A062709 -id:A062709 - cp's OEIS frontend

A254464 a(n) = 212^n + 104^n + 153^n + 36^n + 6*5^n + 7^n + 28.

84, 210, 714, 2982, 14178, 73470, 404634, 2331462, 13906578, 85232910, 533860554, 3403329942, 22012307778, 144090486750, 952693102074, 6352175272422, 42655384385778, 288161867586990, 1956674663089194, 13344181547374902, 91343993647708578, 627261876368085630

Offset: 0

Luciano Ancora, Jan 31 2015

This is the sequence of seventh terms of "third partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers.
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A062709, A254362, A254363, A254364, A254463.

Table[21 2^n + 10 4^n + 15 3^n + 3 6^n + 6 5^n + 7^n + 28, {n, 0, 24}] (* Michael De Vlieger, Jan 31 2015 *)
LinearRecurrence[{28,-322,1960,-6769,13132,-13068,5040},{84,210,714,2982,14178,73470,404634},30] (* Harvey P. Dale, May 17 2019 *)

vector(30, n, n--; 21*2^n + 10*4^n + 15*3^n + 3*6^n + 6*5^n + 7^n + 28) \\ Colin Barker, Jan 31 2015

From Colin Barker, Jan 31 2015: (Start)
G.f.: -6*(40188*x^6 - 74058*x^5 + 52931*x^4 - 19005*x^3 + 3647*x^2 - 357*x + 14)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)).
a(n) = 28*a(n-1) - 322*a(n-2) + 1960*a(n-3) - 6769*a(n-4) + 13132*a(n-5) - 13068*a(n-6) + 5040*a(n-7). (End)
E.g.f.: exp(x)*(exp(x)*(exp(5*x) + 3*exp(4*x) + 6*exp(3*x) + 10*exp(2*x) + 15*exp(x) + 21) + 28). - Elmo R. Oliveira, Sep 16 2024

A369414 Irregular triangle read by rows: row n lists the values of the vertices at the n-th level of the MI graph (see comments).

Original entry on oeis.org

1, 2, 4, 8, 5, 16, 13, 10, 7, 32, 29, 26, 23, 20, 17, 14, 11, 64, 61, 58, 55, 52, 49, 46, 43, 40, 37, 34, 31, 28, 25, 22, 19, 128, 125, 122, 119, 116, 113, 110, 107, 104, 101, 98, 95, 92, 89, 86, 83, 80, 77, 74, 71, 68, 65, 62, 59, 56, 53, 50, 47, 44, 41, 38, 35

Offset: 0

Paolo Xausa, Jan 24 2024

The vertices of the graph consist of all of the positive integers that are not divisible by 3. A vertex v (for v >= 4) has 2*v as left child and 2*v - 3 as right child (see example).
Matos and Antunes (1998) use this graph to illustrate the fact that, for a string (theorem) S belonging to the MIU formal system containing no U characters, the length of the path from vertex v (where v is the number of I characters in S) to the root corresponds to the number of times step 2 of their algorithm for generating "normal" proofs (described in A369409) is applied.
See A368946 for the description of the MIU formal system.

			The first levels of the graph are shown below. Cf. Matos and Antunes (1998), p. 7, figure 1.
                           +--1
                           |
                        +--2
                        |
            +-----------4-----------+
            |                       |
      +-----8-----+           +-----5-----+
      |           |           |           |
   +-16--+     +-13--+     +-10--+     +--7--+
   |     |     |     |     |     |     |     |
  32    29    26    23    20    17    14    11
                       ...
Written as an irregular triangle, the sequence begins:
  [0]  1;
  [1]  2;
  [2]  4;
  [3]  8  5;
  [4] 16 13 10  7;
  [5] 32 29 26 23 20 17 14 11;
  ...

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41.

Paolo Xausa, Table of n, a(n) for n = 0..16384 (rows 0..15 of the triangle, flattened).
Armando B. Matos and Luis Filipe Antunes, