A034262 -id:A034262 - cp's OEIS frontend

A221524 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 2 or more.

0, 0, 0, 0, 2, 0, 0, 6, 2, 0, 0, 12, 10, 4, 0, 0, 20, 30, 36, 6, 0, 0, 30, 68, 144, 94, 10, 0, 0, 42, 130, 400, 536, 274, 16, 0, 0, 56, 222, 900, 1940, 2172, 768, 26, 0, 0, 72, 350, 1764, 5368, 9982, 8544, 2182, 42, 0, 0, 90, 520, 3136, 12458, 33380, 50400, 33960, 6170, 68, 0, 0

Offset: 1

R. H. Hardin Jan 19 2013

Table starts
.0...0......0.......0.........0..........0...........0...........0............0
.0...2......6......12........20.........30..........42..........56...........72
.0...2.....10......30........68........130.........222.........350..........520
.0...4.....36.....144.......400........900........1764........3136.........5184
.0...6.....94.....536......1940.......5368.......12458.......25544........47776
.0..10....274....2172......9982......33380.......90684......212812.......447962
.0..16....768....8544.....50400.....205080......654864.....1763328......4184064
.0..26...2182...33960....256018....1264378.....4738970....14629962.....39113752
.0..42...6170..134480...1297924....7787228....34274630...121342546....365574840
.0..68..17476..533248...6584320...47975704...247928860..1006508448...3416978176
.0.110..49470.2113456..33394958..295543282..1793345580..8348594292..31937713030
.0.178.140066.8377808.169387004.1820672982.12971955294.69248649436.298515152986

			Some solutions for n=6 k=4
..1....0....4....4....3....3....0....4....2....0....4....0....0....1....3....0
..3....3....0....0....0....1....2....0....4....3....2....4....2....3....1....3
..4....0....3....2....4....1....4....2....4....1....4....1....3....0....2....0
..1....0....4....4....1....4....2....3....2....2....2....1....0....0....4....4
..4....4....2....1....0....3....1....0....4....4....4....4....0....2....1....3
..2....1....4....4....2....1....4....3....0....2....2....2....3....4....3....1

R. H. Hardin, Table of n, a(n) for n = 1..1518

Column 2 is A006355
Row 2 is A002378(n-1)
Row 3 is A034262(n-1)
Row 4 is A035287

Empirical for column k:
k=2: a(n) = a(n-1) +a(n-2)
k=3: a(n) = a(n-1) +4*a(n-2) +3*a(n-3) +a(n-4)
k=4: a(n) = 2*a(n-1) +6*a(n-2) +6*a(n-3) +4*a(n-4) +4*a(n-6)
k=5: a(n) = 2*a(n-1) +11*a(n-2) +20*a(n-3) +17*a(n-4) -3*a(n-5) +a(n-6)
k=6: a(n) = 3*a(n-1) +14*a(n-2) +29*a(n-3) +28*a(n-4) +a(n-5) +27*a(n-6) +8*a(n-7) +2*a(n-8)
k=7: a(n) = 3*a(n-1) +21*a(n-2) +58*a(n-3) +79*a(n-4) +32*a(n-5) +23*a(n-6) +4*a(n-7) +8*a(n-8)
Empirical for row n:
n=2: a(n) = n^2 - n
n=3: a(n) = n^3 - 3*n^2 + 4*n - 2
n=4: a(n) = n^4 - 2*n^3 + n^2
n=5: a(n) = n^5 - n^4 - 10*n^3 + 38*n^2 - 60*n + 40 for n>2
n=6: a(n) = n^6 - 20*n^4 + 83*n^3 - 182*n^2 + 236*n - 148 for n>3
n=7: a(n) = n^7 + n^6 - 29*n^5 + 109*n^4 - 204*n^3 + 202*n^2 - 80*n for n>2

A344331 Side s of squares of type 1 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 68, 70, 78, 80, 90, 100, 110, 120, 130, 136, 140, 150, 156, 160, 170, 180, 190, 200, 204, 210, 220, 222, 230, 234, 240, 250, 260, 270, 272, 280, 290, 300, 310, 312, 320, 330, 340, 350, 360, 370, 380, 390, 400, 408, 410, 420, 430, 440, 444, 450, 460, 468, 470

