: -1/(2*x^6+6*x^5+12*x^4+x^3+2*x^2-1).

A247268 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Y, U, X.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 0, 4, 5, 38, 22, 13, 90, 144, 457, 408, 386, 1267, 2230, 5912, 6481, 7098, 18896, 35433, 79634, 101232, 127501, 288304, 546652, 1113907, 1560356, 2148298, 4408181, 8335234, 15954116, 23827541, 35011426, 67591204, 126376945, 232719926

Offset: 0

Alois P. Heinz, Nov 30 2014

nonn

			a(3) = 1, a(5) = 2:
._____.     ._________.   ._________.
| ._. |     |_. .___| |   | |___. ._|
|_| |_|     | |_| |_. |   | ._| |_| |
|_. ._|  ,  | |_. ._| |   | |_. ._| |
| |_| |     | ._|_| |_|   |_| |_|_. |
|_____|     |_|_______|   |_______|_|  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A247124, A247125.

gf:= -(x^40 +12*x^39 +36*x^38 -5*x^36 -2*x^35 +12*x^34 +54*x^33 +4*x^32 -21*x^31 -23*x^30 +4*x^29 +20*x^28 +4*x^27 -4*x^25 -7*x^24 -6*x^23 -3*x^22 +33*x^21 -7*x^20 -10*x^19 -12*x^18 -9*x^17 +12*x^16 +16*x^15 +3*x^14 -2*x^13 -2*x^12 -2*x^11 -3*x^10 +5*x^9 -2*x^6 -7*x^5 -x^4 +1) /
(x^43 +12*x^42 +36*x^41 -3*x^40 -29*x^39 -58*x^38 +12*x^37 +67*x^36 +4*x^35 -123*x^34 -99*x^33 +8*x^32 +23*x^31 -145*x^30 -52*x^29 -52*x^28 -35*x^27 -112*x^26 -99*x^25 -28*x^24 -7*x^23 -15*x^22 -99*x^21 -42*x^20 +22*x^19 +36*x^18 +26*x^17 -4*x^16 +6*x^15 +31*x^14 +5*x^13 +11*x^12 +14*x^11 +23*x^10 -5*x^9 -7*x^8 -x^7 +2*x^6 +9*x^5 +x^4 +x^3 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);

G.f.: see Maple program.

A247706 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape P; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 0, 3, 0, 2, 16, 20, 20, 0, 135, 204, 140, 16, 6, 944, 1432, 1164, 296, 170, 0, 4814, 8796, 8452, 4068, 1708, 92, 20, 26435, 58656, 66994, 41648, 17494, 2700, 762, 0, 158761, 410000, 520728, 371456, 175810, 46648, 12876, 440, 62, 978044, 2783560, 3836254, 3107308, 1696312, 609772, 172724, 18220, 3160, 0

Offset: 0

Alois P. Heinz, Sep 22 2014

Sum_{k>0} k * T(n,k) = A247739(n).

			T(2,2) = 2:
.___.   .___.
|   |   |   |
| ._|   |_. |
|_| |   | |_|
|   |   |   |
|___|   |___| .
Triangle T(n,k) begins:
00 :      1;
01 :      1,      0;
02 :      3,      0,      2;
03 :     16,     20,     20,      0;
04 :    135,    204,    140,     16,      6;
05 :    944,   1432,   1164,    296,    170,     0;
06 :   4814,   8796,   8452,   4068,   1708,    92,    20;
07 :  26435,  58656,  66994,  41648,  17494,  2700,   762,   0;
08 : 158761, 410000, 520728, 371456, 175810, 46648, 12876, 440, 62;

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Pentomino

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247770.
Even bisection of main diagonal gives A247076.
Cf. A247739.

A247711 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape X; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 5, 55, 1, 493, 8, 3930, 76, 27207, 734, 9, 207118, 7414, 157, 1622723, 71986, 2064, 8, 12544364, 638499, 22232, 259, 95912510, 5558790, 222964, 3898, 50, 732066083, 47971603, 2179607, 49537, 948, 8, 5616480627, 410502410, 20604626, 564498, 13889, 180

Offset: 0

Alois P. Heinz, Sep 23 2014

Sum_{k>0} k * T(n,k) = A247744(n).

			T(3,1) = 1:
._____.
| ._. |
|_| |_|
|_. ._|
| |_| |
|_____|
.
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       55,       1;
04 :      493,       8;
05 :     3930,      76;
06 :    27207,     734,      9;
07 :   207118,    7414,    157;
08 :  1622723,   71986,   2064,    8;
09 : 12544364,  638499,  22232,  259;
10 : 95912510, 5558790, 222964, 3898, 50;

Alois P. Heinz, Rows n = 0..185, flattened
Wikipedia, Pentomino

Row sums give A174249 or A233427(n,5).
Columns k=0-1 give: A247775, A247828.
Cf. A247744.

A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

Original entry on oeis.org

1, 1, 3, 5, 13, 52, 123, 366, 909, 2444, 7108, 19157, 53957, 146826, 400704, 1115852, 3059907, 8475420, 23369304, 64225984, 177572352, 488839323, 1349102071, 3722419367, 10255126169, 28303059509, 78013005366, 215160477217, 593488173404, 1636220978049

Offset: 0

Alois P. Heinz, Nov 25 2015

nonn

			a(4) = 13:
._______.      ._______.      ._______.      ._______.
| | | | |      |   |   |      |   | | |      |   ._| |
| | | | |      | ._| ._|      | ._| | |      |___|   |
| | | | |      |_| |_| |      |_| | | |      |   |___|
| | | | | (1)  |   |   | (4)  |   | | | (6)  | ._|   | (2)
|_|_|_|_|      |___|___|      |_ _|_|_|      |_|_____|    .
a(5) = 52:
._________.
|   |_.   |
| ._| |___|
|_|_   _| |
|   |_|   | (2)  ...
|_____|___|          .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A278330.

