A034262 -id:A034262 - cp's OEIS frontend

A216155 Numbers n such that floor(sqrt(n + n^3)) = 1 + floor(sqrt(n^3)) = 1 + A000093(n).

2, 13, 40, 43, 46, 52, 109, 152, 190, 243, 336, 351, 356, 366, 422, 584, 592, 741, 937, 978, 1011, 1040, 1137, 1330, 1355, 1362, 1376, 1398, 1434, 2063, 2320, 2520, 2553, 2660, 2665, 2928, 2940, 2993, 3067, 3075, 3092, 3296, 3532, 3631, 3703, 3712, 3730

Offset: 1

Zak Seidov, Sep 02 2012

nonn

The sequence is infinite. For values of n not in the sequence we have floor(sqrt(n+n^3)) = floor(sqrt(n^3)) = A000093(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000093 (floor(n^(3/2))).

Cf. A000196, A000578, A034262, A247628 (subsequence).

a216155 n = a216155_list !! (n-1)
a216155_list = filter
   (\x -> a000196 (a034262 x) == a000196 (a000578 x) + 1) [1..]
-- Reinhard Zumkeller, Sep 26 2014

Select[Range[10000], Floor[Sqrt[# + #^3]] - Floor[Sqrt[#^3]] == 1 &]

A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 1, 3, 3, 1, 0, 0, 2, 4, 4, 4, 1, 0, 1, 2, 5, 9, 5, 5, 1, 0, 0, 3, 6, 10, 16, 6, 6, 1, 0, 1, 1, 7, 12, 17, 25, 7, 7, 1, 0, 0, 2, 8, 13, 20, 26, 36, 8, 8, 1, 0, 1, 2, 9, 27, 21, 30, 37, 49, 9, 9, 1, 0, 0, 3, 10, 28, 64, 31, 42, 50, 64, 10, 10, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

For any n >= 0, the n-th row, k -> T(n, k), corresponds to a polynomial in k with coefficients in {0, 1}.

For any k > 1, the k-th column, n -> T(n, k), corresponds to sums of distinct powers of k.

			Array T(n, k) begins:
  n\k|  0  1   2   3   4    5    6    7    8    9    10    11    12
  ---+-------------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0     0     0     0
    1|  1  1   1   1   1    1    1    1    1    1     1     1     1
    2|  0  1   2   3   4    5    6    7    8    9    10    11    12
    3|  1  2   3   4   5    6    7    8    9   10    11    12    13
    4|  0  1   4   9  16   25   36   49   64   81   100   121   144
    5|  1  2   5  10  17   26   37   50   65   82   101   122   145
    6|  0  2   6  12  20   30   42   56   72   90   110   132   156
    7|  1  3   7  13  21   31   43   57   73   91   111   133   157
    8|  0  1   8  27  64  125  216  343  512  729  1000  1331  1728
    9|  1  2   9  28  65  126  217  344  513  730  1001  1332  1729
   10|  0  2  10  30  68  130  222  350  520  738  1010  1342  1740
   11|  1  3  11  31  69  131  223  351  521  739  1011  1343  1741
   12|  0  2  12  36  80  150  252  392  576  810  1100  1452  1872

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Index entries for sequences related to binary expansion of n

See A342707 for a similar sequence.

Cf. A000035, A000120, A000583, A000695, A001093, A002061, A002378, A002522, A002523, A005836, A007088, A011379, A027444, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A034262, A053698, A071568, A098547, A104258.

T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^e); v }

T(n, n) = A104258(n).
T(n, 0) = A000035(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A005836(n).
T(n, 4) = A000695(n).
T(n, 5) = A033042(n).
T(n, 6) = A033043(n).
T(n, 7) = A033044(n).
T(n, 8) = A033045(n).
T(n, 9) = A033046(n).
T(n, 10) = A007088(n).
T(n, 11) = A033047(n).
T(n, 12) = A033048(n).
T(n, 13) = A033049(n).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^2.
T(5, k) = k^2 + 1 = A002522(k).
T(6, k) = k^2 + k = A002378(k).
T(7, k) = k^2 + k + 1 = A002061(k).
T(8, k) = k^3.
T(9, k) = k^3 + 1 = A001093(k).
T(10, k) = k^3 + k = A034262(k).
T(11, k) = k^3 + k + 1 = A071568(k).
T(12, k) = k^3 + k^2 = A011379(k).
T(13, k) = k^3 + k^2 + 1 = A098547(k).
T(14, k) = k^3 + k^2 + k = A027444(k).
T(15, k) = k^3 + k^2 + k + 1 = A053698(k).
T(16, k) = k^4 = A000583(k).
T(17, k) = k^4 + 1 = A002523(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A345285 Sides of primary squares of type 1 (A344331). A primary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b, so that the numbers of small and large squares are equal.

