A052548 -id:A052548 - cp's OEIS frontend

A335003 Triangle read by rows where the n-th row is the cycle trajectory of 2^n+1 in the divide-or-choose 2 rule.

3, 5, 10, 9, 36, 18, 17, 136, 68, 34, 33, 528, 264, 132, 66, 65, 2080, 1040, 520, 260, 130, 129, 8256, 4128, 2064, 1032, 516, 258, 257, 32896, 16448, 8224, 4112, 2056, 1028, 514, 513, 131328, 65664, 32832, 16416, 8208, 4104, 2052, 1026, 1025, 524800, 262400, 131200, 65600, 32800, 16400, 8200, 4100, 2050

Offset: 1

Michel Marcus, May 22 2020

The divide-or-choose-2 rule is a quadratic Collatz-type recursion where the map is defined with f(n) = n/2 if n is even, and f(n) = binomial(n, 2) if n is odd.

			Triangle begins:
   3;
   5, 10;
   9, 36, 18;
  17, 136, 68, 34;
  33, 528, 264, 132, 66;
  ...

Michel Marcus, Rows n = 1..100 of the triangle, flattened
Hassan Sedaghat, Bounded Orbits of Quadratic Collatz-type Recursions, arXiv:2004.07357 [math.DS], 2020.
Hassan Sedaghat, Dual Nature of Orbits of the Divide-or-Choose 2 Rule: A Quadratic Collatz-type Recursion, arXiv:2005.10712 [math.NT], 2020.

Cf. A000051 (1st column), A052548 (right diagonal).

f[n_] := If[EvenQ[n], n/2, Binomial[n, 2]]; row[n_] := NestWhileList[f, n, f[#] != n &]; Join @@ Table[row[2^n + 1], {n, 1, 10}] (* Amiram Eldar, May 22 2020 *)

f(n) = if (n%2, binomial(n, 2), n/2);
row(n) = my(m=2^n+1, v=vector(n)); v[1] = m; for (i=2, n, v[i] = f(v[i-1])); v;

A340161 a(n) is the smallest number k for which the set {k + 1, k + 2, ..., k + k} contains exactly n elements with exactly three 1-bits (A014311).

Original entry on oeis.org

1, 4, 6, 7, 10, 11, 13, 18, 19, 21, 25, 34, 35, 37, 41, 49, 66, 67, 69, 73, 81, 97, 130, 131, 133, 137, 145, 161, 193, 258, 259, 261, 265, 273, 289, 321, 385, 514, 515, 517, 521, 529, 545, 577, 641, 769, 1026, 1027, 1029, 1033, 1041, 1057, 1089, 1153, 1281, 1537

Offset: 0

Marius A. Burtea, Dec 30 2020

When n = m*(m-1)/2 + 1, m >= 2 (A000124 \ {1}), then a(n) = k = 2^m+2, m >= 2 (A052548 \ {3, 4}), and only for these values of k, there exists only one set, {k+1, k+2, ..., 2k}, that contains exactly n elements whose binary representation has exactly three 1's (see A340068). - Bernard Schott, Jan 03 2021
From David A. Corneth, Jan 03 2021: (Start)
a(n) = A018900(n) + 1 for n >= 1.
Proof: Let T(k) be the number of values in {k+1, k+2, ..., k+k} that have exactly 3 ones in their binary expansion. Let h(k) be 1 if k has exactly 3 ones in its binary expansion and 0 otherwise and let w(k) be the binary weight of k (cf. A000120). Then T(k + 1) = T(k) + h(2*k + 1) + h(2*k + 2) - h(k + 1) = T(k) + h(2*k + 1) + h(2*(k + 1)) - h(k + 1) but as h(2^m * k) = h(k) two terms cancel and we have T(k + 1) = T(k) + h(2*k + 1). If w(2*k + 1) = w(k) + 1 = 3 then w(k) = 2 which holds for k in A018900. (End)

			For k in {1, 2, 3}, the sets are {1, 2}, {3, 4} and {4, 5, 6}, which do not contain numbers in A014311, so a(0) = 1.
For k = 4, the set is {5, 6, 7, 8} with 7 = A014311(1), so a(1) = 4.
For k = 6, the set {7, 8, 9, 10, 11, 12} contains the elements 7 = A014311(1) and 11 = A014311(2), so a(2) = 6.

