A219158 -id:A219158 - cp's OEIS frontend

A113881 Table of smallest number of squares, T(m,n), needed to tile an m X n rectangle, read by antidiagonals.

1, 2, 2, 3, 1, 3, 4, 3, 3, 4, 5, 2, 1, 2, 5, 6, 4, 4, 4, 4, 6, 7, 3, 4, 1, 4, 3, 7, 8, 5, 2, 5, 5, 2, 5, 8, 9, 4, 5, 3, 1, 3, 5, 4, 9, 10, 6, 5, 5, 5, 5, 5, 5, 6, 10, 11, 5, 3, 2, 5, 1, 5, 2, 3, 5, 11, 12, 7, 6, 6, 5, 5, 5, 5, 6, 6, 7, 12, 13, 6, 6, 4, 6, 4, 1, 4, 6, 4, 6, 6, 13, 14, 8, 4, 6, 2, 3, 7, 7, 3, 2, 6, 4, 8, 14

Offset: 1

Devin Kilminster (devin(AT)27720.net), Jan 27 2006

a(n) = A338573(n) for n <= 105, as stated by R. J. Mathar. These sequences are essentially different though, because a(13433) = T(67,98) = T(98,67) = a(13464), but A338573(13433) != A338573(13464). The relationship between the tiling problem and resistor networks is remarkable. There are explanations in M. Ortolano et al., 2013. - Rainer Rosenthal, Nov 09 2020

			T(n,n) = 1 (1 n X n square).
T(n,1) = n (n 1 X 1 squares).
T(6,7) = 6 (2 3 X 3, 1 4 X 4, 1 2 X 2, 2 1 X 1).
T(11,13) = 6 (1 7 X 7, 1 6 X 6, 1 5 X 5, 2 4 X 4 1 1 X 1).
Table T(m,n) begins:
:   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
:   2, 1, 3, 2, 4, 3, 5, 4, 6,  5, ...
:   3, 3, 1, 4, 4, 2, 5, 5, 3,  6, ...
:   4, 2, 4, 1, 5, 3, 5, 2, 6,  4, ...
:   5, 4, 4, 5, 1, 5, 5, 5, 6,  2, ...
:   6, 3, 2, 3, 5, 1, 5, 4, 3,  4, ...
:   7, 5, 5, 5, 5, 5, 1, 7, 6,  6, ...
:   8, 4, 5, 2, 5, 4, 7, 1, 7,  5, ...
:   9, 6, 3, 6, 6, 3, 6, 7, 1,  6, ...
:  10, 5, 6, 4, 2, 4, 6, 5, 6,  1, ...

Alois P. Heinz, Antidiagonals n = 1..350, flattened (using data from A219158)
Bertram Felgenhauer, Filling rectangles with integer-sided squares
Richard J. Kenyon, Tiling a rectangle with the fewest squares, Combin. Theory Ser. A 76 (1996), no. 2, 272-291.
M. Ortolano, M. Abrate, and L. Callegaro, On the synthesis of Quantum Hall Array Resistance Standards, arXiv preprint arXiv:1311.0756 [physics.ins-det], 2013.
Mark Walters, Rectangles as sums of squares, Discrete Math. 309 (2009), no. 9, 2913-2921.

Rows (or columns) m=1-10 give: A001477, A030451, A226576, A226577, A226578, A226579, A226580, A226581, A226582, A226583.

Cf. A219158, A219924, A226545, A338573.

