A087289 -id:A087289 - cp's OEIS frontend

A087290 Number of pairs of polynomials (f,g) in GF(3)[x] satisfying deg(f) <= n, deg(g) <= n and gcd(f,g) = 1.

8, 56, 488, 4376, 39368, 354296, 3188648, 28697816, 258280328, 2324522936, 20920706408, 188286357656, 1694577218888, 15251194969976, 137260754729768, 1235346792567896, 11118121133111048, 100063090197999416, 900567811781994728, 8105110306037952536

Offset: 0

W. Edwin Clark, Aug 29 2003

An unpublished result due to Stephen Suen, David desJardins, and W. Edwin Clark. This is the case k = 2, q = 3 of their formula q^((n+1)*k) * (1 - 1/q^(k-1) + (q-1)/q^((n+1)*k)) for the number of ordered k-tuples (f_1, ..., f_k) of polynomials in GF(q)[x] such that deg(f_i) <= n for all i and gcd(f_1, ..., f_k) = 1.

			a(0) = 8 since there are eight pairs, (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2) of polynomials (f,g) in GF(3)[x] of degree at most 0 such that gcd(f,g) = 1.

Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A087289, A087291, A087292.

2*3^(2Range[0,30]+1)+2 (* or *) LinearRecurrence[{10,-9},{8,56},30] (* Harvey P. Dale, Mar 07 2012 *)

a(n) = 2*3^(2*n+1) + 2.
a(n) = 10*a(n-1) - 9*a(n-2), a(0)=8, a(1)=56. - Harvey P. Dale, Mar 07 2012
G.f.: 8*(1-3*x)/((1-x)*(1-9*x)). - Colin Barker, Apr 16 2012

More terms from Harvey P. Dale, Mar 07 2012

A087291 Number of pairs of polynomials (f,g) in GF(2)[x] satisfying 1 <= deg(f) <= n, 1 <= deg(g) <= n and gcd(f,g) = 1.

Original entry on oeis.org

0, 2, 18, 98, 450, 1922, 7938, 32258, 130050, 522242, 2093058, 8380418, 33538050, 134184962, 536805378, 2147352578, 8589672450, 34359214082, 137437904898, 549753716738, 2199019061250

Offset: 0

W. Edwin Clark, Aug 29 2003

Unpublished result due to Stephen Suen, David desJardins, and W. Edwin Clark. This is the case k = 2, q = 2 of their formula (q^(n+1)-q)^k*(1-1/(q^(k-1))) for the number of ordered k-tuples (f_1, ..., f_k) of polynomials in GF(q)[x] such that 1 <= deg(f_i) <= n for all i and gcd(f_1, ..., f_k) = 1.

			a(1) = 2 since gcd(x,x+1) = 1 and gcd(x+1,x) = 1 and no other pair (f,g) of polynomials in GF(2)[x] of degree 1 satisfy gcd(f,g) = 1.

Index entries for linear recurrences with constant coefficients, signature (7, -14, 8).

Cf. A087289, A087290, A087292.

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

G.f.: 2*x*(1+2*x)/((1-x)*(1-2*x)*(1-4*x)). - Colin Barker, Feb 22 2012

A087292 Number of pairs of polynomials (f,g) in GF(3)[x] satisfying 1 <= deg(f) < =n, 1 <= deg(g) <= n and gcd(f,g) = 1.

Original entry on oeis.org

0, 24, 384, 4056, 38400, 351384, 3179904, 28671576, 258201600, 2324286744, 20919997824, 188284231896, 1694570841600, 15251175838104, 137260697334144, 1235346620381016, 11118120616550400, 100063088648317464, 900567807132948864, 8105110292090814936

Offset: 0

W. Edwin Clark, Aug 29 2003

Unpublished result due to Stephen Suen, David desJardins, and W. Edwin Clark. This is the case k = 2, q = 3 of their formula (q^(n+1)-q)^k*(1-1/(q^(k-1))) for the number of ordered k-tuples (f_1, ..., f_k) of polynomials in GF(q)[x] such that 1 <= deg(f_i) <= n for all i and gcd(f_1, ..., f_k) = 1.

			There are 6 polynomials in GF(3)[x] of degree 1. a(1) = 24 since the 6*4 = 24 ordered pairs (f,g) where g is not equal to f or 2f are the only ordered pairs of polynomials of degree 1 satisfying gcd(f,g) = 1.

Index entries for linear recurrences with constant coefficients, signature (13,-39,27).

Cf. A087289, A087290, A087291.

a(n) = 6*(3^n-1)^2.

