A162147 -id:A162147 - cp's OEIS frontend

A005475 a(n) = n(5n+1)/2.

0, 3, 11, 24, 42, 65, 93, 126, 164, 207, 255, 308, 366, 429, 497, 570, 648, 731, 819, 912, 1010, 1113, 1221, 1334, 1452, 1575, 1703, 1836, 1974, 2117, 2265, 2418, 2576, 2739, 2907, 3080, 3258, 3441, 3629

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 0, in the direction 0, 11, ..., and the line from 3, in the direction 3, 24, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. - Omar E. Pol, Sep 26 2011

For n >= 3, a(n) is the sum of the numbers appearing in the 3rd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

G. C. Greubel, Table of n, a(n) for n = 0..5000
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Leo Tavares, Illustration: Hexagonal Trapeziums.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A110449, A130520, A162147, A202803.

Cf. similar sequences listed in A022289.

seq(binomial(5*n+1,2)/5, n=0..34); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(2*n+j, j=1..n): seq(a(n), n=0..38); # Zerinvary Lajos, Apr 29 2007

Table[n (5 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)

a(n)=n*(5*n+1)/2; \\ Joerg Arndt, Mar 27 2013

a(n) = A110449(n, 2) for n>1.
a(n) = a(n-1) + 5*n - 2 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A130520(5*n+2). - Philippe Deléham, Mar 26 2013
a(n) = A202803(n)/2. - Philippe Deléham, Mar 27 2013
a(n) = A162147(n) - A162147(n-1). - J. M. Bergot, Jun 21 2013
a(n) = A000217(3*n) - A000217(2*n). - Bruno Berselli, Oct 13 2016
From G. C. Greubel, Aug 23 2017: (Start)
G.f.: x*(2*x + 3)/(1-x)^3.
E.g.f.: (x/2)*(5*x+6)*exp(x). (End)
Sum_{n>=1} 1/a(n) = 10+2*gamma+2*Psi(1/5) = 0.57635... see A001620 and A200135. - R. J. Mathar, May 30 2022
Sum_{n>=1} 1/a(n) = 10 - sqrt(1+2/sqrt(5))*Pi - sqrt(5)*log(phi) - 5*log(5)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 10 2022

Incorrect comment deleted and minor errors corrected by Johannes W. Meijer, Feb 04 2010

A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 14, 5, 5, 20, 38, 33, 7, 6, 30, 80, 117, 70, 11, 7, 42, 145, 305, 330, 149, 15, 8, 56, 238, 660, 1072, 906, 298, 22, 9, 72, 364, 1260, 2777, 3622, 2367, 591, 30, 10, 90, 528, 2198, 6174, 11160, 11676, 6027, 1132, 42, 11, 110, 735, 3582, 12292, 28784, 42805, 36450, 14873, 2139, 56

Offset: 1

Christian G. Bower, Sep 07 2002

For k>=1, n->infinity is log(T(n,k)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)). - Vaclav Kotesovec, Mar 08 2015

			Square array T(n,k) begins:
  1,  2,   3,    4,    5, ...
  2,  6,  12,   20,   30, ...
  3, 14,  38,   80,  145, ...
  5, 33, 117,  305,  660, ...
  7, 70, 330, 1072, 2777, ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-10: A000041, A005380, A217093, A255050, A255052, A270239, A270240, A270241, A270242, A270243.
Rows 1-3: A000027, A002378, A162147.
Main diagonal: A075197.
Cf. A255903.

with(numtheory):
A:= proc(n, k) option remember; local d, j;
      `if`(n=0, 1, add(add(d*binomial(d+k-1, k-1),
       d=divisors(j)) *A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 26 2012

Transpose[Table[nn=6;p=Product[1/(1- x^i)^Binomial[i+n,n],{i,1,nn}];Drop[CoefficientList[Series[p,{x,0,nn}],x],1],{n,0,nn}]]//Grid  (* Geoffrey Critzer, Sep 27 2012 *)

T(n,k) = Sum_{i=0..k} C(k,i) * A255903(n,i). - Alois P. Heinz, Mar 10 2015

A035005 Number of possible queen moves on an n X n chessboard.

Original entry on oeis.org

0, 12, 56, 152, 320, 580, 952, 1456, 2112, 2940, 3960, 5192, 6656, 8372, 10360, 12640, 15232, 18156, 21432, 25080, 29120, 33572, 38456, 43792, 49600, 55900, 62712, 70056, 77952, 86420, 95480, 105152, 115456, 126412, 138040, 150360

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

The number of (2 to n) digit sequences that can be found reading in any orientation, including diagonals, in an (n X n) grid. - Paul Cleary, Aug 12 2005

			3 X 3 board: queen has 8*6 moves and 1*8 moves, so a(3)=56.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033586 (King), A035006 (Rook), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A162147.

