A042948 -id:A042948 - cp's OEIS frontend

A338062 Numbers k such that the Enots Wolley sequence A336957(k) is odd.

1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 128, 129, 132, 133, 136, 137

Offset: 1

N. J. A. Sloane, Oct 18 2020

nonn

a(n) = A042948(n) for n<=1065, but then the two sequences start to differ. - R. J. Mathar, Nov 06 2020

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957, A338063-A338069.

M = 1000;
A[1] = 1; A[2] = 2;
Clear[B]; B[_] = 0;
For[n = 3, True, n++, For[k = 3, k <= M, k++, If[B[k] == 0 && GCD[k, A[n-1]] > 1 && GCD[k, A[n-2]] == 1, If[Length[FactorInteger[k][[All, 1]] ~Complement~ FactorInteger[A[n-1]][[All, 1]]] > 0, A[n] = k; B[k] = 1; Break[]]]]; If[k > M, Break[]]];
Reap[For[k = 1, k <= M, k++, If[OddQ[A[k]], Sow[k]]]][[2, 1]] (* Jean-François Alcover, Oct 23 2020, after Maple code in A336957 *)

A052951 Expansion of (1 + x - 2x^2)/(1 - 2x)^2.

Original entry on oeis.org

1, 5, 14, 36, 88, 208, 480, 1088, 2432, 5376, 11776, 25600, 55296, 118784, 253952, 540672, 1146880, 2424832, 5111808, 10747904, 22544384, 47185920, 98566144, 205520896, 427819008, 889192448, 1845493760, 3825205248, 7918845952

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals binomial transform of A042948 starting with "1": (1, 4, 5, 8, 9, 12, 13, ...) = terms > 0, == 0 or 1 mod 4. - Gary W. Adamson, Feb 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1021.
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A042948, A118413.

Concatenation([1], List([1..40], n-> 2^(n-1)*(2*n+3) )); # G. C. Greubel, Oct 21 2019

I:=[1, 5, 14]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

spec:= [S,{S=Prod(Union(Sequence(Union(Z,Z)),Z),Sequence(Union(Z,Z)))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0, 1, 2^(n-1)*(2*n+3)), n=0..40); # G. C. Greubel, Oct 21 2019

CoefficientList[Series[(1+x-2*x^2)/(1-2*x)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 22 2012 *)
LinearRecurrence[{4,-4}, {1,5,14}, 40] (* G. C. Greubel, Oct 21 2019 *)

x='x+O('x^40); Vec((1+x-2*x^2)/(1-2*x)^2) \\ Altug Alkan, Mar 03 2018

[1]+[2^(n-1)*(2*n+3) for n in (1..40)] # G. C. Greubel, Oct 21 2019

G.f.: (1+x-2*x^2)/(1-2*x)^2.
a(n) = 4*(a(n-1) - a(n-2)).
a(n) = (n+1)*2^n + 2^(n-1), n > 0.
a(n) = A118413(n+1,n-1) for n > 2. - Reinhard Zumkeller, Apr 27 2006
E.g.f.: (1/2)*(-1 + exp(2*x)*(3 + 4*x)). - Stefano Spezia, Oct 22 2019
From Amiram Eldar, Oct 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 4*sqrt(2)*arcsinh(1) - 11/3.
Sum_{n>=0} (-1)^n/a(n) = 13/3 - 4*sqrt(2)*arccot(sqrt(2)). (End)

A266180 Decimal representation of the n-th iteration of the "Rule 6" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 6, 16, 96, 256, 1536, 4096, 24576, 65536, 393216, 1048576, 6291456, 16777216, 100663296, 268435456, 1610612736, 4294967296, 25769803776, 68719476736, 412316860416, 1099511627776, 6597069766656, 17592186044416, 105553116266496, 281474976710656

Offset: 0

Robert Price, Dec 22 2015

A001025 is a subsequence. - Altug Alkan, Dec 23 2015

Rules 38, 134 and 166 also generate this sequence.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 22.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,16).