G.f.: -24*x*(3*x+1)/((x-1)*(3*x-1)*(9*x-1)). [Colin Barker, Sep 05 2012]

A199493 a(n) = 2*8^n+1.

Original entry on oeis.org

3, 17, 129, 1025, 8193, 65537, 524289, 4194305, 33554433, 268435457, 2147483649, 17179869185, 137438953473, 1099511627777, 8796093022209, 70368744177665, 562949953421313, 4503599627370497, 36028797018963969

Offset: 0

Vincenzo Librandi, Nov 07 2011

An Engel expansion of 4 to the base 8 as defined in A181565, with the associated series expansion 4 = 8/3 + 8^2/(3*17) + 8^3/(3*17*129) + 8^4/(3*17*129*1025) + .... Cf. A087289 and A199552. - Peter Bala, Oct 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A087289, A181565, A199552.

Magma
```
[2*8^n+1: n in [0..30]];
```

2*8^Range[0, 20] + 1 (* Wesley Ivan Hurt, Jul 23 2025 *)

a(n) = 8*a(n-1)-7.
a(n) = 9*a(n-1)-8*a(n-2).
G.f.: (3-10*x)/((1-x)*(1-8*x)).

A199552 4*8^n+1.

Original entry on oeis.org

5, 33, 257, 2049, 16385, 131073, 1048577, 8388609, 67108865, 536870913, 4294967297, 34359738369, 274877906945, 2199023255553, 17592186044417, 140737488355329, 1125899906842625, 9007199254740993, 72057594037927937

Offset: 0

Vincenzo Librandi, Nov 08 2011

An Engel expansion of 2 to the base 8 as defined in A181565, with the associated series expansion 2 = 8/5 + 8^2/(5*33) + 8^3/(5*33*257) + 8^4/(5*33*257*2049) + .... Cf. A087289 and A199493. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

A087289, A199493.

Magma
```
[4*8^n+1: n in [0..30]];
```

4*8^Range[0,20]+1 (* or *) LinearRecurrence[{9,-8},{5,33},20] (* Harvey P. Dale, Mar 18 2018 *)

a(n) = 8*a(n-1)-7.
a(n) = 9*a(n-1)-8*a(n-2).
G.f.: (5-12*x)/((1-x)*(1-8*x)).

A228767 Second bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

Original entry on oeis.org

-2, -9, -45, -231, -1161, -5643, -26637, -122895, -557073, -2490387, -11010069, -48234519, -209715225, -905969691, -3892314141, -16642998303, -70866960417, -300647710755, -1271310319653, -5360119185447, -22539988369449, -94557999988779, -395824185999405

Offset: 1

Michel Marcus, following a suggestion of Paul Curtz, Sep 03 2013

sign

The sequence to be transformed is A176328/A176591, its inverse binomial transform begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, -5643, 33578309/2730, ...
Its first bisection is constituted of fractional numbers, with denominators A176591, whereas this bisection is constituted of integers only.
It appears that a(1) = -2 and a(n) = -1 * A005408(n-1) * A087289(n-2) for n>1.

fr(n) = if (n==0, 1, (-1)^n*(subst(bernpol(n), x, 1) + subst(bernpol(n), x, 2))/2);
ibtfr(n) = sum(k = 0, n, (-1)^(n-k)*binomial(n, k) * fr(k));
lista(nn) = {forstep(n=1, nn, 2, print1(ibtfr(n), ", "););} \\ Michel Marcus, Sep 03 2013

Conjecture: G.f. -x*(2-11*x+21*x^2-2*x^3+8*x^4)/((1-x)^2*(1-4*x)^2). [Bruno Berselli, Sep 03 2013]

Conjecture: a(n) = (8+4^n)*(1-2*n)/8 for n>1, a(1)=-2. [Bruno Berselli, Sep 03 2013]

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

Original entry on oeis.org

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset: 0

Peter M. Chema, Feb 28 2017

Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021
Final digit alternates 4 and 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Eric Weisstein's World of Mathematics, Tetrix
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017

Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017

def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n+1) = a(n) + A002023(n).
a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Entry revised by Editors of OEIS, Mar 01 2017

A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 9, 6, 1, 1, 33, 90, 13, 1, 1, 129, 2220, 1162, 24, 1, 1, 513, 59178, 265132, 17435, 48, 1, 1, 2049, 1594836, 67180330, 49163241, 310193, 86, 1, 1, 8193, 43048770, 17181660628, 152662629227, 13121450895, 6286826, 160

Offset: 0

Seiichi Manyama, Nov 12 2017

			Square array begins:
    1,     1,        1,            1,               1, ...
    1,     1,        1,            1,               1, ...
    3,     9,       33,          129,             513, ...
    6,    90,     2220,        59178,         1594836, ...
   13,  1162,   265132,     67180330,     17181660628, ...
   24, 17435, 49163241, 152662629227, 476855156157129, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..2 give A000219, A294813, A294954.
Rows n=0+1, 2 give A000012, A087289.
Cf. A294609, A294758, A294808.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j)) * A(n-j,k) for n > 0.

