A014820 -id:A014820 - cp's OEIS frontend

A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66

Offset: 1

Max Alekseyev, Feb 24 2018

			Array starts:
  n=1:  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  n=2:  1,  2,  3,   4,   5,    6,    7,    8,     9,    10,    11, ...
  n=3:  1,  3,  6,  10,  15,   21,   28,   36,    45,    55,    66, ...
  n=4:  1,  4,  9,  16,  25,   36,   49,   64,    81,   100,   121, ...
  n=5:  1,  5, 15,  35,  70,  126,  210,  330,   495,   715,  1001, ...
  n=6:  1,  6, 19,  37,  61,   91,  127,  169,   217,   271,   331, ...
  n=7:  1,  7, 28,  84, 210,  462,  924, 1716,  3003,  5005,  8008, ...
  n=8:  1,  8, 33,  96, 225,  456,  833, 1408,  2241,  3400,  4961, ...
  n=9:  1,  9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...
  n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951,  6531, 10201, 15231, ...
  ...

Max Alekseyev, Table of n, a(n) for n = 1..351

Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).
Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).
Diagonal: A299754 (n=k).
Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

From Chai Wah Wu, May 28 2018: (Start)
The following are all conjectures.
For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).
For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.
(End)

Row 6 corrected by Max Alekseyev, Aug 14 2022

A061926 Square table by antidiagonals where odd rows are partial sums of previous row, even rows are sums of pairs of values in previous row and initial row is 0 and 1 alternating.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 2, 3, 3, 1, 0, 0, 3, 4, 6, 4, 1, 0, 1, 3, 5, 10, 9, 5, 1, 0, 0, 4, 6, 15, 16, 14, 6, 1, 0, 1, 4, 7, 21, 25, 30, 19, 7, 1, 0, 0, 5, 8, 28, 36, 55, 44, 26, 8, 1, 0, 1, 5, 9, 36, 49, 91, 85, 70, 33, 9, 1, 0, 0, 6, 10, 45, 64, 140, 146, 155, 96, 42, 10

Offset: 0

Henry Bottomley, May 17 2001

			From _Sean A. Irvine_, Mar 14 2023: (Start)
Table begins:
  0 1 0  1  0   1   0   1
  0 1 1  2  2   3   3   4
  0 1 2  3  4   5   6   7
  0 1 3  6 10  15  21  28
  0 1 4  9 16  25  36  49
  0 1 5 14 30  55  91 140
  0 1 6 19 44  85 146 231
  0 1 7 26 70 155 301 532
(End)

Rows are A000035, A004526, A001477, A000217, A000290, A000330, A005900, A006325, A014820, A061927. Columns are A000004, A000012, A001477, A061925.

T(0, 2*k) = 0, T(0, 2*k+1) = 1, T(n, 0) = 0, T(2*n, k) = T(2*n-1, k-1) + T(2*n-1, k), T(2*n+1, k) = T(2*n+1, k-1) + T(2*n, k). - Sean A. Irvine, Mar 14 2023

A070212 Number of 5 X 5 pandiagonal magic squares with sum n.

Original entry on oeis.org

1, 10, 55, 220, 715, 2001, 4995, 11385, 24090, 47905, 90376, 162955, 282490, 473110, 768570, 1215126, 1875015, 2830620, 4189405, 6089710, 8707501, 12264175, 17035525, 23361975, 31660200, 42436251, 56300310, 73983205, 96354820, 124444540, 159463876, 202831420, 256200285, 321488190

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 07 2002

In contrast to other definitions, a magic square may contain here any nonnegative integers, not necessarily distinct. For example, the 10 solutions for n = 1 are the 10 permutation matrices of size 5 X 5 which are pandiagonal in the sense that any of the 10 (principal or broken) diagonals has exactly one 1 and four 0's. - M. F. Hasler, Oct 23 2018

Muniru A Asiru, Table of n, a(n) for n = 0..5000
M. Ahmed, J. De Loera, and R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares, arXiv:math/0201108 [math.CO], 2002.
Maya Ahmed, Jesús De Loera and Raymond Hemmecke, Polyhedral cones of magic cubes and squares, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 25-41.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Index entries related to magic squares.

Cf. A027567, A014820, A111158, A053494.

a:=[1, 10, 55, 220, 715, 2001, 4995, 11385, 24090];;  for n in [10..36] do a[n]:=9*a[n-1]-36*a[n-2]+84*a[n-3]-126*a[n-4]+126*a[n-5]-84*a[n-6]+36*a[n-7]-9*a[n-8]+a[n-9]; od; a; # Muniru A Asiru, Oct 23 2018

seq(coeff(series(-(x^4+x^3+x^2+x+1)/(x-1)^9,x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 23 2018

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,10,55,220,715,2001,4995,11385,24090},40] (* Harvey P. Dale, Mar 13 2018 *)

apply( A070212(n)=1/8064*(n+4)*(n+3)*(n+2)*(n+1)*(n^2+5*n+8)*(n^2+5*n+42), [0..20]) \\ Edited by M. F. Hasler, Oct 23 2018

a(n) = (1/8064) * (n+4)*(n+3)*(n+2)*(n+1)*(n^2+5n+8)*(n^2+5n+42).