A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

Original entry on oeis.org

1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984, 7019, 11148, 35686, 62181, 182776, 339350, 942507, 1841208, 4887096, 9921685, 25442304, 53190380, 132928715, 284198328, 696276202, 1514363221, 3654567764, 8053235650, 19212546163, 42762014028, 101125071372

Offset: 0

Alois P. Heinz, Nov 18 2016

			a(2) = 2,          a(3) = 1:
.___.   .___.      ._____.
|   |   |   |      | ._. |
| ._|   |_. |      |_| |_|
|_| |   | |_|      |_   _|
|   |   |   |      | |_| |
|___|   |___|      |_____| .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,2,8,4,21,-8,-4,-6,0,-16,-8).

Cf. A079978, A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A264812.

a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-8, -16, 0, -6, -4, -8, 21, 4, 8, 2, 2, 0][j], 0)))^n.
    <<1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984>>)[1, 1]:
seq(a(n), n=0..35);

G.f.: -(4*x^6+x^3-1) / (8*x^12 +16*x^11 +6*x^9 +4*x^8 +8*x^7 -21*x^6 -4*x^5 -8*x^4 -2*x^3 -2*x^2+1).

a(n) mod 2 = A079978(n).

A278456 Number of tilings of a 5 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 2, 50, 1954, 56864, 1532496, 42238426, 1178422563, 32890293494, 917103556607, 25552076570350, 711923354658732, 19838824712825618, 552851181380560869, 15406086995815163663, 429312063890812931103, 11963383230714027535776, 333377000620725693771782

Offset: 0

Alois P. Heinz, Nov 22 2016

nonn

			a(1) = 2:
._.   ._.
|_|   | |
|_|   | |
|_|   | |
|_|   | |
|_|   |_|  .

Alois P. Heinz, Table of n, a(n) for n = 0..692
Wikipedia, Pentomino

Column k=5 of A278657.

Cf. A174249, A233427, A273477.

A077909 Expansion of 1/((1-x)(1+x+x^2+2x^3)).

Original entry on oeis.org

1, 0, 0, -1, 2, 0, 1, -4, 4, -1, 6, -12, 9, -8, 24, -33, 26, -40, 81, -92, 92, -161, 254, -276, 345, -576, 784, -897, 1266, -1936, 2465, -3060, 4468, -6337, 7990, -10588, 15273, -20664, 26568, -36449, 51210, -67896, 89585, -124108, 170316, -225377, 303278, -418532, 566009, -754032, 1025088

Offset: 0

N. J. A. Sloane, Nov 17 2002

The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
  .___.     ._____.  ._____.
  | .. |     | .. | |  | | ._. |
  || ||     || || |  | || ||
  |. .|  ,  | .| .|  |. |. |
  | || |     | | || |  | |_| | |
  |___|     ||____|  |___|_|. - Alois P. Heinz, Jan 03 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,0,-1,2)

Partial sums of A077976.

Cf. A174249, A233427, A234312, A234931, A247126.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
        <<1, 0, 0, -1>>)[1, 1]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 20 2013

CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* Jean-François Alcover, Jun 08 2015, after Arkadiusz Wesolowski *)

Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ Joerg Arndt, Aug 28 2013

a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - Johannes W. Meijer, Aug 28 2013

G.f.: 1/(1 + x^3 - 2*x^4). - Arkadiusz Wesolowski, Nov 20 2013

A247124 Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, U, X.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 14, 21, 37, 63, 122, 221, 374, 656, 1147, 2066, 3699, 6477, 11407, 20099, 35656, 63323, 111775, 197352, 348556, 616560, 1091570, 1929721, 3410509, 6028021, 10658114, 18851012, 33331681, 58927069, 104177155, 184188343, 325686763, 575858676

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

			a(4) = 3:
._______.   ._______.   ._______.
| | | | |   | | ._. |   | ._. | |
| | | | |   | |_| |_|   |_| |_| |
| | | | |   | |_. ._|   |_. ._| |
| | | | |   | | |_| |   | |_| | |
|_|_|_|_|   |_|_____|   |_____|_|  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A247125, A247268, A264812.

gf:= -(x-1)^2 *(x^4+x^3+x^2+x+1)^2 /
     (x^15 +x^13 +x^11 -3*x^10 -2*x^8 -2*x^6 +6*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

cp's OEIS Frontend

A054318 a(n)-th star number (A003154) is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI

Sage

Formula

Extensions

A247125 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A247268 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Y, U, X.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A247706 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape P; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A247711 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape X; triangle T(n,k), n>=0, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

Views

Author

Keywords

Examples

Links

Crossrefs

A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A278456 Number of tilings of a 5 X n rectangle using pentominoes of any shape and monominoes.

Views

Author

Keywords

Examples

Links

Crossrefs

A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).

Views

A077909 Expansion of 1/((1-x)(1+x+x^2+2x^3)).