A103220 -id:A103220 - cp's OEIS frontend

A015237 a(n) = (2n - 1)n^2.

0, 1, 12, 45, 112, 225, 396, 637, 960, 1377, 1900, 2541, 3312, 4225, 5292, 6525, 7936, 9537, 11340, 13357, 15600, 18081, 20812, 23805, 27072, 30625, 34476, 38637, 43120, 47937, 53100, 58621, 64512

Offset: 0

N. J. A. Sloane

Structured hexagonal prism numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Number of divisors of 60^(n-1) for n>0. - J. Lowell, Aug 30 2008
The sum of the 2*n+1 numbers between n*(n+1) and (n+1)*(n+2) gives a(n+1). - J. M. Bergot, Apr 17 2013
Partial sums of A080859. - J. M. Bergot, Jul 03 2013
a(n) = number of 2 X 2 matrices having all elements in {0..n} with determinant = permanent. - Indranil Ghosh, Dec 26 2016
Number of additions and multiplications needed to multiply two n X n matrices naively. - Charles R Greathouse IV, Jan 19 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
M. Janjic and B. Petkovic, A counting function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100177 (structured prisms); A100145 (more on structured numbers).
Cf. A000578, A045991, A000384, A080859 (first diffs), A103220 (partial sums).
Cf. similar sequences, with the formula (k*n-k+2)*n^2/2, listed in A262000.
Cf. A036970, A272378.

List([0..40],n->(2*n-1)*n^2); # Muniru A Asiru, Feb 05 2019

[(2*n-1)*n^2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

[(2*n-1)*n^2$n=0..40]; # Muniru A Asiru, Feb 05 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[2]==12, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3] + 12}, a, {n, 30}] (* G. C. Greubel, Jul 31 2015 *)
Table[(2 n - 1) n^2, {n, 0, 40}] (* Bruno Berselli, Sep 08 2015 *)

a(n)=(2*n-1)*n^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000578(n) + A045991(n). - Zerinvary Lajos, Jun 11 2008
a(n) = A199771(2*n-1) for n > 0. - Reinhard Zumkeller, Nov 23 2011
G.f.: x*(1+8*x+3*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 12, a(0)=1, a(1)=1, a(2)=12. - G. C. Greubel, Jul 31 2015
E.g.f.: x*(2*x^2 + 5*x + 1)*exp(x). - G. C. Greubel, Jul 31 2015
a(n) = Sum_{i=0..n-1} n*(4*i+1) for n>0. - Bruno Berselli, Sep 08 2015
Sum_{n>=1} 1/a(n) = 4*log(2) - Pi^2/6. - Vaclav Kotesovec, Oct 04 2016
a(n) = Sum_{i=n^2-n+1..n^2+n-1} i. - Wesley Ivan Hurt, Dec 27 2016
From Peter Bala, Jan 30 2019: (Start)
Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have (2*x - 1)*x^2 = Sum_{n >= 0} ((n+1)^5 + n^5)*a(n,x) and (2*x - 1)*x = Sum_{n >= 0} ((n+1)^4 - n^4)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 2. See the Bala link in A036970. Cf. A272378. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - Pi^2/12 - 2*log(2). - Amiram Eldar, Jul 12 2020

A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 3, 52, 370, 1620, 5285, 14168, 33012, 69240, 133815, 242220, 415558, 681772, 1076985, 1646960, 2448680, 3552048, 5041707, 7018980, 9603930, 12937540, 17184013, 22533192, 29203100, 37442600, 47534175, 59796828, 74589102, 92312220, 113413345, 138388960

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A099152, A047659, A103220, A202655, A202656, A202657.

Rest@ CoefficientList[Series[-x^3*(17 x^3 + 69 x^2 + 31 x + 3)/(x - 1)^7, {x, 0, 32}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 1/6*(n-2)*(n-1)*n*(n^3-5*n^2+8*n-3).

G.f.: -x^3*(17*x^3 + 69*x^2 + 31*x + 3)/(x-1)^7.

A202655 Number of ways to place 4 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 7, 223, 2429, 15045, 66122, 230074, 675798, 1745318, 4073993, 8764753, 17630795, 33522531, 60756612, 105666148, 177293340, 288246972, 455749371, 702898611, 1060173961, 1567213681, 2274896558, 3247759614, 4566786770, 6332604226, 8669120733, 11727651845

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7,-19,21,6,-42,42,-6,-21,19,-7,1).

Cf. A099152, A061994, A103220, A202654, A202656, A202657.

