A007466 -id:A007466 - cp's OEIS frontend

A008794 Squares repeated; a(n) = floor(n/2)^2.

0, 0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 36, 36, 49, 49, 64, 64, 81, 81, 100, 100, 121, 121, 144, 144, 169, 169, 196, 196, 225, 225, 256, 256, 289, 289, 324, 324, 361, 361, 400, 400, 441, 441, 484, 484, 529, 529, 576, 576

Offset: 0

N. J. A. Sloane

Also number of non-attacking kings on (n-1) X (n-1) board (cf. A030978). - Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002
Also the independence number and clique covering number of the (n-1) X (n-1) king graph. - Eric W. Weisstein, Jun 20 2017
Maximum number of 2 X 2 tiles that fit on an n X n board. - Jon Perry, Aug 10 2003
(n)-(1) + (n-1)-(2) + (n-3)-(3) + ... + (n-r)-(r) ... n terms. E.g., 5-1+4-2+3 = 9, 6-1+5-2+4-3 = 9, 7-1+6-2+5-3+4 = 16, 8-1+7-2+6-3+5-4 = 16. - Amarnath Murthy, Jul 24 2005
The smallest possible number of white cells in a solution to an n X n nurikabe grid. - Tanya Khovanova, Feb 24 2009
(1 + x + 4*x^2 + 4*x^3 + 9*x^4 + ...) = (1/(1-x))*(1 + 3*x^2 + 5*x^4 + 7*x^6 + ...). - Gary W. Adamson, Apr 07 2010
If the set {1,2,...,n} is divided in half (a part having size ceiling(n/2) and the rest), then a(n+1) is the largest possible difference between the totals of these parts. - Vladimir Shevelev, Oct 14 2017
a(n+1) is the sum of the smallest parts of the partitions of 2n into two odd parts. - Wesley Ivan Hurt, Dec 06 2017
a(n-1) is the largest number of single cells of an n X n grid that share no edge or vertex with each other or those of the grid perimeter. - Stefano Spezia, Jul 30 2021
The binomial transform is 0, 0, 1, 4, 14, 44, 128, 352, 928, 2368, 5888... (see A007466).  - R. J. Mathar, Feb 25 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Stefano Spezia, Illustration of initial terms
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Kings Problem.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A030978, A032766, A059841, A086832, A109613, A182579, A189889.

Flat(List([0..24],n->[n^2,n^2])); # Muniru A Asiru, Oct 09 2018

[(2*n-1)*(-1)^n/8+(2*n^2-2*n +1)/8: n in [0..60]]; // Vincenzo Librandi, Aug 21 2011

A008794:=n->floor(n/2)^2: seq(A008794(n), n=0..50); # Wesley Ivan Hurt, Dec 08 2017

With[{sq = Range[0, 30]^2}, Riffle[sq, sq]] (* Harvey P. Dale, Nov 20 2015 *)
Table[Floor[n/2]^2, {n, 0, 49}] (* Michael De Vlieger, Oct 21 2016 *)
Table[(2 n - 1) (-1)^n/8 + (2 n^2 - 2 n + 1)/8, {n, 0, 49}] (* Michael De Vlieger, Oct 21 2016 *)
CoefficientList[Series[x^2*(1 + x^2)/((1 - x) (1 - x^2)^2), {x, 0, 49}], x] (* Michael De Vlieger, Oct 21 2016 *)
CoefficientList[Series[((x^2-x)Cosh[x]+(1+x+x^2)Sinh[x])/4,{x,0,50}],x]*Table[k!,{k,0,50}] (* Stefano Spezia, Oct 07 2018 *)

a(n)=(n\2)^2 \\ Charles R Greathouse IV, Sep 24 2015

first(n) = Vec(x^2*(1 + x^2)/((1 - x)*(1 - x^2)^2) + O(x^n), -n) \\ Iain Fox, Dec 08 2017

def A008794(n): return (n//2)**2 # Chai Wah Wu, Jun 07 2022

[((-1)^n*(2*n-1) +(2*n^2-2*n +1))/8 for n in (0..50)] # G. C. Greubel, Sep 11 2019

G.f.: x^2*(1 + x^2)/((1 - x)*(1 - x^2)^2).
a(n) = floor(n/2)^2.
From Paul Barry, May 31 2003: (Start)
a(n) = (2*n - 1)*(-1)^n/8 + (2*n^2 - 2*n + 1)/8.
a(n+1) = Sum_{k=0..n} k*(1-(-1)^k)/2. (End)
a(n+2) = Sum_{k=0..n} A109613(k)*A059841(n-k). - Reinhard Zumkeller, Dec 05 2009
a(n) = A182579(n,n-2) for n > 1. - Reinhard Zumkeller, May 07 2012
3*a(n) = A032766(n)^2 - A032766(n^2). - Bruno Berselli, Oct 21 2016
a(n) = Sum_{i=1..n-1; i odd} i. - Olivier Pirson, Nov 06 2017
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - Iain Fox, Dec 08 2017
E.g.f.: ((x^2 - x)*cosh(x) + (1 + x + x^2)*sinh(x))/4. - Stefano Spezia, Oct 07 2018

A097063 Expansion of (1-2x+3x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 0, 3, 4, 9, 12, 19, 24, 33, 40, 51, 60, 73, 84, 99, 112, 129, 144, 163, 180, 201, 220, 243, 264, 289, 312, 339, 364, 393, 420, 451, 480, 513, 544, 579, 612, 649, 684, 723, 760, 801, 840, 883, 924, 969, 1012, 1059, 1104, 1153, 1200, 1251, 1300, 1353, 1404

Offset: 0

Paul Barry, Jul 22 2004

Partial sums of A097062. Pairwise sums are A002061. Binomial transform is essentially A007466.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

A diagonal of A326296.

A097063:=n->(1/4) + (3/4)*(-1)^n + (1/2)*n^2; seq(A097063(n), n=0..50); # Wesley Ivan Hurt, Mar 08 2014

Table[(1/4) + (3/4)*(-1)^n + (1/2)*n^2, {n, 0, 50}] (* Wesley Ivan Hurt, Mar 08 2014 *)

G.f. : (1-2*x+3*x^2)/((1-x^2)(1-x)^2).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = Sum_{k=0..n} (k^2-k+1)*(-1)^(n-k).
a(2n) = A058331(n); a(2n+1) = A046092(n). - R. J. Mathar, Oct 27 2008
a(n) = binomial(n+1, 2) - ceiling((n+1)/2) + 2((n+1) mod 2). - Wesley Ivan Hurt, Mar 08 2014
a(n) = 2*floor(n/2) + ceiling((n-1)^2/2). - M. Ryan Julian Jr., Sep 10 2019
a(n) = A326296(n + 1, n) for n > 0. - Andrew Howroyd, Sep 23 2019

A141611 Triangle read by rows: T(n, k) = (n-k+1)(k+1)binomial(n, k).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 18, 18, 4, 5, 32, 54, 32, 5, 6, 50, 120, 120, 50, 6, 7, 72, 225, 320, 225, 72, 7, 8, 98, 378, 700, 700, 378, 98, 8, 9, 128, 588, 1344, 1750, 1344, 588, 128, 9, 10, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 10, 11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 22 2008

Read as a square array, this array factorizes as M*transpose(M), where M = ( k*binomial(n, k) )A003506(n,k).%20-%20_Peter%20Bala">{n,k>=1} = A003506(n,k). - _Peter Bala, Mar 06 2017

			Triangle begins as:
   1;
   2,   2;
   3,   8,    3;
   4,  18,   18,    4;
   5,  32,   54,   32,    5;
   6,  50,  120,  120,   50,    6;
   7,  72,  225,  320,  225,   72,    7;
   8,  98,  378,  700,  700,  378,   98,    8;
   9, 128,  588, 1344, 1750, 1344,  588,  128,    9;
  10, 162,  864, 2352, 3780, 3780, 2352,  864,  162,  10;
  11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11;
  ...
From _Peter Bala_, Mar 06 2017: (Start)
Factorization as a square array
  /1         \ /1  2  3  4...\ /1  2   3   4...\
  |2  2      | |   2  6 12...| |2  8  12  32...|
  |3  6  3   |*|      3 12...|=|3 18  54 120...|
  |4 12 12 4 | |         4...| |4 32 120 320...|
  |...       | |             | |...            |
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..5151 (Rows 0..100 of triangle, flattened)

Cf. A003506, A007466 (row sums), A037966, A085373.

