A010701 -id:A010701 - cp's OEIS frontend

A210042 Triangle of coefficients of polynomials u(n,x) jointly generated with A124927; see the Formula section.

1, 3, 5, 2, 7, 6, 2, 9, 12, 8, 2, 11, 20, 20, 10, 2, 13, 30, 40, 30, 12, 2, 15, 42, 70, 70, 42, 14, 2, 17, 56, 112, 140, 112, 56, 16, 2, 19, 72, 168, 252, 252, 168, 72, 18, 2, 21, 90, 240, 420, 504, 420, 240, 90, 20, 2, 23, 110, 330, 660, 924, 924, 660, 330, 110

Offset: 1

Clark Kimberling, Mar 17 2012

Row sums:  A000225
For a discussion and guide to related arrays, see A208510.
u(n,x) = u(n-1,x) + v(n-1,x) + 1,
v(n,x) = x*u(n-1,x) + x*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

			From _Philippe Deléham_, Mar 25 2012: (Start)
(1, 2, -2, 1, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
   1;
   1,  0;
   3,  0,  0;
   5,  2,  0,  0;
   7,  6,  2,  0,  0;
   9, 12,  8,  2,  0,  0;
  11, 20, 20, 10,  2,  0,  0;
  13, 30, 40, 30, 12,  2,  0,  0; (End)

Cf. A124927, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210042 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A124927 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A010701 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 signed *)

First five rows:
  1;
  3,
  5,  2;
  7,  6,  2;
  9, 12,  8,  2;
First three polynomials u(n,x): 1, 3, 5 + 2x.
Also, counting the top row as row 0, row n for n > 0 is as follows: 2n+1, 2C(n,2), 2C(n,3), ..., 2C(n,n).
From Philippe Deléham, Mar 25 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x-y*x+2*x^2)/(1-2*x-y*x+x^2+y*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 3, T(1,1) = T(2,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
G.f.: (1+x-x*y)*x*y/((-1+x)*(x+x*y-1)). - R. J. Mathar, Aug 12 2015

A291971 Triangle read by rows: T(n,k) = 3 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 3, 0, 3, 0, 3, 9, 0, 3, 9, 0, 3, 18, 0, 3, 18, 27, 0, 3, 27, 27, 0, 3, 27, 54, 0, 3, 36, 81, 0, 3, 36, 108, 81, 0, 3, 45, 135, 81, 0, 3, 45, 189, 162, 0, 3, 54, 216, 243, 0, 3, 54, 270, 405, 0, 3, 63, 324, 486, 243, 0, 3, 63, 378, 729, 243, 0, 3, 72, 432, 891

Offset: 0

Seiichi Manyama, Sep 07 2017

			First few rows are:
  1;
  0, 3;
  0, 3;
  0, 3,  9;
  0, 3,  9;
  0, 3, 18;
  0, 3, 18,  27;
  0, 3, 27,  27;
  0, 3, 27,  54;
  0, 3, 36,  81;
  0, 3, 36, 108, 81.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A032308.
Columns 0-1 give A000007, A010701.
Cf. A008289 (m=1), A291970 (m=2), this sequence (m=3).

A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 05 2017

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
The partial sum of this sequence is A184985.
a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

			For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
V. Manturov, Knot Theory, CRC Press, 2004.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.
J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
A. Banerjee, Knot theory [Foil knot family].
D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0,1,2},120,{1}] (* Harvey P. Dale, Feb 20 2023 *)

makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

PARI
```
a(n) = if(n>2, 1, n);
```

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
A005803(a(n)) = A005096(a(n)) = A000007(n).
A107583(a(n)) = A103775(n+5).
a(n+1) = 2^A185012(n+1), with a(0) = 0.
a(n) = A163985(n) mod A004278(n+1).
a(n) = A157928(n) + A171386(n+1).
a(n) = A063524(n) + A157928(n) + A185012(n).
a(n) = A010701(n) - A141044(n) - A179184(n).
G.f.: (x + x^2 - x^3)/(1 - x).
E.g.f.: (2*exp(x) - 2 + x^2)/2.