Rest@ CoefficientList[Series[-x^4*(151 x^6 + 1022 x^5 + 2233 x^4 + 2132 x^3 + 1001 x^2 + 174 x + 7)/((x - 1)^9*(x + 1)^2), {x, 0, 29}], x] (* Michael De Vlieger, Aug 19 2019 *)
LinearRecurrence[{7,-19,21,6,-42,42,-6,-21,19,-7,1},{0,0,0,7,223,2429,15045,66122,230074,675798,1745318},30] (* Harvey P. Dale, Mar 17 2025 *)

a(n) = n^8/24 - 2*n^7/3 + 41*n^6/9 - 257*n^5/15 + 341*n^4/9 - 97*n^3/2 + 2341*n^2/72 - 87*n/10 + (n/2 - 1/2)*floor(n/2).

G.f.: -x^4*(151*x^6 + 1022*x^5 + 2233*x^4 + 2132*x^3 + 1001*x^2 + 174*x + 7)/((x-1)^9*(x+1)^2).

A202656 Number of ways to place 5 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 23, 1104, 16945, 141696, 810746, 3568352, 12948318, 40514560, 112720393, 285073712, 666143975, 1456288512, 3007576740, 5913372864, 11138305068, 20202100224, 35433809451, 60316600080, 99947225741, 161638967424, 255701773822, 396439174560, 603407582570

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -17, 7, 43, -77, 11, 99, -99, -11, 77, -43, -7, 17, -7, 1).

Cf. A099152, A108792, A103220, A202654, A202655, A202657.

Rest@ CoefficientList[Series[-x^5*(1899 x^9 + 16515 x^8 + 60512 x^7 + 116784 x^6 + 137646 x^5 + 98222 x^4 + 41688 x^3 + 9608 x^2 + 943 x + 23)/((x - 1)^11*(x + 1)^4), {x, 0, 27}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = n^10/120 - 2*n^9/9 + 95*n^8/36 - 183*n^7/10 + 14663*n^6/180 - 1201*n^5/5 + 16753*n^4/36 - 25364*n^3/45 + 68293*n^2/180 - 12781*n/120 + (n^3/2 - 6*n^2 + 39*n/2 - 61/4)*floor(n/2).

G.f.: -x^5*(1899*x^9 + 16515*x^8 + 60512*x^7 + 116784*x^6 + 137646*x^5 + 98222*x^4 + 41688*x^3 + 9608*x^2 + 943*x + 23)/((x-1)^11*(x+1)^4).

A202657 Number of ways to place 6 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 83, 6107, 126376, 1377328, 9984758, 54399330, 239675936, 895773148, 2935757573, 8641608781, 23259768860, 58039719112, 135720432200, 299995484600, 631220344328, 1271607596876, 2464466665667, 4613731163831, 8372196591052, 14769606793684, 25395151577010

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (5, -4, -18, 30, 24, -61, -31, 74, 70, -92, -104, 104, 92, -70, -74, 31, 61, -24, -30, 18, 4, -5, 1).

Cf. A099152, A176186, A103220, A202654, A202655, A202656.

Rest@ CoefficientList[Series[-x^6*(31709 x^16 + 377288 x^15 + 2265487 x^14 + 8441426 x^13 + 22166758 x^12 + 43217858 x^11 + 64805639 x^10 + 75943200 x^9 + 70077016 x^8 + 50738668 x^7 + 28477437 x^6 + 12074418 x^5 + 3711058 x^4 + 771370 x^3 + 96173 x^2 + 5692 x + 83)/((x - 1)^13*(x + 1)^6*(x^2 + x + 1)^2), {x, 0, 26}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = n^12/720 - n^11/18 + 73*n^10/72 - 72247*n^9/6480 + 5909*n^8/72 - 320653*n^7/756 + 112795*n^6/72 - 8892919*n^5/2160 + 8086231*n^4/1080 - 5740271*n^3/648 + 2598425*n^2/432 - 13161367*n/7560 + (n^5/4 - 77*n^4/12 + 757*n^3/12 - 7007*n^2/24 + 14581*n/24 - 1677/4)*floor(n/2) + (64*n/9 - 88/9)*floor(n/3) + (8*n/3 - 52/9)*floor((n + 1)/3).