Original entry on oeis.org

10, 30, 68, 78, 130, 160, 222, 290, 300, 350, 480, 510, 520, 738, 742, 810, 820, 1010, 1088, 1218, 1248, 1342, 1530, 1740, 1752, 1820, 1830, 2080, 2210, 2430, 2560, 2590, 2750, 2758, 3270, 3390, 3492, 3552, 3560, 3570, 4112, 4290, 4498, 4640, 4770, 4800, 4930, 5508, 5600, 5850, 6028, 6250

Offset: 1

Bernard Schott, Jun 13 2021

nonn

Notation: s = side of the primary tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
Every term is of the form s = a*b * (a^2+b^2), with 1 <= a < b, and corresponding z = (a*b)^2 * (a^2+b^2) (A345286).
Every such primary square is composed of m = a*b * (a^2+b^2) elementary rectangles of length L = a^2+b^2 and width W = a*b, so with an area A = a*b * (a^2+b^2) = m.
If gcd(a, b) = 1, then primitive sides of square s = a*b * (a^2+b^2) are in A344333 that is a subsequence.
If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.
If q is a term and integer r > 1, then q * r^4 is another term.
Every term is even.

			a(1) = 10 and the primary square 10 X 10 can be tiled with A345286(1) = 20 small squares with side a = 1 and 20 large squares with side b = 2.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|     with 10 elementary 2 X 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
a(6) = 160 is the first side of an primary square that is not primitive and it corresponds to (a,b) = (2,4); the square 160 X 160 can be tiled with A345286(6) = 1280 small squares with side a = 2 and 1280 large squares with side b = 4.

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Index to sequences related to Olympiads.

Cf. A034262, A344330, A344333, A344334, A345286, A345287.

Subsequence of A344331.

A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 2, 22, 30, 20, 5, 1, 0, 4, 34, 93, 68, 30, 6, 1, 0, 2, 78, 246, 276, 130, 42, 7, 1, 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1, 0, 3, 278, 2190, 4180, 3130, 1338, 350, 72, 9, 1

Offset: 0

Peter Luschny, Jun 27 2023

The name expresses the 'row view' of the array. The 'column view' regards the array as the collection of the inverse Möbius transforms of the power sequences k^n = 0^n, 1^n, 2^n, .... (n >= 0). Viewed this way, the array is a generalization of the number of divisors sequence tau (A000005), to which it reduces in the case k = 1.

The array has offset (0, 0). It uses the usual definition of 'k divides n' as described in Apostol, rather than the shortened version, which restricts to values k > 0 as some programs do (but not SageMath). Such a restriction makes sense in the context of rational numbers but not in the case of natural numbers.

			Array A(n, k) starts:
  [0] 1, 1,   1,    1,     1,      1,       1,       1,        1, ... A000012
  [1] 0, 1,   2,    3,     4,      5,       6,       7,        8, ... A001477
  [2] 0, 2,   6,   12,    20,     30,      42,      56,       72, ... A002378
  [3] 0, 2,  10,   30,    68,    130,     222,     350,      520, ... A034262
  [4] 0, 3,  22,   93,   276,    655,    1338,    2457,     4168, ...
  [5] 0, 2,  34,  246,  1028,   3130,    7782,   16814,    32776, ... A131471
  [6] 0, 4,  78,  768,  4180,  15780,   46914,  118048,   262728, ...
  [7] 0, 2, 130, 2190, 16388,  78130,  279942,  823550,  2097160, ... A190578
  [8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...
   A000005,A055895,A363913, ...                             A066108 (diagonal)
.
Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,   1;
  [3] 0, 2,   2,   1;
  [4] 0, 2,   6,   3,    1;
  [5] 0, 3,  10,  12,    4,   1;
  [6] 0, 2,  22,  30,   20,   5,   1;
  [7] 0, 4,  34,  93,   68,  30,   6,  1;
  [8] 0, 2,  78, 246,  276, 130,  42,  7, 1;
  [9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;

Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Cf. A113704 (in compact form A113705), A000005 (column 1), A055895 (column 2), A363913 (column 3), A001477 (row 1), A002378 (row 2), A034262 (row 3), A131471 (row 5), A190578 (row 7), A363912 (row sums), A066108 (main diagonal of array).

Cf. A363734, A363735, A363421.