Offset: 1

Bernard Schott, May 20 2021

nonn

This sequence is relative to the generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008 (see A344330).
There are two types of solutions, the first one is proposed here, while type 2 is described in A344332.
Some notations: s = side of the tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
-> Primitive squares
Side s of primitive squares of type 1 must satisfy the Diophantine equation s^2 = z * (a^2+b^2), with gcd(a, b) = 1, and without using the conditions a^2+b^2 = c^2, when a and b belong to a Pythagorean triple (a, b, c).
In this case, the sides of the primitive squares of type 1 are s = a*b * (a^2+b^2) with 1 <= a < b and gcd(a, b) = 1 (A344333), then corresponding z = (a*b)^2 * (a^2+b^2) (A344334).
Every primitive square is composed of m = a*b * (a^2+b^2) elementary rectangles of length L = a^2+b^2 and width W = a*b, so with an area A = a*b * (a^2+b^2) = m.
In particular: for a = 1, b = n, s = n*(n^2+1) form the subsequence A034262 \ {0, 1} and z = n^2*(n^2+1) form the subsequence A071253 \ {0, 2}).
See example with design for a square of side s = 10 with a = 1, b = 2, m = 10, z = 20.
-> Non-primitive squares
If s is the side of a primitive square of type 1 with z squares of side a and z squares of side b, then every k * s is a non-primitive term that gives one or two distinct tilings of type 1, depending of value of k:
- For every k > 1, the square ks X ks can be tiled with k^2*z squares of side a and k^2*z squares of side b (see example).
- For every k = r^4, r>1, the square ks X ks also can be tiled with z squares of side ka and z squares of side kb.
---> Consequences:
1) For every pair (a, b), 1 <= a < b, there is a square of side s = a*b * (a^2+b^2) that can be tiled with squares of side a and side b so that the number z of squares of side a and side b is the same, this number z = (a*b)^2 * (a^2+b^2).
2) If q is a term and K > 1, K * q is another term.
3) Every term is even.

			Primitive square with s = 10:
   a = 1, b = 2, s = 10, m = 10, z = 20, and
Non-primitive square with s = 20:
   a = 1, b = 2, s = 20, m = 40, z = 80.
      ___ ___ _ ___ ___ _ ___________________
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|___________________|
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |___________________|___________________|
with respectively m = 10 (and m = 40) elementary 2 X 5 rectangles as below:
          ___ ___ _
         |   |   |_|
         |___|___|_|
There are these three possibilities:
- 10 is a primitive term because the square 10 X 10 can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 2 X 2.
- 20 is a non-primitive term because the square 20 X 20 can be tiled with 80 squares of size 1 X 1 and 80 squares of size 2 X 2.
- 30 is a primitive term because the square 30 X 30 can be tiled with 90 squares of size 1 X 1 and 90 squares of size 3 X 3, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 3 X 3,
  but also, 30 is a non-primitive term because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2.

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Index to sequences related to Olympiads.

Cf. A344330, A344332, A344333, A344334.

Cf. A034262, A071253.

isokp1(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->x*y*(x^2+y^2), [1..m]), s); }
isok(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Dec 22 2021

A344333 Primitive side of squares of type 1 (A344331) that are tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

10, 30, 68, 78, 130, 222, 290, 300, 350, 510, 520, 738, 742, 820, 1010, 1218, 1342, 1530, 1740, 1752, 1820, 1830, 2210, 2590, 2750, 2758, 3270, 3390, 3492, 3560, 3570, 4112, 4290, 4498, 4770, 4930, 5850, 6028, 6328, 6870, 6878, 6942, 8020, 8030, 8190, 8610, 9282, 9620, 9962

Offset: 1

Bernard Schott, Jun 01 2021

nonn

Some notations: s = side of the tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
Every term is of the form s = a*b * (a^2+b^2) with gcd(a, b) = 1, then corresponding z = (a*b)^2 * (a^2+b^2) (see A344334).
Every primitive square is composed of m = a*b * (a^2+b^2) elementary rectangles of length L = a^2+b^2 and width W = a*b, so with an area A = a*b * (a^2+b^2) = m.
If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.
Every term is even.

			Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Index to sequences related to Olympiads.

Cf. A034262, A344330, A344331, A344332, A344334.

isok(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->if (gcd(x,y)==1, x*y*(x^2+y^2), 0), [1..m]), s); } \\ Michel Marcus, Dec 22 2021

A105374 a(n) = 4n^3 + 4n.

Original entry on oeis.org

0, 8, 40, 120, 272, 520, 888, 1400, 2080, 2952, 4040, 5368, 6960, 8840, 11032, 13560, 16448, 19720, 23400, 27512, 32080, 37128, 42680, 48760, 55392, 62600, 70408, 78840, 87920, 97672, 108120, 119288, 131200, 143880, 157352, 171640, 186768, 202760, 219640, 237432

Offset: 0

Henry Bottomley, Apr 02 2005

For n > 1, the number of straight lines with n points in a 4-dimensional hypercube with n points on each edge is 4*n^3 + 12*n^2 + 16*n + 8, i.e., A105374(n+1).

			a(5) = 4*5^3 + 4*5 = 500 + 20 = 520.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Essentially row or column of A102728 and A105374.