[(n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011

Table[(n-1)2n^2+(4n^3-6n^2+2n)/3,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,12,56,152},40] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = (n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3.
From Johannes W. Meijer, Feb 04 2010: (Start)
a(n) = A002492(n-1) + A035006(n) since Queen = Bishop + Rook.
a(n) = 4 * A162147(n-1). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=12, a(2)=56, a(3)=152. - Harvey P. Dale, Aug 24 2011
From Colin Barker, Mar 11 2012: (Start)
a(n) = 2*n*(1-6*n+5*n^2)/3.
G.f.: 4*x^2*(3+2*x)/(1-x)^4. (End)
E.g.f.: 2*exp(x)*x^2*(9 + 5*x)/3. - Stefano Spezia, Jul 31 2022

More terms from Erich Friedman

A162148 a(n) = n(n+1)(5*n+7)/6.

Original entry on oeis.org

0, 4, 17, 44, 90, 160, 259, 392, 564, 780, 1045, 1364, 1742, 2184, 2695, 3280, 3944, 4692, 5529, 6460, 7490, 8624, 9867, 11224, 12700, 14300, 16029, 17892, 19894, 22040, 24335, 26784, 29392, 32164, 35105, 38220, 41514, 44992, 48659, 52520, 56580

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Partial sums of A147875.
Equals the fourth right hand column of A175136 for n>=1. - Johannes W. Meijer, May 06 2011
a(n) is the number of triples (w,x,y) havingt all terms in {0,...,n} and x+y>w. - Clark Kimberling, Jun 14 2012

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

Cf. A002412, A002413, A006331, A016061, A033994.
Cf. A162147, A175136, A190048, A190049, A254407.
Related to A000292 and A091894.

[n*(n+1)*(5*n+7)/6: n in [0..50]]; // Vincenzo Librandi, May 07 2011

A162148:= n-> n*(n+1)*(5*n+7)/6; seq(A162148(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[(n(n+1)(5n+7))/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,4,17,44}, 50] (* Harvey P. Dale, May 20 2014 *)

a(n)=n*(n+1)*(5*n+7)/6 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(5*n+7)/6 for n in (0..50)] # G. C. Greubel, Mar 31 2021

a(n) = A162147(n) + A000217(n).
From Johannes W. Meijer, May 06 2011: (Start)
G.f.: x*(4+x)/(1-x)^4.
a(n) = 4*binomial(n+2,3) + binomial(n+1,3).
a(n) = A091894(3,0)*binomial(n+2,3) + A091894(3,1)*binomial(n+1,3). (End)
a(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i).
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4), a(0)=0, a(1)=4, a(2)=17, a(3)=44. - Harvey P. Dale, May 20 2014
E.g.f.: x*(24 +27*x +5*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

Definition rephrased by R. J. Mathar, Jun 27 2009

A185909 Accumulation array of A185908, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 7, 6, 4, 11, 14, 10, 5, 15, 23, 23, 15, 6, 19, 32, 38, 34, 21, 7, 23, 41, 54, 56, 47, 28, 8, 27, 50, 70, 80, 77, 62, 36, 9, 31, 59, 86, 105, 110, 101, 79, 45, 10, 35, 68, 102, 130, 145, 144, 128, 98, 55, 11, 39, 77, 118, 155, 181, 190, 182, 158, 119, 66, 12, 43, 86, 134, 180, 217, 238, 240, 224, 191, 142, 78, 13, 47, 95, 150, 205, 253, 287, 301, 295, 270, 227, 167, 91, 14, 51, 104, 166, 230, 289, 336, 364, 370, 355, 320, 266, 194, 105, 15, 55, 113, 182, 255

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185907 < A185908 < A185909 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
   1,  2,  3,  4,  5
   3,  7, 11, 15, 19
   6, 14, 23, 32, 41
  10, 23, 38, 54, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185907, A185908.
diag (1,7,...): A004068.
diag (2,11,...): A033994.
diag (3,14,...): A162147.

f[n_, 0] := 0; f[0, k_] := 0; (*needed for the weight array*)
f[n_, k_] := Min[n, k] + n - 1;
s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}];
Table[s[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten

A212415 Number of (w,x,y,z) with all terms in {1,...,n} and w=y<=z.