(* *** Warning *** This empirical toy-program is based on the greedy algorithm. Its output was only verified for n+k <= 32. Any use outside this domain might produce only upper bounds instead of minimums. *)
nmax = 31; Clear[T];
Tmin[n_, k_] := Table[{1 + T[ c, k - c] + T[n - c, k], 1 + T[n, k - c] + T[n - c, c]}, {c, 1, k - 1}] // Flatten // Min;
Tmin2[n_, k_] := Module[{n1, n2, k1, k2}, 1 + T[n2, k1 + 1] + T[n - n1, k2] + T[n - n2, k1] + T[n1, k - k1] /. {Reduce[1 <= n1 <= n - 1 && 1 <= n2 <= n - 1 && 1 <= k1 <= k - 1 && 1 <= k2 <= k - 1 && n1 + 1 + n2 == n && k1 + 1 + k2 == k, Integers] // ToRules} // Min];
T[n_, n_] = 1;
T[n_, 1] := n;
T[1, k_] := k;
T[n_, k_ /; k > 1] /; n > k && Divisible[n, k] := n/k;
T[n_, k_ /; k > 1] /; n > k := T[n, k] = If[k >= 5 && n >= 6 && n - k <= 3, Min[Tmin[n, k], Tmin2[n, k], T[k, n - k] + 1], T[k, n - k] + 1];
T[n_, k_ /; k > 1] /; n < k := T[n, k] = T[k, n];
Table[T[n - k + 1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 11 2016, checked against first 496 terms of the b-file *)

A285721 Square array read by antidiagonals: A(n,k) = number of steps in simple Euclidean algorithm for gcd(n,k) to reach the termination test n=k, read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 2, 2, 3, 4, 1, 0, 1, 4, 5, 3, 3, 3, 3, 5, 6, 2, 3, 0, 3, 2, 6, 7, 4, 1, 4, 4, 1, 4, 7, 8, 3, 4, 2, 0, 2, 4, 3, 8, 9, 5, 4, 4, 5, 5, 4, 4, 5, 9, 10, 4, 2, 1, 4, 0, 4, 1, 2, 4, 10, 11, 6, 5, 5, 4, 6, 6, 4, 5, 5, 6, 11, 12, 5, 5, 3, 5, 3, 0, 3, 5, 3, 5, 5, 12, 13, 7, 3, 5, 1, 2, 7, 7, 2, 1, 5, 3, 7, 13, 14, 6, 6, 2, 6, 3, 5, 0, 5, 3, 6, 2, 6, 6, 14

Offset: 1

Antti Karttunen, May 03 2017

			The top left 18 X 18 corner of the array:
   0, 1, 2, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15, 16, 17
   1, 0, 2, 1, 3, 2, 4, 3, 5,  4,  6,  5,  7,  6,  8,  7,  9,  8
   2, 2, 0, 3, 3, 1, 4, 4, 2,  5,  5,  3,  6,  6,  4,  7,  7,  5
   3, 1, 3, 0, 4, 2, 4, 1, 5,  3,  5,  2,  6,  4,  6,  3,  7,  5
   4, 3, 3, 4, 0, 5, 4, 4, 5,  1,  6,  5,  5,  6,  2,  7,  6,  6
   5, 2, 1, 2, 5, 0, 6, 3, 2,  3,  6,  1,  7,  4,  3,  4,  7,  2
   6, 4, 4, 4, 4, 6, 0, 7, 5,  5,  5,  5,  7,  1,  8,  6,  6,  6
   7, 3, 4, 1, 4, 3, 7, 0, 8,  4,  5,  2,  5,  4,  8,  1,  9,  5
   8, 5, 2, 5, 5, 2, 5, 8, 0,  9,  6,  3,  6,  6,  3,  6,  9,  1
   9, 4, 5, 3, 1, 3, 5, 4, 9,  0, 10,  5,  6,  4,  2,  4,  6,  5
  10, 6, 5, 5, 6, 6, 5, 5, 6, 10,  0, 11,  7,  6,  6,  7,  7,  6
  11, 5, 3, 2, 5, 1, 5, 2, 3,  5, 11,  0, 12,  6,  4,  3,  6,  2
  12, 7, 6, 6, 5, 7, 7, 5, 6,  6,  7, 12,  0, 13,  8,  7,  7,  6
  13, 6, 6, 4, 6, 4, 1, 4, 6,  4,  6,  6, 13,  0, 14,  7,  7,  5
  14, 8, 4, 6, 2, 3, 8, 8, 3,  2,  6,  4,  8, 14,  0, 15,  9,  5
  15, 7, 7, 3, 7, 4, 6, 1, 6,  4,  7,  3,  7,  7, 15,  0, 16,  8
  16, 9, 7, 7, 6, 7, 6, 9, 9,  6,  7,  6,  7,  7,  9, 16,  0, 17
  17, 8, 5, 5, 6, 2, 6, 5, 1,  5,  6,  2,  6,  5,  5,  8, 17,  0