Cf. A002522, A006003, A008586, A034262.

I:=[0, 8, 40, 120]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 26 2012

CoefficientList[Series[8*x*(1+x+x^2)/(1-x)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,8,40,120},50] (* Vincenzo Librandi, Jun 26 2012 *)

a(n)=4*n^3+4*n \\ Charles R Greathouse IV, Oct 16 2015

a(n) = A002522(n)*A008586(n).
G.f.: 8*x*(1 + x + x^2)/(1-x)^4. - Colin Barker, May 24 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 26 2012
a(n) = 8*A006003(n). - Bruce J. Nicholson, Apr 18 2017
From Elmo R. Oliveira, Aug 07 2025: (Start)
E.g.f.: 4*x*(1 + x)*(2 + x)*exp(x).
a(n) = 4*A034262(n). (End)

A131471 a(n) = n^5+n.

Original entry on oeis.org

0, 2, 34, 246, 1028, 3130, 7782, 16814, 32776, 59058, 100010, 161062, 248844, 371306, 537838, 759390, 1048592, 1419874, 1889586, 2476118, 3200020, 4084122, 5153654, 6436366, 7962648, 9765650, 11881402, 14348934, 17210396, 20511178, 24300030, 28629182, 33554464

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

Cf. A271208, A271209.

[n^5+n: n in [0..30]]; // Vincenzo Librandi, Oct 01 2011

Table[n^5 + n, {n, 0, 50}] (* Paolo Xausa, Nov 03 2024 *)

a(n) = n^5+n; \\ Michel Marcus, Apr 02 2016

G.f.: 2*x*(1+11*x+36*x^2+11*x^3+x^4)/(-1+x)^6. - R. J. Mathar, Nov 14 2007

a(n) = A271208(n) + 1 = A271209(n) - 1. - Paolo Xausa, Nov 03 2024

A001621 a(n) = n(n + 1)(n^2 + n + 2)/4.

Original entry on oeis.org

0, 2, 12, 42, 110, 240, 462, 812, 1332, 2070, 3080, 4422, 6162, 8372, 11130, 14520, 18632, 23562, 29412, 36290, 44310, 53592, 64262, 76452, 90300, 105950, 123552, 143262, 165242, 189660, 216690, 246512, 279312, 315282, 354620, 397530, 444222, 494912, 549822

Offset: 0

N. J. A. Sloane

Number of integer sequences of length n+1 with sum zero and sum of absolute values 4. - R. H. Hardin, Feb 22 2009

Partial sums of A034262. - Jeremy Gardiner, Jun 23 2013

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002817, A058919, A092365.

Cf. A000124, A000217, A034262.

a[n_]:=Sum[i+i^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)
Array[# (# + 1) (#^2 + # + 2)/4 &, 39, 0] (* or *)
CoefficientList[Series[-2x (x^2 + x + 1)/(x - 1)^5, {x, 0, 38}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 12, 42, 110}, 39] (* Robert G. Wilson v, Aug 05 2018 *)

Equals 2 * A002817 and (A058919(n-1) - 1)/2.
G.f.: (-2*x*(x^2+x+1))/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = A000217(n) * A000124(n). - Torlach Rush, Aug 05 2018
E.g.f.: exp(x)*x*(8 + 16*x + 8*x^2 + x^3)/4. - Stefano Spezia, Oct 08 2022

A131472 a(n) = n^6 + n.

Original entry on oeis.org

0, 2, 66, 732, 4100, 15630, 46662, 117656, 262152, 531450, 1000010, 1771572, 2985996, 4826822, 7529550, 11390640, 16777232, 24137586, 34012242, 47045900, 64000020, 85766142, 113379926, 148035912, 191103000, 244140650, 308915802

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6+n: n in [0..30]]; // _Vincenzo Librandi+, Oct 01 2011

Table[n^6+n,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,2,66,732,4100,15630,46662},60] (* Harvey P. Dale, May 03 2012 *)

G.f.: 2*x*(1 + 26*x + 156*x^2 + 146*x^3 + 31*x^4)/(1 - x)^7. - R. J. Mathar, Nov 14 2007
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=0, a(1)=2, a(2)=66, a(3)=732, a(4)=4100, a(5)=15630, a(6)=46662. - Harvey P. Dale, May 03 2012
E.g.f.: exp(x)*x*(2 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5). - Stefano Spezia, Oct 08 2022