G.f.: -x^6*(31709*x^16 + 377288*x^15 + 2265487*x^14 + 8441426*x^13 + 22166758*x^12 + 43217858*x^11 + 64805639*x^10 + 75943200*x^9 + 70077016*x^8 + 50738668*x^7 + 28477437*x^6 + 12074418*x^5 + 3711058*x^4 + 771370*x^3 + 96173*x^2 + 5692*x + 83)/((x-1)^13*(x+1)^6*(x^2+x+1)^2).

A203246 Second elementary symmetric function of the first n terms of (1,1,2,2,3,3,4,4,...).

Original entry on oeis.org

1, 5, 13, 31, 58, 106, 170, 270, 395, 575, 791, 1085, 1428, 1876, 2388, 3036, 3765, 4665, 5665, 6875, 8206, 9790, 11518, 13546, 15743, 18291, 21035, 24185, 27560, 31400, 35496, 40120, 45033, 50541, 56373, 62871, 69730, 77330, 85330, 94150, 103411, 113575

Offset: 2

Clark Kimberling, Dec 31 2011

Second subdiagonal of A246117. - Peter Bala, Aug 15 2014

Sela Fried, On A203246, 2024.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A203298, A203299, A246117, A212523 (odd bisection), A103220 (even bisection).

f[k_] := Floor[(k + 1)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]  (* A203246 *)

Conjectural o.g.f.: x^2*(1 + 3*x + x^2 + x^3)/((1 - x^2)^3*(1 - x)^2). - Peter Bala, Aug 15 2014
Conjectural closed form: 64*a(n) = 2*n^2 -16*n/3 -3 +16*n^3/3 +2*n^4 +(-1)^n *(3-2*n^2). - R. J. Mathar, Oct 01 2016
Both conjectures are true. See link. - Sela Fried, Dec 22 2024

A103219 Triangle read by rows: T(n,k) = (n+1-k)(4(n+1-k)^2 - 1)/3+2k(n+1-k)^2.

Original entry on oeis.org

1, 10, 3, 35, 18, 5, 84, 53, 26, 7, 165, 116, 71, 34, 9, 286, 215, 148, 89, 42, 11, 455, 358, 265, 180, 107, 50, 13, 680, 553, 430, 315, 212, 125, 58, 15, 969, 808, 651, 502, 365, 244, 143, 66, 17, 1330, 1131, 936, 749, 574, 415, 276, 161, 74, 19, 1771, 1530, 1293

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 26 2005

The triangle is generated from the product B * A of the infinite lower triangular matrices A =
1 0 0 0...
3 1 0 0...
5 3 1 0...
7 5 3 1...
...
and B =
1 0 0 0...
1 3 0 0...
1 3 5 0...
1 3 5 7...
...

			Triangle begins:
1,
10,3,
35,18,5,
84,53,26,7,
165,116,71,34,9,
286,215,148,89,42,11,

Row sums give A103220.
T(n, 0) = (n+1)*(4*(n+1)^2 - 1)/3 = A000447(n+1);
T(n+1, n)= 8*n+2 = A017089(n+1);
Cf. A103218 (for product A*B), A103220.

T[n_, k_] := (n + 1 - k)*(4*(n + 1 - k)^2 - 1)/3 + 2*k*(n + 1 - k)^2; Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Feb 10 2005 *)

T(n,k)=(n+1-k)*(4*(n+1-k)^2-1)/3+2*k*(n+1-k)^2; for(i=0,10, for(j=0,i,print1(T(i,j),","));print())

A185375 a(n) = n(n-1)(2n+1)(2n-1)(2n-3)(10*n-17)/90.

Original entry on oeis.org

0, 0, 1, 91, 966, 5082, 18447, 53053, 129948, 282948, 562989, 1043119, 1824130, 3040830, 4868955, 7532721, 11313016, 16556232, 23683737, 33201987, 45713278, 61927138, 82672359, 108909669, 141745044, 182443660

Offset: 0

Wesley Transue, Jan 21 2012

Third column (k=2) of A008958.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Third column (k=2) of A008958 Triangle of central factorial numbers.

Cf. A103220.