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array

One less than A072030.
Row 2 & column 2: A028242 (but with starting offset 1).
Row 3 & column 3 (from zero onward) seems to be A226576.
Cf. A003989, A285722, A285732.
Compare also to arrays A049834, A113881, A219158.

def A(n, k): return 0 if n==k else 1 + A(abs(n - k), min(n, k))
for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

(define (A285721 n) (A285721bi (A002260 n) (A004736 n)))
(define (A285721bi row col) (cond ((= row col) 0) ((> row col) (+ 1 (A285721bi (- row col) col))) (else (+ 1 (A285721bi row (- col row))))))
;; Alternatively:
(define (A285721bi row col) (if (= row col) 0 (+ 1 (A285721bi (abs (- row col)) (min col row)))))
;; Another implementation, as an one-dimensional sequence:
(definec (A285721 n) (if (zero? (A285722 n)) 0 (+ 1 (A285721 (A285722 n)))))

If n = k, then A(n,k) = 0, if n > k, then A(n,k) = 1 + A(n-k,k), otherwise [when n < k], A(n,k) = 1 + A(n,k-n).
Or alternatively, when n <> k, A(n,k) = 1 + A(abs(n-k),min(n,k)).
A(n,k) = A072030(n,k)-1.
As an one-dimensional sequence:
a(n) = 0 if A285722(n) = 0, otherwise a(n) = 1 + a(A285722(n)). [Here A285722 is also used as an one-dimensional sequence.]

A030451 a(2n) = n, a(2n+1) = n+2.

Original entry on oeis.org

0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 12, 11, 13, 12, 14, 13, 15, 14, 16, 15, 17, 16, 18, 17, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22, 24, 23, 25, 24, 26, 25, 27, 26, 28, 27, 29, 28, 30, 29, 31, 30, 32, 31, 33, 32, 34, 33, 35, 34, 36, 35, 37, 36, 38, 37

Offset: 0

Daniel Smith (2true(AT)gte.net)

Previous name was: Once started, this mixes the natural numbers and the natural numbers shifted by 1.
Smallest number of integer-sided squares needed to tile a 2 X n rectangle. a(5) = 4:
..._...
|   |   |_|
|_|___||. - _Alois P. Heinz, Jun 12 2013

Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A168361 (first differences), A198442 (partial sums).
Cf. A008619, A110570.
Row m=2 of A113881, A219158.
Essentially the same as A028242.

a:= n-> iquo(n, 2, 'r') +[0, 2][r+1]:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 12 2013

Riffle[# + 1, #] &@ Range[0, 37] (* or *)
Table[3/4 - (-1)^n 3/4 + n/2, {n, 0, 72}] (* or *)
CoefficientList[Series[(2 x - x^2)/((1 - x) (1 - x^2)), {x, 0, 72}], x] (* Michael De Vlieger, Apr 25 2016 *)

PARI
```
a(n)=n\2+2*(n%2)
```

a(n) = 3/4 -(-1)^n*3/4 +n/2.
G.f.: (2*x-x^2)/((1-x)*(1-x^2)).
a(2n) = n, a(2n+1) = n+2.
a(n+2) = a(n)+1.
a(n) = -a(-3-n).
a(n) = A110570(n,2) for n>1. - Reinhard Zumkeller, Jul 28 2005
a(n) = (n+1)-a(n-1) with n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{k=1..n} (-1)^(n+k)*(k+1). - Arkadiusz Wesolowski, Nov 23 2012
a(n+1) = (a(0) + a(1) + ... + a(n))/a(n) for n>0. This formula with different initial conditions produces A008619. - Ivan Neretin, Apr 25 2016
E.g.f.: (x*exp(x) + 3*sinh(x))/2. - Ilya Gutkovskiy, Apr 25 2016
Sum_{n>=1} (-1)^n/a(n) = 1. - Amiram Eldar, Oct 04 2022

New name (using existing formula) from Joerg Arndt, Apr 26 2016

A226576 Smallest number of integer-sided squares needed to tile a 3 X n rectangle.