Essentially a duplicate of A018900 - N. J. A. Sloane, Jan 23 2021.
Cf. A014311, A340068.
Cf. also A000124, A052548 \ {3} (is a subsequence).

fb:=func; a:=[]; for n in [0..64] do k:=1; while #[s:s in [k+1..2*k]|fb(s)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

first(n) = {my(res = vector(n), t = 1); res[1] = 1; for(i = 2, oo, if(hammingweight(2*i-1) == 3, t++; if(t > n, return(res)); res[t] = i))} \\ David A. Corneth, Jan 03 2021

from math import isqrt, comb
def A340161(n): return 1+(1<<(m:=isqrt(n<<3)+1>>1))+(1<<(n-1-comb(m,2))) if n else 1 # Chai Wah Wu, Mar 10 2025

From Bernard Schott, Jan 03 2021: (Start)
a(m*(m-1)/2 + 1) = 2^m + 2 for m >= 2.
a(m*(m-1)/2 + 2) = 2^m + 3 for m >= 2.
a(n) = A018900(n) + 1 for n >= 1 (see A340068). (End)

A363299 a(n) is the sum of the n-th powers of the terms of row 4 of Pascal's triangle.

Original entry on oeis.org

5, 16, 70, 346, 1810, 9826, 54850, 312706, 1810690, 10601986, 62563330, 371185666, 2210336770, 13194911746, 78901035010, 472332468226, 2829699842050, 16961019183106, 101697395621890, 609909495824386, 3658357463318530, 21945746733400066, 131656888214355970, 789870960541958146

Offset: 0

J. Lowell, May 26 2023

			a(2) = 1^2 + 4^2 + 6^2 + 4^2 + 1^2 = 1 + 16 + 36 + 16 + 1 = 70.

Index entries for linear recurrences with constant coefficients, signature (11,-34,24).

Cf. A007318.

Cf. A000012 (row 0), A007395 (row 1), A052548 (row 2), A115099 (row 3).

Table[6^n + 2*(4^n + 1), {n, 0, 24}] (* Amiram Eldar, May 27 2023 *)

def A363299(n): return 2+(((1<Chai Wah Wu, Jun 27 2023

a(n) = 2 + 2*4^n + 6^n.
From Natalia L. Skirrow, Jun 25 2023: (Start)
G.f.: (5-39*x+64*x^2)/((1-x)*(1-4*x)*(1-6*x)).
E.g.f.: 2*e^x + 2*e^(4*x) + e^(6*x).
(End)

A384441 Binary XOR of n and the prime factors of n.

Original entry on oeis.org

1, 0, 0, 6, 0, 7, 0, 10, 10, 13, 0, 13, 0, 11, 9, 18, 0, 19, 0, 19, 17, 31, 0, 25, 28, 21, 24, 25, 0, 26, 0, 34, 41, 49, 33, 37, 0, 55, 41, 47, 0, 44, 0, 37, 43, 59, 0, 49, 54, 53, 33, 59, 0, 55, 57, 61, 41, 37, 0, 56, 0, 35, 59, 66, 73, 72, 0, 87, 81, 70, 0, 73, 0, 109

Offset: 1

Karl-Heinz Hofmann, May 30 2025

			For n = 12 the prime factors are {2,3} -> a(12) = 12 XOR 2 XOR 3 = 13.
a(13) = 13 XOR 13 = 0.

Alois P. Heinz, Table of n, a(n) for n = 1..16384
Karl-Heinz Hofmann, Graph up to n = 2^17
Karl-Heinz Hofmann, Zoom trip of the graph with number of prime factors of n colored.
Wikipedia, Bitwise operation: XOR

Cf. A000040, A052548, A178910, A293212.

f:= l-> `if`(l=[], 0, Bits[Xor](l[1], f(l[2..-1]))):
a:= n-> f([n, map(i-> i[1], ifactors(n)[2])[]]):
seq(a(n), n=1..74);  # Alois P. Heinz, May 30 2025

a[n_] := BitXor @@ Join[{n}, FactorInteger[n][[;; , 1]]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, May 30 2025 *)

a(n) = my(f=factor(n)[,1]); my(b=n); for (k=1, #f, b=bitxor(b, f[k])); b; \\ Michel Marcus, May 30 2025

from sympy import primefactors
def A384441(n):
    result = n
    for pf in primefactors(n): result ^= pf
    return result

a(n) = XOR(n,A293212(n)).
a(n) = 0 <=> n is prime.
a(2^n) = A052548(n) for n>=2.

