A027480 -id:A027480 - cp's OEIS frontend

A275740 Sums of the next n consecutive nonsquare integers.

0, 2, 8, 21, 46, 83, 136, 210, 306, 426, 575, 758, 972, 1223, 1519, 1855, 2236, 2669, 3156, 3694, 4290, 4956, 5678, 6467, 7332, 8269, 9278, 10368, 11548, 12804, 14148, 15593, 17126, 18753, 20485, 22325, 24262, 26308, 28481, 30756, 33148

Offset: 0

Olivier Gérard, Aug 07 2016

Row sums of nonsquare integers (A000037), seen as a regular triangle:
.
2   | 2,
8   | 3,  5,
21  | 6,  7,  8,
46  | 10, 11, 12, 13,
83  | 14, 15, 17, 18, 19,
136 | 20, 21, 22, 23, 24, 26,
210 | 27, 28, 29, 30, 31, 32, 33,
306 | 34, 35, 37, 38, 39, 40, 41, 42,
...
The equivalent for all integers are A006003 (starting from 1), A229183 (starting from 2) and A027480 (starting from 0).
There are several sequences close to nonsquares whose sum of groups of n terms starts like this sequence, notably A028761, A158276, A167759.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000037, A006003, A229183, A027480.

R:= 0: s:= 1:
for n from 1 to 100 do
  if floor(sqrt(s+n)) = floor(sqrt(s)) then
    R:= R, n*s + n*(n+1)/2; s:= s+n;
  else
    R:= R, n*s + n*(n+1)/2 - floor(sqrt(s+n))^2 + s+n+1; s:= s+n+1;
  fi
od:
R; # Robert Israel, Oct 02 2022

Table[Sum[
  i + Floor[1/2 + Sqrt[i]], {i, n (n - 1)/2 + 1, (n + 1) (n)/2}], {n,
  0, 40}]
Join[{0},Module[{nn=1000,nsi,len},nsi=Select[Range[nn],!IntegerQ[Sqrt[#]]&];len=Floor[ (Sqrt[ 8*Length[nsi]+1]-1)/2];Total/@TakeList[nsi,Range[len]]]] (* Harvey P. Dale, Jan 04 2024 *)

Definition clarified by Harvey P. Dale, Jan 04 2024

A294033 Triangle read by rows, expansion of exp(xz)z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 2, -3, 6, 3, -8, -12, 12, 4, 25, -40, -30, 20, 5, 96, 150, -120, -60, 30, 6, -427, 672, 525, -280, -105, 42, 7, -2176, -3416, 2688, 1400, -560, -168, 56, 8, 12465, -19584, -15372, 8064, 3150, -1008, -252, 72, 9, 79360, 124650, -97920, -51240, 20160, 6300, -1680, -360, 90, 10, -555731, 872960, 685575, -359040, -140910, 44352, 11550, -2640, -495, 110, 11

Offset: 1

Peter Luschny, Oct 24 2017

			Triangle starts:
  [1][   1]
  [2][   2,   2]
  [3][  -3,   6,    3]
  [4][  -8, -12,   12,    4]
  [5][  25, -40,  -30,   20,    5]
  [6][  96, 150, -120,  -60,   30,  6]
  [7][-427, 672,  525, -280, -105, 42, 7]

T(n, 0) = signed A065619. Row sums of abs(T(n,k)) = A231179.
Diagonals A000027, A002378, A027480, A162668.
A003506 (m=1), this seq. (m=2), A294034 (m=3).
Cf. A000111, A247453.

gf := exp(x*z)*z*(tanh(z)+sech(z)):
s := n -> n!*coeff(series(gf,z,n+2),z,n):
C := n -> PolynomialTools:-CoefficientList(s(n),x):
ListTools:-FlattenOnce([seq(C(n), n=1..7)]);
# Alternatively:
T := (n, k) -> `if`(n = k+1, n,
(k+1)*binomial(n,k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):
for n from 1 to 7 do seq(T(n,k), k=0..n-1) od;

L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];
p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];
Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten

T(n, k) = (k+1)*binomial(n,k+1)*2^(n-k-1)*(Euler(n-k-1, 1/2) + Euler(n-k-1, 1)) for 0 <= k <= n-2.

