A027444 -id:A027444 - cp's OEIS frontend

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.

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).

A348461 Size of largest bipartite biregular Moore graph of diameter 4 and degrees n and n.

Original entry on oeis.org

8, 30, 80, 170, 312

Offset: 2

N. J. A. Sloane, Oct 31 2021

a(7) >= 516, a(8) = 800, a(9) = 1170, a(10) = 1640.

G. Araujo-Pardo, C. Dalfó, M. Á. Fiol and N. López, Bipartite biregular Moore graphs, arXiv:2103.11443 [math.CO], 2021.
G. Araujo-Pardo, C. Dalfó, M. Á. Fiol and N. López, Bipartite biregular Moore graphs, Discrete Math., 334 (2021), # 112582.

Cf. A027444, A053698, A348462, A348463.

Empirical observation: For the terms a(2)-a(6) and a(8)-a(10)  a(n) = 2*(A027444(n-1) + 1).  It is unknown whether this is also valid for n = 7 and n > 10. - Hugo Pfoertner, Oct 31 2021
Is this the same as 2*A053698(n-1)? If not, where is the first place these sequences differ? - Omar E. Pol, Oct 31 2021
a(n) <= 2*A053698(n-1) (the Moore bound). - Pontus von Brömssen, Oct 31 2021

A318765 a(n) = (n + 2)*(n^2 + n - 1).

Original entry on oeis.org

-2, 3, 20, 55, 114, 203, 328, 495, 710, 979, 1308, 1703, 2170, 2715, 3344, 4063, 4878, 5795, 6820, 7959, 9218, 10603, 12120, 13775, 15574, 17523, 19628, 21895, 24330, 26939, 29728, 32703, 35870, 39235, 42804, 46583, 50578, 54795, 59240, 63919, 68838, 74003, 79420, 85095

Offset: 0

Bruno Berselli, Sep 04 2018

First differences are in A004538.

a(n) is divisible by 11 for n = 3, 7, 9, 14, 18, 20, 25, 29, 31, 36, 40, ... with formula (1/3)*(11*m + (1 + (m mod 3))*(-1)^((m-1) mod 3) + 8), m >= 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of similar sequences (row k=3, m>1).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A004538.
Subsequence of A047216.
Similar sequences (see Table in Links section): A011379, A027444, A033445, A034262, A045991, A069778.

GAP
```
List([0..50], n -> (n+2)*(n^2+n-1));
```

[(n+2)*(n^2+n-1) for n in 0:50] |> println

Magma
```
[(n+2)*(n^2+n-1): n in [0..50]];
```

seq((n+2)*(n^2+n-1),n=0..43); # Paolo P. Lava, Sep 04 2018

Table[(n + 2) (n^2 + n - 1), {n, 0, 50}]

Maxima
```
makelist((n+2)*(n^2+n-1), n, 0, 50);
```
PARI
```
vector(50, n, n--; (n+2)*(n^2+n-1))
```
Python
```
[(n+2)*(n**2+n-1) for n in range(50)]
```
Sage
```
[(n+2)*(n^2+n-1) for n in (0..50)]
```

O.g.f.: (-2 + 11*x - 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-2 + 5*x + 6*x^2 + x^3)*exp(x).
a(n) = -A033445(-n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020

A119576 (n+n^2+n^3)(binomial(2n,n)).

Original entry on oeis.org

0, 6, 84, 780, 5880, 39060, 238392, 1369368, 7516080, 39819780, 205079160, 1032047016, 5094629904, 24743027400, 118504436400, 560749834800, 2625519143520, 12179090862180, 56029885342200, 255864364648200

Offset: 0

Zerinvary Lajos, May 31 2006

Cf. A000984, A027444.

[seq ((n+n^2+n^3)*(binomial(2*n,n)),n=0..25)];

A119577 (n+n^2+n^3)(binomial(2n,n))/2.

Original entry on oeis.org

0, 3, 42, 390, 2940, 19530, 119196, 684684, 3758040, 19909890, 102539580, 516023508, 2547314952, 12371513700, 59252218200, 280374917400, 1312759571760, 6089545431090, 28014942671100, 127932182324100

Offset: 0

Zerinvary Lajos, May 31 2006

Cf. A000984, A027444.

[seq ((n+n^2+n^3)*(binomial(2*n,n))/2,n=0..29)];

Table[(n+n^2+n^3) Binomial[2n,n]/2,{n,0,20}] (* Harvey P. Dale, May 26 2020 *)

cp's OEIS Frontend

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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A348461 Size of largest bipartite biregular Moore graph of diameter 4 and degrees n and n.

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A318765 a(n) = (n + 2)*(n^2 + n - 1).

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GAP

Julia

Magma

Maple

Mathematica

Maxima

PARI

Python

Sage

Formula

A119576 (n+n^2+n^3)*(binomial(2*n,n)).

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Maple

A119577 (n+n^2+n^3)*(binomial(2*n,n))/2.

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Author

Keywords

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Programs

Maple

Mathematica

A119576 (n+n^2+n^3)(binomial(2n,n)).

A119577 (n+n^2+n^3)(binomial(2n,n))/2.