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661

Offset: 1

Clark Kimberling, Jul 02 2025

nonn

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).
The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:
    U             V         volume   squared diagonal length, ||g(n)||^2
(1, 0, 0)     (0, 1, 0)     A000079           A052548
(1, 0, 0)     (0, 1, 1)     A008776         3*A052919
(1, 0, 0)     (1, 0, 1)     A000351           A178676
(1, 0, 0)     (1, 1, 1)     A167747         5*A204061
(1, 0, 0)     (0, 2, 0)     A005054         4*A199215
(1, 0, 0)     (1, 2, 0)     A013731         8*A199552
(1, 0, 0)     (2, 1, 0)     A011557        10*A000533
(1, 0, 0)     (1, 1, 2)     A067403        18*A135423
(1, 0, 0)     (2, 1, 1)     A334603        11*A199750
(1, 0, 1)     (0, 1, 0)     A008776           this sequence
(1, 1, 0)     (0, 1, 1)     A081341         6*A199318
(1, 1, 0)     (1, 1, 1)     A270369         9*A199559
(1, 2, 3)     (3, 2, 1)   2*A009992   48 + 96*A009992

			Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000079, A052548, A008776.

f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
f[n_] := f[n] = h[n - 1];
g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
h[n_] := h[n] = Cross[f[n], g[n]];
v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
Table[v[n], {n, 1, 16}]  (* A008776 *)
u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
Join[{1},Table[1+2*(3^(n-1)+1),{n,40}]] (* or *) LinearRecurrence[{4,-3},{1,5,9},50] (* Harvey P. Dale, Jul 20 2025 *)

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.

A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 7, 4, 3, 3, 15, 8, 5, 4, 4, 31, 16, 9, 6, 5, 5, 63, 32, 17, 10, 7, 6, 6, 127, 64, 33, 18, 11, 8, 7, 7, 255, 128, 65, 34, 19, 12, 9, 8, 8, 511, 256, 129, 66, 35, 20, 13, 10, 9, 9, 1023, 512, 257, 130, 67, 36, 21, 14, 11, 10, 10, 2047, 1024, 513, 258, 131, 68, 37, 22

Offset: 0

Ross La Haye, Feb 23 2004

			{0};
{1,1};
{3,2,2};
{7,4,3,3};
{15,8,5,4,4};
{31,16,9,6,5,5};
{63,32,17,10,7,6,6};
a(5,3) = 34 because 2^5 + (3-1) = 34.

Rows: a(0, k) = A001477(k), a(1, k) = A000027(k+1) etc. etc. Columns: a(n, 0) = A000225(n). a(n, 1) = A000079(n). a(n, 2) = A000051(n). a(n, 3) = A052548(n). a(n, 4) = A062709(n). Diagonals: a(n, n+3) = A052968(n+1). a(n, n+2) = A005126(n). a(n, n+1) = A006127(n). a(n, n) = A052944(n). a(n, n-1) = A083706(n-1). Also note that the sums of the antidiagonals = the partial sums of the main diagonal, i.e., a(n, n).

Flatten[ Table[ Table[ a[i, n - i], {i, n, 0, -1}], {n, 0, 11}]] (* both from Robert G. Wilson v, Feb 26 2004 *)
Table[a[n, k], {n, 0, 10}, {k, 0, 10}] // TableForm (* to view the table *)

For k > 0, a(n, k)= a(n, k-1) + 1.

a(n, k) = 2^n + (k-1).

More terms from Robert G. Wilson v, Feb 23 2004

A108155 Numbers of the forms 2^n-2, 2^n-1, 2^n, 2^n+1 and 2^n+2.