Cf. A001025, A266178, A266179, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

rule=6; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)
LinearRecurrence[{0,16},{1,6},30] (* Harvey P. Dale, May 25 2016 *)

print([int(4**(n-1)*(5-(-1)**n)) for n in range(30)]) # Karl V. Keller, Jr., Jun 03 2021

From Colin Barker, Dec 23 2015 and Apr 13 2019: (Start)
a(n) = 4^(n-1)*(5-(-1)^n).
a(n) = 16*a(n-2) for n>1.
G.f.: (1+6*x) / ((1-4*x)*(1+4*x)).
(End)

A361236 Array read by antidiagonals: T(n,k) is the number of noncrossing k-gonal cacti with n polygons up to rotation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 5, 11, 1, 1, 1, 1, 8, 33, 49, 1, 1, 1, 1, 9, 63, 230, 204, 1, 1, 1, 1, 12, 105, 664, 1827, 984, 1, 1, 1, 1, 13, 159, 1419, 7462, 15466, 4807, 1, 1, 1, 1, 16, 221, 2637, 21085, 90896, 137085, 24739, 1

Offset: 0

Andrew Howroyd, Mar 05 2023

The number of noncrossing k-gonal cacti is given by column 2*(k-1) of A070914. This sequence enumerates these cacti with rotations being considered equivalent.
Equivalently, T(n,k) is the number of connected acyclic k-uniform noncrossing antichains with n blocks covering (k-1)*n+1 nodes where the intersection of two blocks is at most 1 node modulo cyclic rotation of the nodes.
Noncrossing trees correspond to the case of k = 2.

			=====================================================
n\k | 1     2       3        4        5         6 ...
----+------------------------------------------------
  0 | 1     1       1        1        1         1 ...
  1 | 1     1       1        1        1         1 ...
  2 | 1     1       1        1        1         1 ...
  3 | 1     4       5        8        9        12 ...
  4 | 1    11      33       63      105       159 ...
  5 | 1    49     230      664     1419      2637 ...
  6 | 1   204    1827     7462    21085     48048 ...
  7 | 1   984   15466    90896   334707    941100 ...
  8 | 1  4807  137085  1159587  5579961  19354687 ...
  9 | 1 24739 1260545 15369761 96589350 413533260 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Wikipedia, Cactus graph.
Index entries for sequences related to cacti.

Columns k=1..4 are A000012, A296532, A361237, A361238.
Row n=3 is A042948.
Cf. A070914, A303694, A303912, A361239, A361242.

\\ here u is Fuss-Catalan sequence with p = 2*k-1.
u(n,k,r) = {r*binomial(n*(2*k-1) + r, n)/(n*(2*k-1) + r)}
T(n,k) = if(n==0, 1, u(n, k, 1)/((k-1)*n+1) + sumdiv(gcd(k,n-1), d, if(d>1, eulerphi(d)*u((n-1)/d, k, 2*k/d)/k)))

T(0,k) = T(1,k) = T(2,k) = 1.

A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 11, 13, 16, 17, 19, 21, 24, 25, 27, 29, 32, 33, 35, 37, 40, 41, 43, 45, 48, 49, 51, 53, 56, 57, 59, 61, 64, 65, 67, 69, 72, 73, 75, 77, 80, 81, 83, 85, 88, 89, 91, 93, 96, 97, 99, 101, 104, 105, 107, 109, 112, 113, 115, 117, 120, 121, 123

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A004526, A016813, A042948, A047470, A140081, A140201, A173562.

[n : n in [0..150] | n mod 8 in [0, 1, 3, 5]]; // Wesley Ivan Hurt, Jun 01 2016

A047624:=n->(8*n-11-I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047624(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Table[(8n-11-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,5,8},100] (* G. C. Greubel, Jun 01 2016 *)

From Reinhard Zumkeller, Feb 21 2010: (Start)
a(n+1) = A173562(n) - A173562(n-1);
a(n+1) - a(n) = A140081(n-1) + 1;
a(n) = A140201(n-1) + A042948(A004526(n-1)). (End)
G.f.: x^2*(1+2*x+2*x^2+3*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-11-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A016813(k-1) for k>0, a(2k-1) = A047470(k). (End)
E.g.f.: (6 + sin(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016
Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*Pi/16 + (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 20 2021

A140201 Partial sums of A140081.