A141611:= func< n,k | (k+1)*(n-k+1)*Binomial(n,k) >;
[A141611(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 22 2024

T[n_, m_]:= (n-m+1)*(m+1)*Binomial[n,m];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

T(n,m)=(n - m + 1)*(m + 1)*binomial(n, m) \\ Charles R Greathouse IV, Feb 15 2017

def A141611(n,k): return (k+1)*(n-k+1)*binomial(n,k)
flatten([[A141611(n,k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Sep 22 2024

T(n, k) = (k+1)*(n-k+1)*binomial(n,k).
Sum_{k=0..n} T(n, k) = A007466(n+1) (row sums).
O.g.f.: (1 - (1 + t)*x + 2*t*x^2)/(1 - (1 + t)*x)^3 = 1 + (2 + 2*t)*x + (3 + 8*t + 3*t^2)*x^2 + (4 + 18*t + 18*t^2 + 4*t^3)*x^3 + .... - Peter Bala, Mar 06 2017
From G. C. Greubel, Sep 22 2024: (Start)
T(2*n, n) = A037966(n+1).
T(2*n-1, n) = 2*A085373(n-1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) - 2*[n=2]. (End)

Offset corrected by G. C. Greubel, Sep 22 2024

A053545 Comparisons needed for Batcher's sorting algorithm applied to 2^n items.

Original entry on oeis.org

0, 1, 5, 19, 63, 191, 543, 1471, 3839, 9727, 24063, 58367, 139263, 327679, 761855, 1753087, 3997695, 9043967, 20316159, 45350911, 100663295, 222298111, 488636415, 1069547519, 2332033023, 5066719231, 10972299263, 23689428991, 51002736639, 109521666047, 234612588543, 501437431807

Offset: 0

N. J. A. Sloane, Mar 21 2000

Appears to be number of edges in graph where nodes are binary vectors of length n, two nodes u, v being joined by an edge if there's a vector of length n-1 that can be reached from u by deleting a bit and from v by deleting a bit. An independent set in this graph is a code that will correct single deletions.
Binomial transform of A005893: (1, 4, 10, 20, 34, 52, 74, ...) = (1, 5, 19, 63, 191, ...). - Gary W. Adamson, Apr 28 2008
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)= (-1)^n*coeff(charpoly(A,x),x^2). - Milan Janjic, Jan 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
I. Wegener, The Complexity of Boolean Functions, Wiley, 1987, see p. 151, (2.7).
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

The size of a maximal independent set in this graph (1, 1, 2, 2, 4, 6, 10, ...) agrees with A000016 for n <= 7 (and probably for n=8).

Cf. A005893, A007466.

[2^(n-2)*(n^2-n+4)-1: n in [0..30]]; // Vincenzo Librandi, Oct 10 2011

A053545:=n->2^(n - 2)*(n^2 - n + 4) - 1; seq(A053545(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2014

Table[2^(n-2)(n^2-n+4)-1,{n,0,30}] (* or *) LinearRecurrence[{7,-18,20,-8},{0,1,5,19},30] (* Harvey P. Dale, Apr 03 2013 *)

a(n)=2^(n-2)*(n^2-n+4)-1 \\ Charles R Greathouse IV, Feb 19 2017

G.f.: x*(1-2x+2x^2)/((1-x)*(1-2x)^3).
a(n) = 2^(n-2)*(n^2-n+4)-1.
Partial sums of A007466. - J. M. Bergot, Jan 20 2013
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4); a(0)=0, a(1)=1, a(2)=5, a(3)=19. - Harvey P. Dale, Apr 03 2013

A228643 Triangle read by rows: T(n,1) = n * (n - 1) + 1 and for k: 1 < k <= n: T(n,k) = T(n,k-1) + T(n-1,k-1).

Original entry on oeis.org

1, 3, 4, 7, 10, 14, 13, 20, 30, 44, 21, 34, 54, 84, 128, 31, 52, 86, 140, 224, 352, 43, 74, 126, 212, 352, 576, 928, 57, 100, 174, 300, 512, 864, 1440, 2368, 73, 130, 230, 404, 704, 1216, 2080, 3520, 5888, 91, 164, 294, 524, 928, 1632, 2848, 4928, 8448

Offset: 1

J. M. Bergot and Reinhard Zumkeller, Aug 29 2013

T(n,1) = A002061(n); T(n,2) = A005893(n-1) for n > 1;

T(n,n) = A007466(n).

			.   1:    1
.   2:    3,  4
.   3:    7, 10, 14
.   4:   13, 20, 30, 44
.   5:   21, 34, 54, 84, 128
.   6:   31, 52, 86,140, 224, 352
.   7:   43, 74,126,212, 352, 576, 928
.   8:   57,100,174,300, 512, 864,1440,2368
.   9:   73,130,230,404, 704,1216,2080,3520, 5888
.  10:   91,164,294,524, 928,1632,2848,4928, 8448,14336
.  11:  111,202,366,660,1184,2112,3744,6592,11520,19968,34304
.  12:  133,244,446,812,1472,2656,4768,8512,15104,26624,46592,80896.