Original entry on oeis.org

-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 30, 31, 32, 33, 34, 62, 63, 64, 65, 66, 126, 127, 128, 129, 130, 254, 255, 256, 257, 258, 510, 511, 512, 513, 514, 1022, 1023, 1024, 1025, 1026, 2046, 2047, 2048, 2049, 2050, 4094, 4095, 4096, 4097, 4098, 8190, 8191, 8192, 8193, 8194, 16382, 16383, 16384

Offset: 1

Henrik Lundquist (sploinker(AT)sploink.dk), Jun 06 2005

sign

This number sequence is designed for a quick test of e.g. arithmetic in binary systems.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,2,2,2,2,2).

Cf. A107684.

Union of A000918, A000225, A000079, A000051, and A052548.

A267615 a(n) = 2^n + 11.

Original entry on oeis.org

12, 13, 15, 19, 27, 43, 75, 139, 267, 523, 1035, 2059, 4107, 8203, 16395, 32779, 65547, 131083, 262155, 524299, 1048587, 2097163, 4194315, 8388619, 16777227, 33554443, 67108875, 134217739, 268435467, 536870923, 1073741835, 2147483659, 4294967307, 8589934603, 17179869195, 34359738379

Offset: 0

Ilya Gutkovskiy, Jan 18 2016

Recurrence relation b(n) = 3*b(n - 1) - 2*b(n - 2) for n>1, b(0) = k, b(1) = k + 1, gives the closed form b(n) = 2^n + k - 1.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. sequences with closed form 2^n + k - 1: A168616 (k=-4), A028399 (k=-3), A036563 (k=-2), A000918 (k=-1), A000225 (k=0), A000079 (k=1), A000051 (k=2), A052548 (k=3), A062709 (k=4), A140504 (k=5), A168614 (k=6), A153972 (k=7), A168415 (k=8), A242475 (k=9), A188165 (k=10), A246139 (k=11), this sequence (k=12).

Cf. A156940.

[2^n+11: n in [0..30]]; // Vincenzo Librandi, Jan 19 2016

Table[2^n + 11, {n, 0, 35}]
LinearRecurrence[{3, -2}, {12, 13}, 40] (* Vincenzo Librandi, Jan 19 2016 *)

a(n) = 2^n + 11; \\ Altug Alkan, Jan 18 2016

G.f.: (12 - 23*x)/(1 - 3*x + 2*x^2).
a(n) = 3*a(n - 1) - 2*a(n - 2) for n>1, a(0)=12, a(1)=13.
a(n) = A000079(n) + A010850(n).
Sum_{n>=0} 1/a(n) = 0.367971714327125...
Lim_{n->oo} a(n + 1)/a(n) = 2.
E.g.f.: exp(2*x) + 11*exp(x). - Elmo R. Oliveira, Nov 08 2023

A376323 (1/5) times obverse convolution (3)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 6, 48, 576, 11520, 414720, 28200960, 3722526720, 967856947200, 499414184755200, 513397781928345600, 1053492248516965171200, 4319318218919557201920000, 35401132122264690826936320000, 580153753219673753271832412160000, 19012798800515148242224491811307520000

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000051, A010701, A052548, A374848, A375053, A139030.

s[n_] := 3; t[n_] := 2^n + 1;
u[n_] := (1/5) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2) * QPochhammer[-4, 1/2, n+1]/5, {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

a(n) = 2 a(n-1)*A052548(n-1) for n>=1.

cp's OEIS Frontend

A335003 Triangle read by rows where the n-th row is the cycle trajectory of 2^n+1 in the divide-or-choose 2 rule.

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A340161 a(n) is the smallest number k for which the set {k + 1, k + 2, ..., k + k} contains exactly n elements with exactly three 1-bits (A014311).

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A363299 a(n) is the sum of the n-th powers of the terms of row 4 of Pascal's triangle.

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A384441 Binary XOR of n and the prime factors of n.

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A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

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A091264 Matrix defined by a(n,k) = 2^n + (k-1), read by antidiagonals.

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Extensions

A108155 Numbers of the forms 2^n-2, 2^n-1, 2^n, 2^n+1 and 2^n+2.

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A267615 a(n) = 2^n + 11.

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