[n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90 : n in [0..50]]; // Wesley Ivan Hurt, Apr 23 2021

Table[n*(n - 1)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(10*n - 17)/90, {n, 0, 50}] (* G. C. Greubel, Jun 28 2017 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,1,91,966,5082,18447},30] (* Harvey P. Dale, Oct 10 2021 *)

a(n) = binomial(2*n+1,5)*(10*n-17)/3  \\ Michel Marcus, Jun 18 2013

a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90.
a(n) = binomial(2*n+1,5)*(10*n-17)/3.
From G. C. Greubel, Jun 28 2017: (Start)
G.f.: x^2*(1 + 84*x + 350*x^2 + 196*x^3 + 9*x^4)/(1 - x)^7.
E.g.f.: (1/90)*x^2*(45 + 1320 x + 2280 x^2 + 864 x^3 + 80 x^4)*exp(x). (End)
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). - Wesley Ivan Hurt, Apr 23 2021
a(n) = Sum_{1 <= i <= j <= n-1} (2*i - 1)^2 * (2*j - 1)^2. - Peter Bala, Sep 03 2023

A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 1, 9, 9, 3, 1, 16, 48, 32, 0, 1, 25, 150, 250, 75, 15, 1, 36, 360, 1200, 1224, 288, 0, 1, 49, 735, 4165, 8869, 6321, 931, 133, 1, 64, 1344, 11648, 43136, 64512, 33024, 4096, 0, 1, 81, 2268, 27972, 160866, 423306, 469800

Offset: 1

Walter Trump, Mar 09 2021

T(0,0):=1 for combinatorial reasons.
A semi-queen can only move horizontal, vertical and parallel to the main diagonal of the board. Moves parallel to the secondary diagonal are not allowed.
Instead of a board on a torus, you can imagine that the semi-queens can leave a flat board on one side and re-enter the board on the other side.

			  1;
  1,  1;
  1,  4,   0;
  1,  9,   9,   3;
  1, 16,  48,  32,  0;
  1, 25, 150, 250, 75, 15;

Walter Trump, Table of n, a(n) for n = 1..222
Walter Trump, Semi-queen problem

Cf. A006717, A099152, A103220, A202654, A202655, A202656, A202657.

T(n,0) = 1.
T(n,1) = n^2.
T(n,2) = n^2*(n-1)*(n-2)/2.
T(n,3) = n^2*(n-1)*(n-2)*(n^2-6n+10)/6.
T(2n+1,2n+1) = A006717(n).
T(2n,2n) = 0.

A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 4, 9, 9, 24, 25, 16, 45, 60, 49, 25, 72, 105, 112, 81, 36, 105, 160, 189, 180, 121, 49, 144, 225, 280, 297, 264, 169, 64, 189, 300, 385, 432, 429, 364, 225, 81, 240, 385, 504, 585, 616, 585, 480, 289, 100, 297, 480, 637, 756, 825

Offset: 1

Gary W. Adamson, Jun 10 2004

Matrix square of A158405.

			[1 0 0 / 1 3 0 / 1 3 5]^2 = [1 0 0 / 4 9 0 / 9 24 25]. Delete the zeros and
read by rows:
1;
4, 9;
9, 24, 25;
16,45, 60, 49;
25,72,105,112, 81;

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966.

Cf. A094728, A095871, A095872.

A095873 := proc(n,k)
        (2*k-1)*(n+k-1)*(n-k+1) ;
end proc:
seq(seq(A095873(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Oct 30 2011

Table[(2k-1)(n+k-1)(n-k+1),{n,10},{k,n}]//Flatten (* Harvey P. Dale, May 03 2018 *)

T(n,k) = (2*k-1)*A094728(n,k).

Sum_{k=1..n} T(n,k)= n*(n+1)*(3*n^2+n-1)/6 = A103220(n). - R. J. Mathar, Oct 30 2011

Definition in closed form by R. J. Mathar, Oct 30 2011

cp's OEIS Frontend

A015237 a(n) = (2*n - 1)*n^2.

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A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

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A202655 Number of ways to place 4 nonattacking semi-queens on an n X n board.

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A202656 Number of ways to place 5 nonattacking semi-queens on an n X n board.

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A202657 Number of ways to place 6 nonattacking semi-queens on an n X n board.

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A203246 Second elementary symmetric function of the first n terms of (1,1,2,2,3,3,4,4,...).

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A103219 Triangle read by rows: T(n,k) = (n+1-k)*(4*(n+1-k)^2 - 1)/3+2*k*(n+1-k)^2.

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A185375 a(n) = n*(n-1)*(2*n+1)*(2*n-1)*(2*n-3)*(10*n-17)/90.

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A015237 a(n) = (2n - 1)n^2.

A103219 Triangle read by rows: T(n,k) = (n+1-k)(4(n+1-k)^2 - 1)/3+2k(n+1-k)^2.

A185375 a(n) = n(n-1)(2n+1)(2n-1)(2n-3)(10*n-17)/90.

A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.