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):
for n from 0 to 8 do seq(A(n, k), k = 0..8) od;
# If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...
InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):
# ... the transposed array is given by applying InvMoebius to the powers r^m:
seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);
# For instance we see that the number of divisors is the inverse
# Moebius transform of the constant sequence s = 1.

def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))
for n in srange(9): print([A(n, k) for k in (0..8)])

A(n, k) = Sum_{j=0..n} divides(j, n) * k^j, where divides(k, n) <-> [k = n or (k > 0 and n mod k = 0)], and '[ ]' denotes the Iverson bracket.

The columns are the inverse Möbius transforms of the powers x^n, x >= 0.

A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 6, 1, 1, 3, 2, 3, 1, 3, 2, 1, 3, 1, 2, 1, 1, 3, 6, 1, 1, 3, 6, 1, 1, 1, 6, 3, 1, 3, 2, 3, 3, 3, 2, 1, 1, 3, 2, 1, 1, 1, 6, 3, 3, 3, 2, 3, 1, 1, 6, 1, 3, 3, 2, 1, 1, 9, 2, 1, 3, 1, 6, 1, 1, 3, 6, 3, 1, 1, 6, 1, 3, 1, 6, 1, 1, 9, 2, 3, 1, 3, 6, 3, 1, 1, 2, 3, 1, 3, 2, 1, 3, 3, 6, 1, 3, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Shadow transform of A034262. - Michel Marcus, Jun 06 2013

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
N. J. A. Sloane, Transforms.

Cf. A087688, A000089, A034262.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[p == 2, If[e == 1, 2, 1], If[Mod[p, 4] == 1, 3, 1]], {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019 *)

a(n)={my(v=vector(n)); sum(i=0, n-1, lift(Mod(i,n)^3 + i) == 0)} \\ Andrew Howroyd, Jul 15 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, if(e==1, 2, 1), if(p%4==1, 3, 1)))} \\ Andrew Howroyd, Jul 15 2018

Multiplicative with a(2^1) = 2, a(2^e) = 1 for e > 1, a(p^e) = 3 for p mod 4 == 1, a(p^e) = 1 for p mod 4 == 3. - Andrew Howroyd, Jul 15 2018

More terms from David Wasserman, Jun 17 2005

A131473 a(n) = n^6 - n.

Original entry on oeis.org

0, 0, 62, 726, 4092, 15620, 46650, 117642, 262136, 531432, 999990, 1771550, 2985972, 4826796, 7529522, 11390610, 16777200, 24137552, 34012206, 47045862, 63999980, 85766100, 113379882, 148035866, 191102952, 244140600, 308915750

Offset: 0

Mohammad K. Azarian, Jul 27 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6-n: n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A131473:=n->n^6-n; seq(A131473(n), n=0..30); # Wesley Ivan Hurt, Feb 25 2014

f[n_]:=n^6-n;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
Array[#^6-#&,60,0] (* Harvey P. Dale, Aug 10 2011 *)

[lucas_number1(3,n^3,n) for n in range(0, 27)] # Zerinvary Lajos, May 16 2009

A180499 n^3 + n-th cubefree number.

Original entry on oeis.org

2, 10, 30, 68, 130, 222, 350, 521, 739, 1011, 1343, 1741, 2211, 2759, 3392, 4114, 4932, 5852, 6880, 8022, 9284, 10673, 12193, 13852, 15654, 17606, 19714, 21985, 24423, 27035, 29827, 32805, 35975, 39343

Offset: 1

Jonathan Vos Post, Jan 20 2011

First differs from n^3 + n (A034262) at n=8 because 8 is the first positive integer which is not cubefree.
What cubes appear in this sequence?
No cubes appear in this sequence: the n-th cubefree number is asymptotically zeta(3)*n, putting members of this sequence strictly between n^3 and (n+1)^3. (Lacking effective error bounds this actually only shows that there are finitely many.) - Charles R Greathouse IV, Jan 21 2011

			a(8) = 8^3 + 8th number that is not divisible by any cube > 1 = 8^3 + 9 = 521.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000578, A004709, A034262, A161203.

#[[1]]+#[[2]]^3&/@Module[{cf=Select[Range[50],Max[FactorInteger[#][[All,2]]] < 3&]},Thread[{cf,Range[Length[cf]]}]] (* Harvey P. Dale, Jun 28 2020 *)

from sympy import mobius, integer_nthroot
def A180499(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return n**3+m # Chai Wah Wu, Aug 12 2024

a(n) = n^3 + A004709(n) = A000578(n) + A004709(n).

A181427 a(n) = n + [n^2 if n is odd or n^3 if n is even].

Original entry on oeis.org

2, 10, 12, 68, 30, 222, 56, 520, 90, 1010, 132, 1740, 182, 2758, 240, 4112, 306, 5850, 380, 8020, 462, 10670, 552, 13848, 650, 17602, 756, 21980, 870, 27030, 992, 32800, 1122, 39338, 1260, 46692, 1406, 54910, 1560, 64040, 1722, 74130, 1892, 85228, 2070

Offset: 1

Dinesh Panchamia (dgpanchamia(AT)gmail.com), Oct 19 2010

a(2*k+1) = 2*A000384(k+1) (k in A001477). - Bruno Berselli, Oct 20 2010

			For n=5, 5+5^2=30 and n=6 6+6^3=222.

B. Berselli, Table of n, a(n) for n = 1..10000. [From _Bruno Berselli_, Oct 20 2010]
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A002939, A034262, A065679.

Cf. A000384, A001477.

If[OddQ[ # ],#+#^2,#+#^3]&/@Range[50] (* Harvey P. Dale, Nov 03 2010 *)

a(n) = n + n^(2*(n mod 2)+3*(1-(n mod 2))).
a(n) = n + n^((5+(-1)^n)/2) = n*(1+A065679(n)).
G.f.: 2*x*(1+5*x+2x^2+14*x^3-3*x^4+5*x^5)/(1-x^2)^4.
a(n)-4*a(n-2)+6*a(n-4)-4*a(n-6)+a(n-8) = 0 for n>8.
a(2*n) = A034262(2*n). a(2*n+1) = A002939(n+1).

Formulas and more terms from R. J. Mathar and Bruno Berselli, Oct 19 2010

A246654 T(n,k) = 2(K(n,2)I(k,2) - (-1)^(n+k)I(n,2)K(k,2)), where I(n,x) and K(n,x) are Bessel functions; triangle read by rows for 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 10, 7, 3, 1, 0, 43, 30, 13, 4, 1, 0, 225, 157, 68, 21, 5, 1, 0, 1393, 972, 421, 130, 31, 6, 1, 0, 9976, 6961, 3015, 931, 222, 43, 7, 1, 0, 81201, 56660, 24541, 7578, 1807, 350, 57, 8, 1, 0, 740785, 516901, 223884, 69133, 16485, 3193, 520, 73, 9, 1, 0

Offset: 0

Peter Luschny, Sep 12 2014

			T(n, k) as a rectangular matrix (for n >= 0). Only the lower infinite triangle (0 <= k <=n) constitutes the sequence although T(n,k) is defined for all (n,k) in Z^2.
[   0,    1,   -1,   3, -10,  43, -225, 1393, -9976]
[   1,    0,    1,  -2,   7, -30,  157, -972,  6961]
[   1,    1,    0,   1,  -3,  13,  -68,  421, -3015]
[   3,    2,    1,   0,   1,  -4,   21, -130,   931]
[  10,    7,    3,   1,   0,   1,   -5,   31,  -222]
[  43,   30,   13,   4,   1,   0,    1,   -6,    43]
[ 225,  157,   68,  21,   5,   1,    0,    1,    -7]
[1393,  972,  421, 130,  31,   6,    1,    0,     1]
[9976, 6961, 3015, 931, 222,  43,    7,    1,     0]
The diagonals d(n,k) = T(n+k-floor(n/2),k-floor(n/2)) are represented by polynomials described in A246656.
   n\k:    0   1    2     3    4     p_n(x)
-------------------------------------------------------
d(0,k):    0,  0,   0,    0,   0, .. 0                   A000004
d(1,k):    1,  1,   1,    1,   1, .. 1                   A000012
d(2,k):  [0],  1,   2,    3,   4, .. x                   A001477
d(3,k):  [1],  3,   7,   13,  21, .. x^2+x+1             A002061
d(4,k):  [0,  2],  10,   30,  68, .. x^3+x               A034262
d(5,k):  [1,  7],  43,  157, 421, .. x^4+2*x^3+2*x^2+x+1

T(n+0,0) = A001040(n).
T(n+1,1) = A001053(n+1).
T(n+2,2) = A058307(n).
T(n+3,3) = A058308(n).
T(n+4,4) = A058309(n).
Cf. A001040, A001053, A001477, A002061, A034262, A058307, A058308, A058309, A246656.