Reinhard Zumkeller, Rows n = 1..100 of table, flattened

a228643 n k = a228643_tabl !! (n-1) !! (k-1)
a228643_row n = a228643_tabl !! (n-1)
a228643_tabl = map fst $ iterate
   (\(row, x) -> (scanl (+) (x * (x - 1) + 1) row, x + 1)) ([1], 2)

A340257 a(n) = 2^n * (1+n*(n+1)/2).

Original entry on oeis.org

1, 4, 16, 56, 176, 512, 1408, 3712, 9472, 23552, 57344, 137216, 323584, 753664, 1736704, 3964928, 8978432, 20185088, 45088768, 100139008, 221249536, 486539264, 1065353216, 2323644416, 5049942016, 10938744832, 23622320128, 50868518912, 109253230592, 234075717632

Offset: 0

Alois P. Heinz, Jan 02 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Partial sums of A080929.

Cf. A000079, A000124, A001787, A001815, A036289, A087431, A340298.

a:= n-> 2^n*(1+n*(n+1)/2):
seq(a(n), n=0..30);

Table[2^n (1+(n(n+1))/2),{n,0,30}] (* or *) LinearRecurrence[{6,-12,8},{1,4,16},30] (* Harvey P. Dale, Jan 19 2023 *)

G.f.: (4*x^2-2*x+1)/(1-2*x)^3.
E.g.f.: exp(2*x)*(2*x^2+2*x+1).
a(n) = A000079(n) + A001815(n+1).
a(n) = A000079(n) * A000124(n).
a(n) = 2*a(n-1) + n*2^n = 2*a(n-1) + A036289(n), assuming a(-1) = 1/2.
a(n) = A340298(2^n).
a(n) = 2 * A087431(n) for n > 0.
a(n) = 4 * A007466(n) for n > 0.

A087431 a(n) = 0^n/2 + 2^n*(n^2+n+2)/4.

Original entry on oeis.org

1, 2, 8, 28, 88, 256, 704, 1856, 4736, 11776, 28672, 68608, 161792, 376832, 868352, 1982464, 4489216, 10092544, 22544384, 50069504, 110624768, 243269632, 532676608, 1161822208, 2524971008, 5469372416, 11811160064, 25434259456

Offset: 0

Paul Barry, Sep 02 2003

Binomial transform of A080335 (with additional leading 1).

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

LinearRecurrence[{6,-12,8},{1,2,8,28},30] (* Harvey P. Dale, Nov 26 2015 *)

a(n) = 2*A007466(n) for n >= 1.

G.f.: (1-4*x+8*x^2-4*x^3)/(1-2*x)^3. - Colin Barker, Mar 18 2012

A383623 a(n) = 4^n - (n^2 + 3n + 4)2^(n-2).

Original entry on oeis.org

0, 0, 2, 20, 128, 672, 3168, 14016, 59648, 247808, 1014272, 4113408, 16588800, 66674688, 267444224, 1071497216, 4289921024, 17168596992, 68694441984, 274822594560, 1099389992960, 4397780172800, 17591605133312, 70367481692160, 281472242024448

Offset: 0

Enrique Navarrete, May 03 2025

a(n) is the number of strings of length n defined on {0,1,2,3} that contain at least two 2s or at least two 3s (or both).

			a(3)=20 since the strings are 220 (3 of this type), 221 (3 of this type), 223 (3 of this type), 330 (3 of this type), 331 (3 of this type), 332 (3 of this type), 222 and 333.

Index entries for linear recurrences with constant coefficients, signature (10,-36,56,-32).

Cf. A000302, A007466.

E.g.f.: exp(4*x) - exp(2*x)*(1+x)^2.
a(n) = 4^n - A007466(n+1).
G.f.: 2*x^2/((1 - 2*x)^3*(1 - 4*x)). - Stefano Spezia, May 03 2025

cp's OEIS Frontend

A008794 Squares repeated; a(n) = floor(n/2)^2.

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A097063 Expansion of (1-2*x+3*x^2)/((1+x)*(1-x)^3).

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A141611 Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).

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A053545 Comparisons needed for Batcher's sorting algorithm applied to 2^n items.

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A228643 Triangle read by rows: T(n,1) = n * (n - 1) + 1 and for k: 1 < k <= n: T(n,k) = T(n,k-1) + T(n-1,k-1).

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A340257 a(n) = 2^n * (1+n*(n+1)/2).

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A087431 a(n) = 0^n/2 + 2^n*(n^2+n+2)/4.

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A097063 Expansion of (1-2x+3x^2)/((1+x)*(1-x)^3).

A141611 Triangle read by rows: T(n, k) = (n-k+1)(k+1)binomial(n, k).

A383623 a(n) = 4^n - (n^2 + 3n + 4)2^(n-2).