A303749 First differences of A302774; Number of terms in A303762 that have prime(n) as their largest prime factor (A006530).

Original entry on oeis.org

1, 2, 4, 7, 16, 19, 52, 55, 160, 163, 484, 487, 1456, 1459, 4372, 4375, 13120, 13123, 39364, 39367, 118096, 118099, 354292, 354295

Offset: 1

Antti Karttunen, May 05 2018

nonn

For n >= 1, the difference A000079(n-1) - a(n): 0, 0, 0, 1, 0, 13, 12, 73, 96, 349, 540, 1561, 2640, 6733, 12012, 28393, 52416, 117949, 222780, 484921, 930480, 1979053, 3840012, ..., indicates how many squarefree numbers A303762 misses in each round. The first of these is 70 missed at the round 4.

The first differences of these terms is: 1, 2, 3, 9, 3, 33, 3, 105, 3, 321, 3, 969, 3, 2913, 3, 8745, 3, 26241, 3, 78729, 3, 236193, 3, ... which after the first two initial terms seem to be an interleaving of sequences A010701 and A036543.

Cf. A010701, A036543, A302774, A303762.

a(n) = A302774(n+1) - A302774(n).

A341694 Square array T(n, k) read by antidiagonals upwards, n, k > 0: T(n, k) = A227736(n, k) for k = 1..A005811(n), and T(n, k) = T(n, k - A005811(n)) + ... + T(n, k-1) for k > A005811(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 2, 5, 1, 3, 2, 1, 4, 2, 8, 1, 3, 3, 3, 3, 7, 2, 13, 1, 1, 1, 3, 5, 5, 11, 2, 21, 1, 1, 2, 4, 3, 8, 9, 18, 2, 34, 1, 2, 1, 1, 5, 3, 13, 17, 29, 2, 55, 1, 2, 1, 1, 4, 9, 3, 21, 31, 47, 2, 89, 1

Offset: 1

Rémy Sigrist, Feb 17 2021

This table contains all Fibonacci sequences of order m > 0 with positive terms:
- order 1 corresponds to constant sequences (n in A126646),
- order 2 corresponds to Fibonacci-like sequences (n in A043569),
- order 3 corresponds to tribonacci-like sequences (n in A043570),
- order 4 corresponds to tetranacci-like sequences (n in A043571).
For any n > 0, the row A341746(n) corresponds to the n-th row from which the first term has been removed.

			Array T(n, k) begins:
  n\k|  1  2  3  4  5   6   7   8   9   10   11   12   13    14
  ---+---------------------------------------------------------
    1|  1  1  1  1  1   1   1   1   1    1    1    1    1     1 --> A000012
    2|  1  1  2  3  5   8  13  21  34   55   89  144  233   377 --> A000045
    3|  2  2  2  2  2   2   2   2   2    2    2    2    2     2 --> A007395
    4|  2  1  3  4  7  11  18  29  47   76  123  199  322   521 --> A000032
    5|  1  1  1  3  5   9  17  31  57  105  193  355  653  1201 --> A000213
    6|  1  2  3  5  8  13  21  34  55   89  144  233  377   610 --> A000045
    7|  3  3  3  3  3   3   3   3   3    3    3    3    3     3 --> A010701
    8|  3  1  4  5  9  14  23  37  60   97  157  254  411   665 --> A104449
    9|  1  2  1  4  7  12  23  42  77  142  261  480  883  1624 --> A275778
   10|  1  1  1  1  4   7  13  25  49   94  181  349  673  1297 --> A000288

Rémy Sigrist, Table of n, a(n) for n = 1..11325
Rémy Sigrist, Colored representation of the table for n, k = 1..1024
Rémy Sigrist, PARI program for A341694
Wikipedia, Fibonacci numbers of higher order
Index entries for sequences related to Fibonacci numbers

Cf. A005811, A043569, A043570, A043571, A126646, A136480, A227736, A341699, A341746.