T := (n, k) -> (BesselK(n,2)*BesselI(k,2) - (-1)^(n+k)*BesselI(n,2) *BesselK(k,2))*2; seq(lprint(seq(round(evalf(T(n,k),99)), k=0..n)), n=0..8);
# Recurrence
T := proc(n,k) option remember; local m; m := n-1;
if  k > m or k < 0 then 0 elif k = m then 1 else T(m-1,k) + m*T(m,k) fi end:
seq(print(seq(T(n,k), k=0..n)), n=0..8);

T[n_, k_] := T[n, k] = With[{m = n - 1}, If[k > m || k < 0, 0, If[k == m, 1, T[m - 1, k] + m*T[m, k]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2019 *)

def A246654_col(n, k): # k-th column of the triangle
    if n < 2: return n
    return hypergeometric([(1-n)/2, 1-n/2], [1-n, 1+k, 1-n-k], 4) *rising_factorial(k+1,n-1)
for k in range(6): [round(A246654_col(n,k).n(100)) for n in (0..10)]

T(n+k,k) = hypergeom([(1-n)/2, 1-n/2], [1-n, 1+k, 1-n-k], 4)* Pochhammer(k+1, n-1).
Recurrence: T(n,k) = T(n-2,k)+(n-1)*T(n-1,k), T(n,n)=0, T(n,n-1)=1.
T(n,k) = T(n,-k) = T(-n,k) = T(-n,-k).

A246656 Triangle read by rows: T(n, k) is the coefficient of x^k of the polynomial p_n(x) representing the n-th diagonal of A246654.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 1, 0, 0, 3, 0, -1, 0, 1, 0, 1, 8, 5, -5, 0, 3, 1, 0, 0, -18, 0, 29, 0, -8, 0, 1, 0, 1, -80, -13, 121, 29, -35, -7, 4, 1, 0, 0, 357, 0, -513, 0, 182, 0, -22, 0, 1, 0, 1, 1865, 344, -2686, -484, 945, 175, -114, -21, 5, 1, 0

Offset: 0

Peter Luschny, Sep 13 2014

			The first few polynomials and their coefficients:
             0;               0;
            1, 0;             1;
          0, 1, 0;            x;
         1, 1, 1, 0;          x*(x+1)+1;
       0, 1, 0, 1, 0;         x*(x^2+1);
      1, 1, 2, 2, 1, 0;       x*(x+1)*(x^2+x+1)+1;
    0, 3, 0, -1, 0, 1, 0;     x*(x^4-x^2+3);
  1, 8, 5, -5, 0, 3, 1, 0;    x*(x+1)*(x^4+2*x^3-2*x^2-3*x+8)+1;
0,-18, 0, 29, 0, -8, 0, 1,0;  x*(x^6-8*x^4+29*x^2-18);
The values of some polynomials:
------------------------------------------------
     n:    -4    -3   -2  -1   0   1    2     3
------------------------------------------------
p_0(n):     0,    0,   0,  0,  0,  0,   0,    0,   A000004
p_1(n):     1,    1,   1,  1,  1,  1,   1,    1,   A000012
p_2(n):    -4,   -3,  -2, -1,  0,  1,   2,    3,   A001477
p_3(n):    13,    7,   3,  1,  1,  3,   7,   13,   A002061
p_4(n):   -68,  -30, -10, -2,  0,  2,  10,   30,   A034262
p_5(n):   157,   43,   7,  1,  1,  7,  43,  157,
p_6(n):  -972, -225, -30, -3,  0,  3,  30,  225,

Cf. A246654, A001477, A002061, A034262.

with(Student[NumericalAnalysis]):
poly := proc(n) local B; if n = 0 then return 0 fi;
B := (n,k) -> round(evalf(2*(BesselK(n,2)*BesselI(k,2)
-(-1)^(n+k)*BesselI(n,2)*BesselK(k,2)),64));
[seq([k+iquo(n,2),B(k+n,k)], k=-iquo(n,2)..n-1)];
PolynomialInterpolation(%, independentvar=x);
expand(Interpolant(%)) end:
A246656_row := n -> seq(coeff(poly(n),x,j), j=0..n);
seq(print(A246656_row(n)), n=0..11);

cp's OEIS Frontend

A216155 Numbers n such that floor(sqrt(n + n^3)) = 1 + floor(sqrt(n^3)) = 1 + A000093(n).

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A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

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A345285 Sides of primary squares of type 1 (A344331). A primary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b, so that the numbers of small and large squares are equal.

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A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.

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A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).

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A131473 a(n) = n^6 - n.

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A180499 n^3 + n-th cubefree number.

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A181427 a(n) = n + [n^2 if n is odd or n^3 if n is even].

A246654 T(n,k) = 2(K(n,2)I(k,2) - (-1)^(n+k)I(n,2)K(k,2)), where I(n,x) and K(n,x) are Bessel functions; triangle read by rows for 0 <= k <= n.