T(n, k) is the coefficient of x^k of the polynomial p(n) = n*Sum_{k=1..n} binomial(n-1, k-1)*L(k-1)*x^(n-k) and L(n) = (-1)^binomial(n,2)*A000111(n). In particular n divides T(n, k).

A373288 T(n, k) is the total number of symmetric peaks in all partitions of n with exactly k blocks, n >= 3, 2 <= k <= n-1.

Original entry on oeis.org

1, 3, 2, 8, 12, 3, 20, 54, 30, 4, 48, 215, 205, 60, 5, 112, 799, 1185, 580, 105, 6, 256, 2842, 6230, 4585, 1365, 168, 7, 576, 9812, 30828, 32256, 14140, 2828, 252, 8, 1280, 33165, 146355, 210378, 128037, 37170, 5334, 360, 9, 2816, 110361, 674535, 1301860, 1060815, 420756, 86730, 9360, 495, 10

Offset: 3

W. Asakly, Jun 01 2024

			The triangle T(n, k) begins:
   3|   1
   4|   3      2
   5|   8     12      3
   6|  20     54     30      4
   7|  48    215    205     60      5
   8| 112    799   1185    580    105      6
   9| 256   2842   6230   4585   1365    168      7
  10| 576   9812  30828  32256  14140   2828    252      8
.
T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:
The canonical form of a set partition of [n] is an n-tuple indicating the block in which each integer occurs. The symmetric peaks in the canonical sequential form are listed:
  (1, 2, 1, 1, 3) -> 1 symmetric peak   (1, 2, 1)
  (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)
  (1, 2, 1, 2, 3) -> 1 symmetric peak,  (1, 2, 1)
  (1, 2, 1, 3, 2) -> 1 symmetric peak,  (1, 2, 1)
  (1, 2, 1, 3, 3) -> 1 symmetric peak,  (1, 2, 1)
  (1, 2, 1, 3, 1) -> 2 symmetric peaks, (1, 2, 1) and (1, 3, 1)
  (1, 2, 2, 3, 2) -> 1 symmetric peak,  (2, 3, 2)
  (1, 2, 3, 2, 1) -> 1 symmetric peak,  (2, 3, 2)
  (1, 2, 3, 2, 2) -> 1 symmetric peak,  (2, 3, 2)
  (1, 2, 3, 2, 3) -> 1 symmetric peak,  (2, 3, 2).

W. Asakly and Noor Kezil, Counting symmetric and non-symmetric peaks in a set partition, arXiv:2401.01687 [math.CO], 2024.

Cf. A008277 (Stirling2).

Cf. A001792 (1st column), A027480 (subdiagonal).

T := (n, k) -> (k-1) * Stirling2(n-1, k) + add(binomial(j, 2) * add(j^(i-3) * Stirling2(n-i, k),i=3..n-k), j = 2..k): seq(print(seq(T(n, k), k = 2..n-1)), n = 3..10);  # Peter Luschny, Jun 06 2024

T[n_, k_] := (k-1) * StirlingS2[n-1, k] + Sum[Binomial[j, 2] * Sum[j^(i-3) * StirlingS2[n-i, k], {i, 3, n-k}], {j, 2, k}];
Table[T[n, k], {n, 3, 12}, {k, 2, n-1}] // Flatten

T(n, k) = (k-1) * stirling(n-1, k, 2) + sum(j=2, k, binomial(j, 2) * sum(i=3, n-k, j^(i-3) * stirling(n-i, k, 2))); \\ Michel Marcus, Jun 06 2024

T(n,k) = (k-1) * Stirling2(n-1, k) + Sum_{j=2..k} binomial(j, 2) * Sum_{i=3..n-k} j^(i-3) * Stirling2(n-i, k).