Original entry on oeis.org

0, 1, 2, 4, 4, 5, 6, 8, 8, 9, 10, 12, 12, 13, 14, 16, 16, 17, 18, 20, 20, 21, 22, 24, 24, 25, 26, 28, 28, 29, 30, 32, 32, 33, 34, 36, 36, 37, 38, 40, 40, 41, 42, 44, 44, 45, 46, 48, 48, 49, 50, 52, 52, 53, 54, 56, 56, 57, 58, 60, 60, 61, 62, 64, 64, 65, 66, 68, 68, 69, 70, 72, 72, 73, 74

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 09 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002265, A004524, A004526, A042948, A047624, A057353, A121262.

I:=[0, 1, 2, 4, 4]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..80]]; // Vincenzo Librandi, Sep 17 2012

A140201:=n->(4*n+1-I^(2*n)+(-I)^(1+n)+I^(1+n))/4: seq(A140201(n), n=0..100); # Wesley Ivan Hurt, Jun 04 2016

Accumulate[PadRight[{}, 68, {0, 1, 1, 2}]] (* Harvey P. Dale, Aug 19 2011 *)

a(n) = A047624(n+1) - A042948(A004526(n)). - Reinhard Zumkeller, Feb 21 2010
a(n) = A002265(n+1) + A057353(n+1). - Reinhard Zumkeller, Feb 26 2011
From Bruno Berselli, Jan 27 2011: (Start)
G.f.: x*(1+x+2*x^2)/((1+x)*(1+x^2)*(1-x)^2).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
a(n) = n + A121262(n+1). (End)
a(n) = n when n+1 is not a multiple of 4, and a(n) = n+1 when n+1 is a multiple of 4. - Dennis P. Walsh, Aug 06 2012
a(n) = A004524(n+1) + A004526(n+1). - Arkadiusz Wesolowski, Sep 17 2012
a(n) = (4*n+1-i^(2*n)+(-i)^(1+n)+i^(1+n))/4 where i=sqrt(-1). - Wesley Ivan Hurt, Jun 04 2016
a(n) = n+1-(sign((n+1) mod 4) mod 3). - Wesley Ivan Hurt, Sep 26 2017

A201630 a(n) = a(n-1) + 2*a(n-2) with n>1, a(0)=2, a(1)=7.

Original entry on oeis.org

2, 7, 11, 25, 47, 97, 191, 385, 767, 1537, 3071, 6145, 12287, 24577, 49151, 98305, 196607, 393217, 786431, 1572865, 3145727, 6291457, 12582911, 25165825, 50331647, 100663297, 201326591, 402653185, 805306367, 1610612737, 3221225471, 6442450945, 12884901887

Offset: 0

Bruno Berselli, Dec 03 2011

B. Satyanarayana and K. S. Prasad, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd. (Eastern Economy Edition), 2009, p. 73 (problem 3.3).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A005010, A042948, A097581.

[n le 2 select 5*n-3 else Self(n-1)+2*Self(n-2): n in [1..33]];

Mathematica
```
LinearRecurrence[{1, 2}, {2,7}, 33]
```

a[0]:2$ a[1]:7$ a[n]:=a[n-1]+2*a[n-2]$ makelist(a[n], n, 0, 32);

v=vector(33); v[1]=2; v[2]=7; for(i=3, #v, v[i]=v[i-1]+2*v[i-2]); v

def A201630(n): return 3*2**n - (-1)**n
print([A201630(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

G.f.: (2+5*x)/((1+x)*(1-2*x)).
a(n) = 3*2^n - (-1)^n.
a(n) = 7 + 2*Sum_{i=0..n-2} a(i), for n>0.
a(n) = A097581(A042948(n+1)).
a(n+2) - a(n) = a(n+1) + a(n) = A005010(n).
E.g.f.: 3*exp(2*x) - exp(-x). - G. C. Greubel, Feb 07 2025

A090964 Permutation of natural numbers generated by 2-rowed array shown below.

Original entry on oeis.org

1, 4, 2, 5, 3, 8, 6, 9, 7, 12, 10, 13, 11, 16, 14, 17, 15, 20, 18, 21, 19, 24, 22, 25, 23, 28, 26, 29, 27, 32, 30, 33, 31, 36, 34, 37, 35, 40, 38, 41, 39, 44, 42, 45, 43, 48, 46, 49, 47, 52, 50, 53, 51, 56, 54, 57, 55, 60, 58, 61, 59, 64, 62, 65, 63, 68, 66, 69, 67, 72, 70, 73

Offset: 0

Giovanni Teofilatto, Feb 29 2004

nonn

1 4 5 8 9 12 13 16 17 20...(A042948)

2 3 6 7 10 11 14 15 18 19...(A042964)

For n > 1, a(n+8) = a(n)+8. - David Wasserman, Feb 23 2006

More terms from David Wasserman, Feb 23 2006

A133907 Least prime number p such that binomial(n+p, p) mod p = 1.