Cf. A000012, A000032, A000045, A000213, A000288, A007395, A010701, A104449, A275778.

PARI
```
See Links section.
```

T(A341746(n), k) = T(n, k+1).

T(n, 1) = A136480(n).

A348643 a(n) = (16n + 1)(2592n^2 + 288n + 7).

Original entry on oeis.org

7, 49079, 361383, 1185751, 2771015, 5366007, 9219559, 14580503, 21697671, 30819895, 42196007, 56074839, 72705223, 92335991, 115215975, 141594007, 171718919, 205839543, 244204711, 287063255, 334664007, 387255799, 445087463, 508407831, 577465735, 652510007, 733789479, 821552983

Offset: 0

Michel Marcus, Oct 27 2021

a(n) is the entry (1,1) of a family of unimodular matrices none of whose entries is 1 or -1, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular.

In these matrices, the entries (1,3) and (3,1) = 2; the entries (2,3) and (3,2) = 3; the entry (3,3) = 0.

			From _Elmo R. Oliveira_, Sep 03 2025: (Start)
G.f.: (7 + 49051*x + 165109*x^2 + 34665*x^3)/(x-1)^4.
E.g.f.: (7 + 49072*x + 131616*x^2 + 41472*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Ajai Choudhry, A diophantine problem concerning third order matrices, arXiv:2110.12643 [math.NT], 2021.
Wikipedia, Unimodular matrix.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000004, A007395, A010701, A348644, A348645, A348646.

a(n) = (16*n + 1)*(2592*n^2 + 288*n + 7);

A348644 a(n) = (18n + 1)(24n + 1)(144*n + 11).

Original entry on oeis.org

11, 73625, 542087, 1778645, 4156547, 8049041, 13829375, 21870797, 32546555, 46229897, 63294071, 84112325, 109057907, 138504065, 172824047, 212391101, 257578475, 308759417, 366307175, 430594997, 501996131, 580883825, 667631327, 762611885, 866198747, 978765161, 1100684375

Offset: 0

Michel Marcus, Oct 27 2021

a(n) is the entry (1,2) of a family of unimodular matrices none of whose entries is 1 or -1, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular.

In these matrices, the entries (1,3) and (3,1) = 2; the entries (2,3) and (3,2) = 3; the entry (3,3) = 0.

Ajai Choudhry, A diophantine problem concerning third order matrices, arXiv:2110.12643 [math.NT], 2021.
Wikipedia, Unimodular matrix.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000004, A007395, A010701, A348643, A348645, A348646.

a(n) = (18*n + 1)*(24*n + 1)*(144*n + 11);

From Elmo R. Oliveira, Sep 03 2025: (Start)
G.f.: (11 + 73581*x + 247653*x^2 + 52003*x^3)/(x-1)^4.
E.g.f.: (11 + 73614*x + 197424*x^2 + 62208*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A348645 a(n) = (12n + 1)(5184n^2 + 540n + 13).

Original entry on oeis.org

13, 74581, 545725, 1786693, 4170733, 8071093, 13861021, 21913765, 32602573, 46300693, 63381373, 84217861, 109183405, 138651253, 172994653, 212586853, 257801101, 309010645, 366588733, 430908613, 502343533, 581266741, 668051485, 763071013, 866698573, 979307413, 1101270781

Offset: 0

Michel Marcus, Oct 27 2021

a(n) is the entry (2,1) of a family of unimodular matrices none of whose entries is 1 or -1, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular.

In these matrices, the entries (1,3) and (3,1) = 2; the entries (2,3) and (3,2) = 3; the entry (3,3) = 0.