G.f.: -(x^4+x^3+x^2+x+1) / (x-1)^9. [Colin Barker, Dec 10 2012]

More terms from Benoit Cloitre, May 12 2002

More terms from M. F. Hasler, Oct 23 2018

A269237 a(n) = (n + 1)^2(5n^2 + 10*n + 2)/2.

Original entry on oeis.org

1, 34, 189, 616, 1525, 3186, 5929, 10144, 16281, 24850, 36421, 51624, 71149, 95746, 126225, 163456, 208369, 261954, 325261, 399400, 485541, 584914, 698809, 828576, 975625, 1141426, 1327509, 1535464, 1766941, 2023650, 2307361, 2619904, 2963169, 3339106, 3749725, 4197096

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

Partial sums of centered dodecahedral numbers (A005904).

OEIS Wiki, Centered Platonic numbers
Eric Weisstein's World of Mathematics, Platonic Solid
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A005904, A006566, A008354, A014820, A037270, A132366.

[(n + 1)^2*(5*n^2 + 10*n + 2)/2 : n in [0..50]]; // Wesley Ivan Hurt, Oct 15 2017

A269237:=n->(n + 1)^2*(5*n^2 + 10*n + 2)/2: seq(A269237(n), n=0..50); # Wesley Ivan Hurt, Oct 15 2017

Table[(n + 1)^2 ((5 n^2 + 10 n + 2)/2), {n, 0, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 34, 189, 616, 1525}, 36]

x='x+O('x^99); Vec((1+29*x+29*x^2+x^3)/(1-x)^5) \\ Altug Alkan, Apr 10 2016

G.f.: (1 + 29*x + 29*x^2 + x^3)/(1 - x)^5.
E.g.f.: exp(x)*(2 + 66*x + 122*x^2 + 50*x^3 + 5*x^4)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
Sum_{n>=0} 1/a(n) = (5 - Pi^2 - sqrt(15)*Pi*cot(sqrt(3/5)*Pi))/9 = 1.0377796966... . - Vaclav Kotesovec, Apr 10 2016

A300624 Figurate numbers based on the 11-dimensional regular convex polytope called the 11-dimensional cross-polytope, or 11-dimensional hyperoctahedron.

Original entry on oeis.org

0, 1, 22, 243, 1804, 10165, 46530, 180775, 614680, 1871145, 5188590, 13286043, 31760676, 71513949, 152784282, 311603535, 609802800, 1150082385, 2098144710, 3714481475, 6399123260, 10753517061, 17664712562, 28418229623, 44847366984, 69528316025, 106032285086

Offset: 0

Alejandro J. Becerra Jr., Aug 14 2018

The 11-dimensional cross-polytope is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 3, 3, 3, 3, 4}. It is the dual of the 11-dimensional hypercube.

Georg Fischer, Table of n, a(n) for n = 0..100 (first 61 terms from Alejandro J. Becerra Jr.)
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Similar sequences: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193 (m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).

[(n*(14175 + 83754*n^2 + 50270*n^4 + 7392*n^6 + 330*n^8 + 4*n^10)) / 155925 : n in [0..40]]; // Wesley Ivan Hurt, Jul 17 2020

concat(0, Vec(x*(1 + x)^10 / (1 - x)^12 + O(x^40))) \\ Colin Barker, Aug 15 2018

a(n) = (n*(14175 + 83754*n^2 + 50270*n^4 + 7392*n^6 + 330*n^8 + 4*n^10)) / 155925 \\ Colin Barker, Aug 15 2018

a(n) = 11-crosspolytope(n).
From Colin Barker, Aug 15 2018: (Start)
G.f.: x*(1 + x)^10 / (1 - x)^12.
a(n) = (n*(14175 + 83754*n^2 + 50270*n^4 + 7392*n^6 + 330*n^8 + 4*n^10)) / 155925.
(End)

A354343 Number of distinct sums of n complex 6th power roots of unity.

Original entry on oeis.org

1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487, 6769, 7057, 7351, 7651, 7957

Offset: 0

Max Alekseyev, Aug 15 2022

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Same as A003215 except for a(1) = 6.
Row 6 of A299807.
Cf. A000012, A000027, A000217, A000290, A000332, A000579, A014820, A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008, A299754.

LinearRecurrence[{3,-3,1},{1,6,19,37,61},60] (* Harvey P. Dale, Nov 03 2024 *)

a(n)=if(n==1, 6, 3*n*(n+1)+1) \\ Charles R Greathouse IV, Aug 15 2022

For n >= 2, a(n) = 3*n^2 + 3*n + 1 = A003215(n).
For n >= 5, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f. (1 + 3*x + 4*x^2 - 3*x^3 + x^4) / (1 - x)^3.

A128089 Denominators in inverse of triangle A128078 by rows, n * each term in n-th row of A126615.

Original entry on oeis.org

1, 4, 4, 6, 18, 9, 8, 24, 48, 16, 10, 30, 60, 100, 25, 12, 36, 72, 120, 180, 36, 14, 42, 84, 140, 210, 294, 49, 16, 48, 96, 160, 240, 336, 448, 64, 18, 54, 108, 180, 270, 378, 504, 648, 81, 20, 60, 120, 200, 300, 420, 560, 720, 900, 100

Offset: 1

Gary W. Adamson, Feb 14 2007

Row sums = A014820: (1, 8, 33, 96, 225, 456, ...).
Denominators of the inverse of A128078: (1/1; 1/4, 1/4; 1/6, 1/18, 1/9; 1/8, 1/24, 1/48, 1/16; ...).
Row sums of this triangle: 1/1, 1/2, 1/3, ...; e.g., (1/8 + 1/24 + 1/48 + 1/16) = 1/4.

			First few rows of the triangle:
   1;
   4,  4;
   6, 18,  9;
   8, 24, 48,  16;
  10, 30, 60, 100,  25;
  12, 36, 72, 120, 180,  36;
  14, 42, 84, 140, 210, 294,  49;
  16, 48, 96, 160, 240, 336, 448,  64;
  ...
Row 4 = (8, 24, 48, 16) = 4 * (2, 6, 12, 4); where (2, 6, 12, 4) = row 4 of A126615.

Cf. A128078, A014820, A006527, A126615, A002260, A128064.

Denominators in inverse triangular matrix of A128078, where A128078 = A002260 * A128064, = (1; -1, 4; -1, -2, 9; -1, -2, -3, 16; ...).

A267218 a(n) is the a(n-1)-st a(n-2)-dimensional orthoplex number, starting with the terms 1, 2.

Original entry on oeis.org

1, 2, 2, 4, 16, 22016

Offset: 1

Robin Powell, Jan 18 2016

nonn

			16 is the 4th 2-orthoplex number = A000290(4).
22016 is the 16th 4-orthoplex number = A014820(16).
The next term will be the 22016th 16-orthoplex number.

OEIS Wiki, Orthoplex numbers

Cf. A000290, A014820, A260161, A051285.

A364429 a(0) = 1, a(n) = (2n^5 + 20n^3 + 23n) 2/15 for n>=1.

Original entry on oeis.org

1, 6, 36, 146, 456, 1182, 2668, 5418, 10128, 17718, 29364, 46530, 71000, 104910, 150780, 211546, 290592, 391782, 519492, 678642, 874728, 1113854, 1402764, 1748874, 2160304, 2645910, 3215316, 3878946, 4648056, 5534766, 6552092, 7713978, 9035328, 10532038, 12221028

Offset: 0

Steven Lu, Jul 24 2023

a(n) is the 6th n-orthoplex (n-dimensional cross-polytope) number.

			a(3) = 146 since the 6th octahedral number is 146; A005900(6) = 146.
a(4) = 456 since the 6th 16-cell number is 456; A014820(5) = 456.

OEIS Wiki, Orthoplex numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005900, A014820, A069038, A069039, A099193, A099195, A099196, A099197, A058331.

Cf. A142978 (column 6 with an initial 1).

Prepend[Table[2/15 (2 x^5 + 20 x^3 + 23 x), {x, 100}], 1]

print([1]+[(2*i**5+20*i**3+23*i)*2//15 for i in range(1,101)])

a(0) = 1, a(n) = (2*n^5 + 20*n^3 + 23*n) * 2/15 for n>=1.

G.f.: (1 + 15*x^2 + 15*x^4 + x^6)/(1 - x)^6. - Stefano Spezia, Jul 24 2023

cp's OEIS Frontend

A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

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A061926 Square table by antidiagonals where odd rows are partial sums of previous row, even rows are sums of pairs of values in previous row and initial row is 0 and 1 alternating.

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A070212 Number of 5 X 5 pandiagonal magic squares with sum n.

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A269237 a(n) = (n + 1)^2*(5*n^2 + 10*n + 2)/2.

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A300624 Figurate numbers based on the 11-dimensional regular convex polytope called the 11-dimensional cross-polytope, or 11-dimensional hyperoctahedron.

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Magma

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A354343 Number of distinct sums of n complex 6th power roots of unity.

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Mathematica

PARI

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A128089 Denominators in inverse of triangle A128078 by rows, n * each term in n-th row of A126615.

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A267218 a(n) is the a(n-1)-st a(n-2)-dimensional orthoplex number, starting with the terms 1, 2.

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A269237 a(n) = (n + 1)^2(5n^2 + 10*n + 2)/2.

A364429 a(0) = 1, a(n) = (2n^5 + 20n^3 + 23n) 2/15 for n>=1.