Original entry on oeis.org

0, 3, 3, 1, 4, 4, 2, 5, 5, 3, 6, 6, 4, 7, 7, 5, 8, 8, 6, 9, 9, 7, 10, 10, 8, 11, 11, 9, 12, 12, 10, 13, 13, 11, 14, 14, 12, 15, 15, 13, 16, 16, 14, 17, 17, 15, 18, 18, 16, 19, 19, 17, 20, 20, 18, 21, 21, 19, 22, 22, 20, 23, 23, 21, 24, 24, 22, 25, 25, 23, 26

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(8) = 5:
  ._._._._._._._._.
  |     |     |   |
  |     |     |___|
  |_____|_____|_|_| .

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

Cf. row m=3 of A113881, A219158.

a:= n-> iquo(n, 3, 'r') +[0, 3, 3][r+1]:
seq(a(n), n=0..80);

CoefficientList[Series[(3 x - 2 x^3)/(1 - x - x^3 + x^4), {x, 0, 70}], x] (* Michael De Vlieger, Oct 01 2017 *)

concat(0, Vec((3*x-2*x^3)/(1-x-x^3+x^4) + O(x^50))) \\ Felix Fröhlich, Oct 02 2017

G.f.: (3*x-2*x^3)/(1-x-x^3+x^4).
a(n) = 1 + a(n-3) for n>2; a(0)=0, a(1)=a(2)=3.
a(n) = (3*n+15+6*cos(2*(n-2)*Pi/3)-8*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - Wesley Ivan Hurt, Oct 01 2017
a(n) = 3*floor((n+2)/3) - 2*floor(n/3). - Ridouane Oudra, Jan 25 2024

A226577 Smallest number of integer-sided squares needed to tile a 4 X n rectangle.

Original entry on oeis.org

0, 4, 2, 4, 1, 5, 3, 5, 2, 6, 4, 6, 3, 7, 5, 7, 4, 8, 6, 8, 5, 9, 7, 9, 6, 10, 8, 10, 7, 11, 9, 11, 8, 12, 10, 12, 9, 13, 11, 13, 10, 14, 12, 14, 11, 15, 13, 15, 12, 16, 14, 16, 13, 17, 15, 17, 14, 18, 16, 18, 15, 19, 17, 19, 16, 20, 18, 20, 17, 21, 19, 21, 18

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(11) = 6:
._._._._._._._._._._._.
|       |       |     |
|       |       |     |
|       |       |_____|
|_______|_______|_|_|_|

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

Row m=4 of A113881, A219158.

a:= n-> iquo(n, 4, 'r') +[0, 4, 2, 4][r+1]:
seq(a(n), n=0..80);

RecurrenceTable[{a[0] == 0, a[1] == 4, a[2] == 2, a[3] == 4, a[n] == 1 + a[n - 4]}, a[n], {n, 0, 80}] (* Bruno Berselli, Jun 12 2013 *)
LinearRecurrence[{1,0,0,1,-1},{0,4,2,4,1},90] (* Harvey P. Dale, Jul 03 2019 *)

makelist(5+(2*n-1-(2+(-1)^n)*(11+2*%i^(n*(n+1))))/8, n, 0, 80); /* Bruno Berselli, Jun 12 2013 */

G.f.: (-3*x^4+2*x^3-2*x^2+4*x)/(x^5-x^4-x+1).
a(n) = 1 + a(n-4) for n>3.
a(n) = 5 + (2*n - 1 - (2 + (-1)^n)*(11 + 2*i^(n*(n+1))))/8, where i=sqrt(-1). [Bruno Berselli, Jun 12 2013]

A226578 Smallest number of integer-sided squares needed to tile a 5 X n rectangle.

