A002478 -id:A002478 - cp's OEIS frontend

A226546 Number of squares in all tilings of a 3 X n rectangle using integer-sided square tiles.

0, 3, 12, 34, 98, 256, 654, 1625, 3964, 9533, 22662, 53373, 124728, 289572, 668514, 1535869, 3513614, 8008090, 18191184, 41200568, 93064834, 209710139, 471520566, 1058065647, 2369890254, 5299215579, 11830941840, 26375563624, 58722396932, 130576680919

Offset: 0

Alois P. Heinz, Jun 10 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-2,-6,-4,-1).

Column k=3 of A226545.

Cf. A002478.

concat(0, Vec(x*(3 + 6*x + x^2) / (1 - x - 2*x^2 - x^3)^2 + O(x^30))) \\ Colin Barker, Jun 07 2020

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

a(n) = 2*a(n-1) + 3*a(n-2) - 2*a(n-3) - 6*a(n-4) - 4*a(n-5) - a(n-6) for n>5. - Colin Barker, Jun 07 2020

A362126 Expansion of 1/(1 - x*(1+x)^2)^2.

Original entry on oeis.org

1, 2, 7, 18, 47, 118, 290, 702, 1677, 3966, 9300, 21654, 50116, 115388, 264475, 603792, 1373621, 3115222, 7045205, 15892794, 35769390, 80337144, 180091131, 403002108, 900370600, 2008572044, 4474586920, 9955434456, 22123162421, 49107537598, 108891513251

Offset: 0

Seiichi Manyama, Apr 08 2023

Index entries for linear recurrences with constant coefficients, signature (2,3,-2,-6,-4,-1).

Column k=2 of A362125.

Cf. A002478, A362084.

my(N=40, x='x+O('x^N)); Vec(1/(1-x*(1+x)^2)^2)

a(n) = 2*a(n-1) + 3*a(n-2) - 2*a(n-3) - 6*a(n-4) - 4*a(n-5) - a(n-6) for n > 5.

a(n) = Sum_{k=0..n} (-1)^k * binomial(-2,k) * binomial(2*k,n-k) = Sum_{k=0..n} (k+1) * binomial(2*k,n-k).

A362144 Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 24, 40, 80, 160, 280, 560, 1120, 2240, 4480, 10080, 20160, 40320, 88704, 177408, 354816, 768768, 1537536, 3075072, 6589440, 13178880, 26357760, 56010240, 112020480, 224040960, 504092160, 1064194560, 2128389120, 4729753600, 9932482560

Offset: 0

Pontus von Brömssen, Apr 10 2023

nonn

Pontus von Brömssen, Table of n, a(n) for n = 0..2000

Third column of A362142.

Cf. A002478, A361219 (rectangular pieces).

a(n) = max_{3*i+2*j<=n} C(i,j,n-3*i-2*j)*2^j, where C(i,j,k) is the trinomial coefficient (i+j+k)!/(i!*j!*k!). (i and j correspond to the number of squares of side lengths 3 and 2, respectively.)

A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

Original entry on oeis.org

1, 3, 12, 37, 111, 315, 864, 2307, 6027, 15471, 39132, 97755, 241606, 591636, 1437078, 3465748, 8305161, 19788957, 46910232, 110686101, 260064912, 608684490, 1419591546, 3300027546, 7648265728, 17676484410, 40747630332, 93704299336, 214999206831, 492262973433

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (3,3,-8,-12,3,17,15,6,1).

Cf. A002478, A362126.

Cf. A000217, A001628, A382614.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^2)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, n-k));

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,n-k).
a(n) = 3*a(n-1) + 3*a(n-2) - 8*a(n-3) - 12*a(n-4) + 3*a(n-5) + 17*a(n-6) + 15*a(n-7) + 6*a(n-8) + a(n-9).
G.f.: 1/(1-x-2*x^2-x^3)^3. - Vincenzo Librandi, Apr 10 2025

A077899 Expansion of (1-x)^(-1)/(1+x-2*x^2+x^3).

Original entry on oeis.org

1, 0, 3, -3, 10, -18, 42, -87, 190, -405, 873, -1872, 4024, -8640, 18561, -39864, 85627, -183915, 395034, -848490, 1822474, -3914487, 8407926, -18059373, 38789713, -83316384, 178955184, -384377664, 825604417, -1773314928, 3808901427, -8181135699, 17572253482, -37743426306

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (0,3,-3,1).