Original entry on oeis.org

2, 3, 5, 2, 2, 7, 11, 2, 2, 3, 3, 2, 2, 17, 17, 2, 2, 3, 3, 2, 2, 23, 29, 2, 2, 5, 3, 2, 2, 31, 37, 2, 2, 37, 37, 2, 2, 3, 41, 2, 2, 43, 47, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 59, 61, 2, 2, 67, 3, 2, 2, 67, 71, 2, 2, 71, 73, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 89, 89, 2, 2, 3, 3, 2, 2, 97

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

Also the least prime number p such that p divides floor(n/p) or p > n.
a(n) = 2 if and only if n is in A042948. - Robert Israel, May 11 2017
Conjecture: a(n) is the smallest prime p such that Sum_{k=1..n} k^(p-1) == n (mod p). Thus a(n) >= A317358(n). - Thomas Ordowski, Jul 29 2018

			a(2)=3, since binomial(2+3,3) mod 3 = 10 mod 3 = 1 and 3 is the minimal prime number with this property.
a(7)=11 because of binomial(7+11, 11) = 31824 = 2893*11 + 1, but binomial(7+k, k) mod k <> 1 for all primes < 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A042948, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133906, A133910, A317358.

f:= proc(n) local m;
  m:= 2:
  while floor(n/m) mod m <> 0 do m:= nextprime(m) od:
  m
end proc:
map(f, [$1..100]); # Robert Israel, May 11 2017

a[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[Mod[Binomial[n+p, p], p] == 1, Return[p]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 05 2023 *)

a(n) = my(p=2); while (binomial(n+p, p) % p != 1, p = nextprime(p+1)); p; \\ Michel Marcus, Dec 17 2022

from sympy import nextprime, ff
def A133907(n):
    p, m = 2, (n+2)*(n+1)>>1
    while m%p != 1:
        q = nextprime(p)
        m = m*ff(n+q,q-p)//ff(q,q-p)
        p = q
    return p # Chai Wah Wu, Feb 22 2023

A140685 Triangle T(n,k) read by rows: T(n,k) = 1 if n is odd and k=(n-1)/2; T(n,k) = 2 otherwise.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Roger L. Bagula and Mats Granvik, Jul 11 2008

Row sums give A042948.

			The triangle starts in row n=1 with columns 0 <= k < n as:
  1;
  2, 2;
  2, 1, 2;
  2, 2, 2, 2;
  2, 2, 1, 2, 2;
  2, 2, 2, 2, 2, 2;
  2, 2, 2, 1, 2, 2, 2;
  2, 2, 2, 2, 2, 2, 2, 2;
  2, 2, 2, 2, 1, 2, 2, 2, 2;
  2, 2, 2, 2, 2, 2, 2, 2, 2, 2;

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 rows of the triangle

Cf. A002024, A002260, A104041.

(define (A140685 n) (A140685tr (A002024 n) (- (A002260 n) 1)))
(define (A140685tr n k) (if (and (odd? n) (= k (/ (- n 1) 2))) 1 2))

Definition simplified by the Assoc. Eds. of the OEIS, Oct 12 2010

More terms from Antti Karttunen, Oct 10 2017

cp's OEIS Frontend

A338062 Numbers k such that the Enots Wolley sequence A336957(k) is odd.

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A052951 Expansion of (1 + x - 2*x^2)/(1 - 2*x)^2.

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A266180 Decimal representation of the n-th iteration of the "Rule 6" elementary cellular automaton starting with a single ON (black) cell.

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A361236 Array read by antidiagonals: T(n,k) is the number of noncrossing k-gonal cacti with n polygons up to rotation.

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A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

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A140201 Partial sums of A140081.

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A201630 a(n) = a(n-1) + 2*a(n-2) with n>1, a(0)=2, a(1)=7.

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A052951 Expansion of (1 + x - 2x^2)/(1 - 2x)^2.