Original entry on oeis.org

0, 5, 4, 4, 5, 1, 5, 5, 5, 6, 2, 6, 6, 6, 7, 3, 7, 7, 7, 8, 4, 8, 8, 8, 9, 5, 9, 9, 9, 10, 6, 10, 10, 10, 11, 7, 11, 11, 11, 12, 8, 12, 12, 12, 13, 9, 13, 13, 13, 14, 10, 14, 14, 14, 15, 11, 15, 15, 15, 16, 12, 16, 16, 16, 17, 13, 17, 17, 17, 18, 14, 18, 18

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(11) = 6:
._._._._._._._._._._._.
|         |     |     |
|         |     |     |
|         |_____|_____|
|         |   |   |   |
|_________|___|___|___|

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

Row m=5 of A113881, A219158.

a:= n-> `if`(n=1, 5, iquo(n, 5, 'r') +[0, 4$3, 5][r+1]):
seq(a(n), n=0..80);

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

a(n) = 1 + a(n-5) for n>6.

A226579 Smallest number of integer-sided squares needed to tile a 6 X n rectangle.

Original entry on oeis.org

0, 6, 3, 2, 3, 5, 1, 5, 4, 3, 4, 6, 2, 6, 5, 4, 5, 7, 3, 7, 6, 5, 6, 8, 4, 8, 7, 6, 7, 9, 5, 9, 8, 7, 8, 10, 6, 10, 9, 8, 9, 11, 7, 11, 10, 9, 10, 12, 8, 12, 11, 10, 11, 13, 9, 13, 12, 11, 12, 14, 10, 14, 13, 12, 13, 15, 11, 15, 14, 13, 14, 16, 12, 16, 15, 14

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(13) = 6:
._._._._._._._._._._._._._.
|           |       |     |
|           |       |     |
|           |       |_____|
|           |_______|     |
|           |   |   |     |
|___________|___|___|_____|

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

Row m=6 of A113881, A219158.

a:= n-> `if`(n=1, 6, iquo(n, 6, 'r') +[0, 4, 3, 2, 3, 5][r+1]):
seq(a(n), n=0..100);

Join[{0,6},LinearRecurrence[{0,1,1,0,-1},{3,2,3,5,1},80]] (* Harvey P. Dale, Jun 03 2014 *)

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

a(n) = 1 + a(n-6) for n>7.

A226580 Smallest number of integer-sided squares needed to tile a 7 X n rectangle.

Original entry on oeis.org

0, 7, 5, 5, 5, 5, 5, 1, 7, 6, 6, 6, 6, 6, 2, 8, 7, 7, 7, 7, 7, 3, 9, 8, 8, 8, 8, 8, 4, 10, 9, 9, 9, 9, 9, 5, 11, 10, 10, 10, 10, 10, 6, 12, 11, 11, 11, 11, 11, 7, 13, 12, 12, 12, 12, 12, 8, 14, 13, 13, 13, 13, 13, 9, 15, 14, 14, 14, 14, 14, 10, 16, 15, 15, 15

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(15) = 8:
._._._._._._._._._._._._._._._.
|             |       |       |
|             |       |       |
|             |       |       |
|             |_______|_______|
|             |     |     |   |
|             |     |     |___|
|_____________|_____|_____|_|_|

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

Row m=7 of A113881, A219158.

a:= n-> `if`(n=1, 7, iquo(n, 7, 'r') +[0, 6, 5$5][r+1]):
seq(a(n), n=0..100);

CoefficientList[Series[x*(x^8 - x^7 - 4*x^6 - 2*x + 7)/(x^8 - x^7 - x + 1), {x, 0, 100}], x] (* Wesley Ivan Hurt, Jan 15 2017 *)

G.f.: x*(x^8-x^7-4*x^6-2*x+7)/(x^8-x^7-x+1).

a(n) = 1 + a(n-7) for n>8.