A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 1, 0, 1, 4, 10, 10, 3, 0, 0, 1, 5, 17, 30, 21, 3, 1, 0, 1, 6, 26, 68, 91, 42, 4, 0, 0, 1, 7, 37, 130, 273, 273, 85, 4, 1, 0, 1, 8, 50, 222, 651, 1092, 820, 170, 5, 0, 0, 1, 9, 65, 350, 1333, 3255, 4369, 2460, 341, 5, 1, 0, 1, 10

Offset: 0

Henry Bottomley, Feb 26 2001

			T(5,3)=10101 base 3=81+9+1=91; T(4,6)=1010 base 6=216+6=222. Table starts {0,0,0,0,...}, {1,1,1,1,...}, {0,1,2,3,...}, {1,2,5,10,...}, ...

Rows include A000004, A000012, A001477, A002522, A034262, A059826, A059830, A059839. Columns include A000035, A008619, A000975, A033113, A033114, A033115, A033116, A033117, A033118, A033119, A056830.

T(n, k)=floor[k^(n+1)/(k^2-1)] =T(n-2, k)+k^(n-1) =k*T(n-1, k)-((-1)^n-1)/2

A062323 Triangle with a(n,n)=1, a(n,k)=(n-1)*a(n-1,k)+a(n-2,k) for n>k.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 7, 10, 7, 3, 1, 30, 43, 30, 13, 4, 1, 157, 225, 157, 68, 21, 5, 1, 972, 1393, 972, 421, 130, 31, 6, 1, 6961, 9976, 6961, 3015, 931, 222, 43, 7, 1, 56660, 81201, 56660, 24541, 7578, 1807, 350, 57, 8, 1, 516901, 740785, 516901, 223884

Offset: 0

Henry Bottomley, Jul 05 2001

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 1, 1, 1;
[3] 2, 3, 2, 1;
[4] 7, 10, 7, 3, 1;
[5] 30, 43, 30, 13, 4, 1;
[6] 157, 225, 157, 68, 21, 5, 1;
[7] 972, 1393, 972, 421, 130, 31, 6, 1;
[8] 6961, 9976, 6961, 3015, 931, 222, 43, 7, 1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Essentially the same as A058294, but more easy seen as a triangle. Columns include A001040, A001053, A058307, A058308, A058309. Other sequences appearing on the right hand side include A000012, A001477, A002061, A034262.

a062323 n k = a062323_tabl !! n !! k
a062323_row n = a062323_tabl !! n
a062323_tabl = map fst $ iterate f ([1], [0,1]) where
   f (us, vs) = (vs, ws) where
     ws = (zipWith (+) (us ++ [0]) (map (* v) vs)) ++ [1]
          where v = last (init vs) + 1
-- Reinhard Zumkeller, Mar 05 2013

a(n, k)=k*a(n, k+1)+a(n, k+2) for n>k.

A190578 a(n) = n^7 + n.

Original entry on oeis.org

0, 2, 130, 2190, 16388, 78130, 279942, 823550, 2097160, 4782978, 10000010, 19487182, 35831820, 62748530, 105413518, 170859390, 268435472, 410338690, 612220050, 893871758, 1280000020, 1801088562, 2494357910, 3404825470, 4586471448

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 12 2011

a(n) = n^7 + n, A005843 for k=1, A002378 for k=2, A034262 for k=3, A091940 for k=4, A131471 for k=5, A131472 for k=6.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A005843, A002378, A034262, A091940, A131471, A131472.

[n^7+n: n in [0..30]]; // Vincenzo Librandi, Sep 30 2011

Mathematica
```
k=7; Table[n^k+n,{n,0,50}]
```

Offset changed from 1 to 0 by Vincenzo Librandi, Sep 30 2011

cp's OEIS Frontend

A221524 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 2 or more.

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A344331 Side s of squares of type 1 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

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A344333 Primitive side of squares of type 1 (A344331) that are tiled with squares of two different sizes so that the number of large or small squares is the same.

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A105374 a(n) = 4*n^3 + 4*n.

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Magma

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A131471 a(n) = n^5+n.

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Magma

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A001621 a(n) = n*(n + 1)*(n^2 + n + 2)/4.

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Mathematica

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A131472 a(n) = n^6 + n.

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Magma

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A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

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A105374 a(n) = 4n^3 + 4n.

A001621 a(n) = n(n + 1)(n^2 + n + 2)/4.