A308704 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d+1).

Original entry on oeis.org

1, 1, 3, 1, 9, 4, 1, 33, 82, 7, 1, 129, 2188, 1033, 6, 1, 513, 59050, 262177, 15626, 12, 1, 2049, 1594324, 67108993, 48828126, 280026, 8, 1, 8193, 43046722, 17179869697, 152587890626, 13060696236, 5764802, 15, 1, 32769, 1162261468, 4398046513153, 476837158203126, 609359740069674, 4747561509944, 134218761, 13

Offset: 1

Seiichi Manyama, Jun 18 2019

			Square array begins:
   1,     1,        1,            1,               1, ...
   3,     9,       33,          129,             513, ...
   4,    82,     2188,        59050,         1594324, ...
   7,  1033,   262177,     67108993,     17179869697, ...
   6, 15626, 48828126, 152587890626, 476837158203126, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened

Columns k=0..3 give A000203, A283498, A283533, A283535.
Row n=1..2 give A000012, A087289.
Cf. A294579, A308698, A308701.

T[n_, k_] := DivisorSum[n, #^(k*# + 1) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j))).

G.f. of column k: Sum_{j>=1} j^(k*j+1) * x^j/(1 - x^j).

A341019 a(n) is the Y-coordinate of the n-th point of the space filling curve M defined in Comments section; A341018 gives X-coordinates.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 6, 5, 4, 5, 6, 7, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 7, 6, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 11, 10, 9, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 12

Offset: 0

Rémy Sigrist, Feb 02 2021

nonn

We define the family {M_n, n >= 0}, as follows:
- M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:
          +
         / \
        /   \
       +     +
      O
- for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:
                           + . . . + . . . +
                           .   B   .   B   .
       + . . . +           .       .       .
       .   B   .           .A     C.A     C.
       .       .    -->    + . . . + . . . +
       .A     C.           .C      .      A.
       + . . . +           .      B.B      .
      O                    .A      .      C.
                           + . . . + . . . +
                          O
- for any n >= 0, M_n has A087289(n) points,
- the space filling curve M is the limit of M_{2*n} as n tends to infinity.
The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059252).

			The curve M starts as follows:
       11+ 13+   +19 +21
        / \ / \ / \ / \
     10+ 12+ 14+18 +20 +22
        \     / \     /
        9+ 15+   +17 +23
        /     \ /     \
      8+  6+   +   +26 +24
        \ / \ 16  / \ /
        7+  5+   +27 +25
            /     \
          4+       +28
            \     /
        1+  3+   +29 +31
        / \ /     \ / \
      0+  2+       +30 +32
- so a(0) = a(2) = a(30) = a(32) = 0,
     a(1) = a(3) = a(29) = a(31) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982.
Larry Riddle, Space Filling Curve
Rémy Sigrist, PARI program for A341019
Index entries for sequences related to coordinates of 2D curves

Cf. A059252, A087289, A341018.

PARI
```
See Links section.
```

A059252(n) = (a(2*n+1)-1)/2.
a(4*n) = 2*A341018(n).
a(16*n) = 4*a(n).

cp's OEIS Frontend

A087290 Number of pairs of polynomials (f,g) in GF(3)[x] satisfying deg(f) <= n, deg(g) <= n and gcd(f,g) = 1.

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A087291 Number of pairs of polynomials (f,g) in GF(2)[x] satisfying 1 <= deg(f) <= n, 1 <= deg(g) <= n and gcd(f,g) = 1.

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A087292 Number of pairs of polynomials (f,g) in GF(3)[x] satisfying 1 <= deg(f) < =n, 1 <= deg(g) <= n and gcd(f,g) = 1.

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A199493 a(n) = 2*8^n+1.

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Magma

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A199552 4*8^n+1.

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Magma

Mathematica

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A228767 Second bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

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Comments

Programs

PARI

Formula

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

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Comments

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Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(k*j)*x^j)^j in powers of x.

Views

Author

Keywords

Examples

A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j in powers of x.