A226581 Smallest number of integer-sided squares needed to tile an 8 X n rectangle.

Original entry on oeis.org

0, 8, 4, 5, 2, 5, 4, 7, 1, 7, 5, 6, 3, 6, 5, 8, 2, 8, 6, 7, 4, 7, 6, 9, 3, 9, 7, 8, 5, 8, 7, 10, 4, 10, 8, 9, 6, 9, 8, 11, 5, 11, 9, 10, 7, 10, 9, 12, 6, 12, 10, 11, 8, 11, 10, 13, 7, 13, 11, 12, 9, 12, 11, 14, 8, 14, 12, 13, 10, 13, 12, 15, 9, 15, 13, 14, 11

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(17) = 8:
._._._._._._._._._._._._._._._._._.
|               |       |         |
|               |       |         |
|               |       |         |
|               |_______|         |
|               |       |_________|
|               |       |     |   |
|               |       |     |___|
|_______________|_______|_____|_|_|

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

Row m=8 of A113881, A219158.

a:= n-> `if`(n=1, 8, iquo(n, 8, 'r') +[0, 6, 4, 5, 2, 5, 4, 7][r+1]):
seq(a(n), n=0..100);

LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{0,8,4,5,2,5,4,7,1,7,5},80] (* Harvey P. Dale, Sep 07 2016 *)

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

a(n) = 1 + a(n-8) for n>9.

A226582 Smallest number of integer-sided squares needed to tile a 9 X n rectangle.

Original entry on oeis.org

0, 9, 6, 3, 6, 6, 3, 6, 7, 1, 6, 7, 4, 7, 7, 4, 7, 8, 2, 7, 8, 5, 8, 8, 5, 8, 9, 3, 8, 9, 6, 9, 9, 6, 9, 10, 4, 9, 10, 7, 10, 10, 7, 10, 11, 5, 10, 11, 8, 11, 11, 8, 11, 12, 6, 11, 12, 9, 12, 12, 9, 12, 13, 7, 12, 13, 10, 13, 13, 10, 13, 14, 8, 13, 14, 11, 14

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(19) = 7:
._._._._._._._._._._._._._._._._._._._.
|                 |         |         |
|                 |         |         |
|                 |         |         |
|                 |         |         |
|                 |_________|_________|
|                 |       |       |   |
|                 |       |       |___|
|                 |       |       |   |
|_________________|_______|_______|___|

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

Row m=9 of A113881, A219158.

a:= n-> `if`(n=1, 9, iquo(n, 9, 'r')+[0, 5, 6, 3, 6, 6, 3, 6, 7][r+1]):
seq(a(n), n=0..100);

G.f.: x*(4*x^10-4*x^9-6*x^8+x^7+3*x^6-3*x^5+3*x^3-3*x^2-3*x+9) / (x^10-x^9-x+1).

a(n) = 1 + a(n-9) for n>10.

cp's OEIS Frontend

A113881 Table of smallest number of squares, T(m,n), needed to tile an m X n rectangle, read by antidiagonals.

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Mathematica

A285721 Square array read by antidiagonals: A(n,k) = number of steps in simple Euclidean algorithm for gcd(n,k) to reach the termination test n=k, read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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Python

Scheme

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A030451 a(2*n) = n, a(2*n+1) = n+2.

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Maple

Mathematica

PARI

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Extensions

A226576 Smallest number of integer-sided squares needed to tile a 3 X n rectangle.

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Maple

Mathematica

PARI

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A226577 Smallest number of integer-sided squares needed to tile a 4 X n rectangle.

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Maple

Mathematica

Maxima

Formula

A226578 Smallest number of integer-sided squares needed to tile a 5 X n rectangle.

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Maple

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A226579 Smallest number of integer-sided squares needed to tile a 6 X n rectangle.

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Maple

Mathematica

Formula

A030451 a(2n) = n, a(2n+1) = n+2.