Cf. A002478.

Vec((1-x)^(-1)/(1+x-2*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (-1)^n*sum(k=0, n, binomial(2*k-1, n-k)); \\ Seiichi Manyama, Aug 13 2024

a(n) = (-1)^n * Sum_{k=0..n} binomial(2*k-1,n-k). - Seiichi Manyama, Aug 13 2024

A121574 Riordan array (1/(1-2x), x(1+x)/(1-2*x)).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 16, 8, 1, 16, 44, 37, 11, 1, 32, 112, 134, 67, 14, 1, 64, 272, 424, 305, 106, 17, 1, 128, 640, 1232, 1168, 584, 154, 20, 1, 256, 1472, 3376, 3992, 2641, 998, 211, 23, 1, 512, 3328, 8864, 12592, 10442, 5221, 1574, 277, 26, 1

Offset: 0

Paul Barry, Aug 08 2006

Row sums are A006190(n+1); diagonal sums are A077939.
Inverse is A121575.
A generalized Delannoy number triangle.
Antidiagonal sums are A002478. - Philippe Deléham, Nov 10 2011.
From Peter Bala, Feb 07 2024: (Start)
The following remarks assume the row indexing starts at n = 1.
The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = 2 + x, R(3,x) = 4 + 5*x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)

			Triangle begins
   1;
   2,   1;
   4,   5,   1;
   8,  16,   8,   1;
  16,  44,  37,  11,   1;
  32, 112, 134,  67,  14,  1;
  64, 272, 424, 305, 106, 17, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.

Cf. Diagonals: A000012, A016789, A080855, A000079, A053220.

T:=Flat(List([0..9],n->List([0..n],k->Sum([0..n-k],j->Binomial(k,j)*Binomial(n-j,k)*2^(n-k-j))))); # Muniru A Asiru, Nov 02 2018

[[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018

T:=(n,k)->add(binomial(k,j)*binomial(n-j,k)*2^(n-k-j),j=0..n-k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Nov 02 2018

Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 02 2018 *)

for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ G. C. Greubel, Nov 02 2018

Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1). - Philippe Deléham, Nov 10 2011
Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = (x + 2)*R(n-1,x) + x*R(n-2,x) with R(1,x) = 1 and R(2,x) = x + 2. - Peter Bala, Feb 07 2024

A360090 a(n) = Sum_{k=0..n} binomial(5*k,n-k).

Original entry on oeis.org

1, 1, 6, 21, 71, 251, 882, 3088, 10829, 37975, 133146, 466852, 1636944, 5739647, 20125051, 70564951, 247423522, 867546829, 3041899638, 10665883415, 37398034921, 131129599227, 459782762029, 1612146986543, 5652708454881, 19820223058176, 69496108849357

Offset: 0

Seiichi Manyama, Jan 25 2023

The number of ways to place non-overlapping Young diagrams of shape (2,1,1,1,1) on an 9 by n rectangle. - Per Alexandersson, Jul 01 2025

Index entries for linear recurrences with constant coefficients, signature (1,5,10,10,5,1).

Cf. A002478, A099234, A099235.

seq(add(binomial(5*k,n-k),k=0..n), n=0..50); # Robert Israel, Jul 09 2025

a(n) = sum(k=0, n, binomial(5*k, n-k));

my(N=30, x='x+O('x^N)); Vec(1/(1-x*(1+x)^5))

a(n) = a(n-1) + 5*a(n-2) + 10*a(n-3) + 10*a(n-4) + 5*a(n-5) + a(n-6).

G.f.: 1/(1 - x*(1+x)^5).

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

Original entry on oeis.org

1, 1, 2, 12, 60, 460, 3900, 39438, 456288, 5896224, 85230000, 1349017560, 23353941600, 437432418696, 8828284404576, 190867622500800, 4401749312069760, 107859517575659520, 2798352667710645120, 76636669899079699776, 2209235394261812751360

Offset: 0

Seiichi Manyama, Apr 12 2023

nonn

Cf. A202152, A362237.

Cf. A002478, A099234, A099235.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*(1+x)^x)))

a(n) = n! * Sum_{i=0..n} Sum_{j=0..n-i} i^j * Stirling1(n-i-j,j)/(n-i-j)!.