A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

0, 3, 12, 40, 98, 253, 540, 1199, 2415, 4893, 9268, 17864, 32421, 59265, 104632, 184338, 315414, 540155, 901845, 1504173, 2461932, 4013511, 6443170, 10314675, 16281749, 25608450, 39838855, 61716941, 94682665, 144726102

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

			Illustration of the first few terms
.
0   | 0
3   | 1,  2
12  | 3,  4,  5
40  | 6,  7,  8,  9,  10
98  | 11, 12, 13, 14, 15, 16, 17
253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
.

Cf. A373300, original version, with positive integers A000027.
Cf. A001477, the nonnegative integers.
Cf. A027480, the sequence of row sums for a regular triangle.

Module[{s = -1},
 Table[s +=
   PartitionsP[
    n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
   s (s - 1)/2, {n, 1, 30}]]

A021449 Decimal expansion of 1/445.

Original entry on oeis.org

0, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1, 1, 2, 3, 5, 9, 5, 5, 0, 5, 6, 1, 7, 9, 7, 7, 5, 2, 8, 0, 8, 9, 8, 8, 7, 6, 4, 0, 4, 4, 9, 4, 3, 8, 2, 0, 2, 2, 4, 7, 1, 9, 1, 0, 1

Offset: 0

N. J. A. Sloane

Expansion in any base b >= 3 of 2/(b(b-1)(b+1)) = 2/(b^3-b). E.g., 1/12 in base 3, 1/30 in base 4, 1/60 in base 5, etc. - Franklin T. Adams-Watters, Nov 07 2006

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

Cf. A027480, A021093.

Join[{0,0},RealDigits[1/445,10,120][[1]]] (* Harvey P. Dale, Aug 25 2015 *)

1/445. \\ Charles R Greathouse IV, Dec 05 2011

From Chai Wah Wu, May 08 2025: (Start)
a(n) = a(n-1) - a(n-22) + a(n-23) for n > 23.
G.f.: x^2*(-2*x^21 - 6*x^20 + 5*x^19 - x^18 - 5*x^17 + 5*x^16 + 4*x^14 - 4*x^13 - 2*x^12 - x^11 - x^10 - x^8 + x^7 + 8*x^6 - 8*x^5 + 6*x^4 - 3*x^3 - 2*x^2 - 2)/(x^23 - x^22 + x - 1). (End)

A180413 Total number of possible knight moves on an n X n X n chessboard, if the knight is placed anywhere.

Original entry on oeis.org

0, 144, 576, 1440, 2880, 5040, 8064, 12096, 17280, 23760, 31680, 41184, 52416, 65520, 80640, 97920, 117504, 139536, 164160, 191520, 221760, 255024, 291456, 331200, 374400, 421200, 471744, 526176, 584640, 647280, 714240, 785664, 861696

Offset: 1

Graziano Aglietti (mg5055(AT)mclink.it), Sep 02 2010

The maximum number of move in tridimensional chessboard is 24, 8 for every dimension. In a vertex the number is smaller.

Binomial transform of [144, 432, 432, 144, 0, 0, 0, ...] = (144, 576, 1440, ...). - Gary W. Adamson, Sep 03 2010

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027480, A135503, A180319, A035008.

Table[24n(n^2-1),{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,144,576,1440},40] (* Harvey P. Dale, Feb 13 2013 *)

a(n)=24*n*(n^2-1) \\ Charles R Greathouse IV, Nov 03 2014

a(n) = 24*n*(n^2-1).
G.f.: 144*x^2/(1-x)^4. - Colin Barker, Mar 17 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=0, a(2)=144, a(3)=576, a(4)=1440. - Harvey P. Dale, Feb 13 2013
E.g.f.: 24 * exp(x) * x^2 * (3 + x). - Vaclav Kotesovec, Feb 15 2015

A304391 Numbers that are the number of permutations of a set of balls of at least three distinct colors.