Original entry on oeis.org

0, 0, 3, 17, 55, 135, 280, 518, 882, 1410, 2145, 3135, 4433, 6097, 8190, 10780, 13940, 17748, 22287, 27645, 33915, 41195, 49588, 59202, 70150, 82550, 96525, 112203, 129717, 149205, 170810, 194680, 220968, 249832, 281435, 315945

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Partial sums of A162147. - J. M. Bergot, Jun 21 2013

			a(3) counts these (w,x,y,z): (1,2,2,2), (1,2,2,3), (1,3,3,3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A211795.

List([0..40], n-> n*(5*n+2)*(n^2-1)/24); # G. C. Greubel, Jul 11 2019

[n*(5*n+2)*(n^2-1)/24: n in [0..40]]; // G. C. Greubel, Jul 11 2019

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < x >= y <= z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212415 *)
Table[n*(5*n+2)*(n^2-1)/4!, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; n*(5*n+2)*(n^2-1)/4!) \\ G. C. Greubel, Jul 11 2019

[n*(5*n+2)*(n^2-1)/24 for n in (0..40)] # G. C. Greubel, Jul 11 2019

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
From Bruno Berselli, May 30 2012: (Start)
G.f.: x^2*(3+2*x)/(1-x)^5.
a(n) = (n-1)*n*(n+1)*(5*n+2)/24. (End)
E.g.f.: x^2*(36 + 32*x + 5*x^2)*exp(x)/24. - G. C. Greubel, Jul 11 2019

A256497 Triangle read by rows, sums of 2 distinct nonzero cubes: T(n,k) = (n+1)^3+k^3, 1 <= k <= n.

Original entry on oeis.org

9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 559, 513, 520, 539, 576, 637, 728, 855, 730, 737, 756, 793, 854, 945, 1072, 1241, 1001, 1008, 1027, 1064, 1125, 1216, 1343, 1512, 1729

Offset: 1

Bob Selcoe, Mar 31 2015

When n=k: T(n,k) = (2n+1)(n^2+n+1). Therefore, T(n,k)/(2n+1) = A002061(n+1).
A002383 is the sequence of all primes of the form T(n,k)/(2n+1), n=k.
When starting at T(n,k) n=k, diagonal sums are n^2*(2n+1)^2. For example, starting at T(4,4) = 189: 189+243+351+513 = 4^2*9^2 = 1296.
Coefficients in T(n,k) are multiples of n+k+1; therefore, coefficients in all diagonals starting at T(n,1) are multiples of n+2.
Let T"(n,k) = T(n,k)/(n+k+1). Then reading T"(n,k) by rows:
i. Row sums are A162147(n). For example, T"(3,k) = [65/5, 72/6, 91/7] = [13,12,13]. 13+12+13 = 38; A162147(3) = 38.
ii. Extend the triangle in A215631 to a symmetric array by reflection about the main diagonal, and let that array be T"215631(n,k). Then the diagonal starting with T"215631(n,1) is row n in T"(n,k). For example, the diagonal starting at T"215631(4,1) = [21,19,19,21]; T"(4,k) = [126/6, 133/7, 152/8, 189/9] = [21,19,19,21].
iii.  Coefficients in T"(n,k) are a permutation of A024612.

			Triangle starts:
n\k   1    2    3    4    5    6    7     8    9   10 ...
1:    9
2:    28  35
3:    65  72   91
4:   126  133  152  189
5:   217  224  243  280  341
6:   344  351  370  407  468  559
7:   513  520  539  576  637  728  855
8:   730  737  756  793  854  945  1072 1241
9:   1001 1008 1027 1064 1125 1216 1343 1512 1729
10:  1332 1339 1358 1395 1456 1547 1674 1843 2060 2331
...
The successive terms are (2^3+1^3), (3^3+1^3), (3^3+2^3), (4^3+1^3), (4^3+2^3), (4^3+3^3), ...

Cf. A024670.

Related: A002061, A002383, A003215, A024612, A162147, A215631.

T(n,k) = (n+1)^3+k^3.

T(n,k) = (2k+1)(k^2+k+1) + Sum_{j=k+1..n} A003215(j), n>=k+1. For example, T(8,4) = 9*21 + 91 + 127 + 169 + 217 = 793.

cp's OEIS Frontend

A005475 a(n) = n*(5*n+1)/2.

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A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.

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A035005 Number of possible queen moves on an n X n chessboard.

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A162148 a(n) = n*(n+1)*(5*n+7)/6.

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A185909 Accumulation array of A185908, by antidiagonals.

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A212415 Number of (w,x,y,z) with all terms in {1,...,n} and w=y<=z.

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A256497 Triangle read by rows, sums of 2 distinct nonzero cubes: T(n,k) = (n+1)^3+k^3, 1 <= k <= n.

A005475 a(n) = n(5n+1)/2.

A162148 a(n) = n(n+1)(5*n+7)/6.