A025234 An L-tile is a 2 X 2 square with the upper 1 X 1 subsquare removed; no rotations are allowed. a(n) = number of tilings of a 4 X n rectangle using tiles that are either 1 X 1 squares or L-tiles.

Original entry on oeis.org

1, 0, 4, 8, 28, 83, 255, 778, 2377, 7259, 22173, 67721, 206844, 631764, 1929609, 5893632, 18001012, 54980764, 167928588, 512906847, 1566579211, 4784826786, 14614369465, 44636891651, 136335139273, 416410496177, 1271848932360, 3884627600872, 11864877355729

Offset: 0

N. J. A. Sloane, Mar 07 2003

E. Deutsch, Counting tilings with L-tiles and squares, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.
Index entries for linear recurrences with constant coefficients, signature (1,5,4,0,-1).

Cf. A002478.

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

A108122 G.f.: (1-2x^2)/(1-x-2x^2-x^3).

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 34, 73, 157, 337, 724, 1555, 3340, 7174, 15409, 33097, 71089, 152692, 327967, 704440, 1513066, 3249913, 6980485, 14993377, 32204260, 69171499, 148573396, 319120654, 685438945, 1472253649, 3162252193, 6792198436, 14588956471, 31335605536

Offset: 0

Roger L. Bagula, Jun 04 2005

The sequence counts row lengths of an array in which rows are obtained by the substitution 1->2, 2->3, 3->1,2,2,3 from previous rows:
1;
2;
3;
1,2,2,3;
2,3,3,1,2,2,3;
3,1,2,2,3,1,2,2,3,2,3,3,1,2,2,3;

Robert Israel, Table of n, a(n) for n = 0..303
Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Cf. A078007, A101399.

a[0],a[1],a[2]:= 1,1,1:
for n from 3 to 100 do
  a[n]:= a[n-1]+2*a[n-2]+a[n-3]
od:
seq(a[i],i=0..100); # Robert Israel, Jun 15 2014

s[1] = {2}; s[2] = {3}; s[3] = {1, 2, 2, 3}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] a0 = Table[Length[p[i]], {i, 0, 20}]
f[n_] := Sum[ 2^i*Binomial[n - 2 m, m - i]*Binomial[n - 2 m + i - 1, n - 2 m - 1], {m, 0, (n - 1)/2}, {i, 0, m}]; f[0] = 1; Array[f, 33, 0] (* or *)
CoefficientList[ Series[(1 - 2 x^2)/(1 - x - 2 x^2 - x^3), {x, 0, 33}], x] (* or *)
LinearRecurrence[ {1, 2, 1}, {1, 1, 1}, 34] (* or *)
Length /@ NestList[ Flatten[ # /. {1 -> 2, 2 -> 3, 3 -> {1, 2, 2, 3}}] &, {1}, 24] (* Robert G. Wilson v, Jun 13 2014 *)

a(n):=sum(sum(2^i*binomial(n-2*m+1,m-i)*binomial(n-2*m+i,n-2*m),i,0,m),m,0,(n)/2); /* Vladimir Kruchinin, Dec 17 2011 */

a(n) = a(n-1) + 2*a(n-2) + a(n-3), starting 1,1,1.
a(n) = A002478(n) - 2*A002478(n-2), n>1.
a(n) = sum(m=0..n/2, sum(i=0..m, 2^i*binomial(n-2*m+1,m-i)*binomial(n-2*m+i,n-2*m))). - Vladimir Kruchinin, Dec 17 2011

More terms from Wesley Ivan Hurt, Jun 14 2014

cp's OEIS Frontend

A226546 Number of squares in all tilings of a 3 X n rectangle using integer-sided square tiles.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362126 Expansion of 1/(1 - x*(1+x)^2)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362144 Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle.

Views

Author

Keywords

Links

Crossrefs

Formula

A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A077899 Expansion of (1-x)^(-1)/(1+x-2*x^2+x^3).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A121574 Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A360090 a(n) = Sum_{k=0..n} binomial(5*k,n-k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

PARI

Formula

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

Views

Author

Keywords

Crossrefs

Programs

A121574 Riordan array (1/(1-2x), x(1+x)/(1-2*x)).

A108122 G.f.: (1-2x^2)/(1-x-2x^2-x^3).