Original entry on oeis.org

6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 105, 110, 120, 132, 140, 156, 168, 180, 182, 210, 240, 252, 272, 280, 306, 336, 342, 360, 380, 420, 462, 495, 504, 506, 552, 560, 600, 630, 650, 660, 702, 720, 756, 812, 840, 858, 870, 930, 990, 992, 1056, 1092, 1120, 1122, 1190

Offset: 1

David A. Corneth, May 24 2018

			There are 30 permutations of 5 balls where 2 are red, 2 are yellow and 1 is blue.

A002378 is a subsequence.

Cf. A027480, A304938.

A027480(k) = k * (k + 1) * (k + 2) / 2 is in the sequence for k >= 2.

A321697 T(j,k) = binomial(j^k,k)/j, j <= m, k <= j, written as triangle T(j,k).

Original entry on oeis.org

1, 1, 3, 1, 12, 975, 1, 30, 10416, 43698160, 1, 60, 63550, 1259394500, 495117695312625, 1, 105, 276060, 19500470010, 39435754026361680, 2386830808433862941972976, 1, 168, 952413, 197321108600, 1595560551370855083, 526069994452248286902543528, 7282228632205891036170867431546711227

Offset: 1

Hugo Pfoertner, Nov 17 2018

Cf. A027480.

f[n_] := Table[Binomial[j^k,k]/j, {j, n, n}, {k,1, j}] ; Flatten[Array[f, 11]] (* Stefano Spezia, Nov 17 2018 *)

for(j=1,7,for(k=1,j,print1(binomial(j^k,k)/j,", ")))

A368608 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x != y and y <= z.

Original entry on oeis.org

2, 1, 4, 5, 2, 1, 6, 9, 8, 4, 2, 1, 8, 13, 14, 12, 6, 4, 2, 1, 10, 17, 20, 20, 16, 9, 6, 4, 2, 1, 12, 21, 26, 28, 26, 21, 12, 9, 6, 4, 2, 1, 14, 25, 32, 36, 36, 33, 26, 16, 12, 9, 6, 4, 2, 1, 16, 29, 38, 44, 46, 45, 40, 32, 20, 16, 12, 9, 6, 4, 2, 1, 18, 33

Offset: 1

Clark Kimberling, Jan 25 2024

Row n consists of 2n positive integers.

			First six rows:
   2   1
   4   5   2   1
   6   9   8   4   2   1
   8  13  14  12   6   4   2  1
  10  17  20  20  16   9   6  4  2  1
  12  21  26  28  26  21  12  9  6  4  2  1
For n=2, there are 3 triples (x,y,z) having x != y and y <= z:
  122:  |x-y| + |y-z| = 1
  211:  |x-y| + |y-z| = 1
  212:  |x-y| + |y-z| = 2
so row 2 of the array is (2,1), representing two 1s and one 2.

Cf. A005443 (column 1), A027480 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368607, A368609.

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] <= #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
v = Flatten[u]  (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]]  (* array *)

cp's OEIS Frontend

A275740 Sums of the next n consecutive nonsquare integers.

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A294033 Triangle read by rows, expansion of exp(x*z)*z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

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A373288 T(n, k) is the total number of symmetric peaks in all partitions of n with exactly k blocks, n >= 3, 2 <= k <= n-1.

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A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

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A021449 Decimal expansion of 1/445.

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A180413 Total number of possible knight moves on an n X n X n chessboard, if the knight is placed anywhere.

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A304391 Numbers that are the number of permutations of a set of balls of at least three distinct colors.

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A321697 T(j,k) = binomial(j^k,k)/j, j <= m, k <= j, written as triangle T(j,k).

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A294033 Triangle read by rows, expansion of exp(xz)z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.