Ajai Choudhry, A diophantine problem concerning third order matrices, arXiv:2110.12643 [math.NT], 2021.
Wikipedia, Unimodular matrix.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000004, A007395, A010701, A348643, A348644, A348646.

a(n) = (12*n + 1)*(5184*n^2 + 540*n + 13);

From Elmo R. Oliveira, Sep 04 2025: (Start)
G.f.: (13 + 74529*x + 247479*x^2 + 51227*x^3)/(x-1)^4.
E.g.f.: (13 + 74568*x + 198288*x^2 + 62208*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A348646 a(n) = (72n + 5)(1296n^2 + 153n + 4).

Original entry on oeis.org

20, 111881, 818606, 2680067, 6256136, 12106685, 20791586, 32870711, 48903932, 69451121, 95072150, 126326891, 163775216, 207976997, 259492106, 318880415, 386701796, 463516121, 549883262, 646363091, 753515480, 871900301, 1002077426, 1144606727, 1300048076, 1468961345, 1651906406

Offset: 0

Michel Marcus, Oct 27 2021

a(n) is the entry (2,2) of a family of unimodular matrices none of whose entries is 1 or -1, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular.

In these matrices, the entries (1,3) and (3,1) = 2; the entries (2,3) and (3,2) = 3; the entry (3,3) = 0.

Ajai Choudhry, A diophantine problem concerning third order matrices, arXiv:2110.12643 [math.NT], 2021.
Wikipedia, Unimodular matrix.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000004, A007395, A010701, A348643, A348644, A348645.

a(n) = (72*n + 5)*(1296*n^2 + 153*n + 4);

From Elmo R. Oliveira, Sep 04 2025: (Start)
G.f.: (20 + 111801*x + 371202*x^2 + 76849*x^3)/(x-1)^4.
E.g.f.: (20 + 111861*x + 297432*x^2 + 93312*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A021097 Decimal expansion of 1/93.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Generalization:
  1/3 = Sum_{i >= 0} 7^i/10^(i+1);
  1/93 = Sum_{i >= 0} 7^i/100^(i+1) (this sequence);
  1/993 = Sum_{i >= 0} 7^i/1000^(i+1);
  1/9993 = Sum_{i >= 0} 7^i/10000^(i+1), etc. - Daniel Forgues, Oct 28 2011
In other words, given n > 1, the decimal expansion of 1/(10^n - 3) contains the first n powers of 7 (including 7^0 = 1) separated by n - 1 zeroes. - Alonso del Arte, Aug 10 2017

			0.010752688172043010752688172...

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

Cf. A010701, A000420.

RealDigits[1/93, 10, 100][[1]] (* Alonso del Arte, Aug 10 2017 *)

cp's OEIS Frontend

A210042 Triangle of coefficients of polynomials u(n,x) jointly generated with A124927; see the Formula section.

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A291971 Triangle read by rows: T(n,k) = 3 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

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Maxima

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Formula

A303749 First differences of A302774; Number of terms in A303762 that have prime(n) as their largest prime factor (A006530).

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A341694 Square array T(n, k) read by antidiagonals upwards, n, k > 0: T(n, k) = A227736(n, k) for k = 1..A005811(n), and T(n, k) = T(n, k - A005811(n)) + ... + T(n, k-1) for k > A005811(n).

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A348643 a(n) = (16*n + 1)*(2592*n^2 + 288*n + 7).

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A348644 a(n) = (18*n + 1)*(24*n + 1)*(144*n + 11).

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PARI

Formula

A348645 a(n) = (12*n + 1)*(5184*n^2 + 540*n + 13).

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PARI

Formula

A348643 a(n) = (16n + 1)(2592n^2 + 288n + 7).

A348644 a(n) = (18n + 1)(24n + 1)(144*n + 11).

A348645 a(n) = (12n + 1)(5184n^2 + 540n + 13).

A348646 a(n) = (72n + 5)(1296n^2 + 153n + 4).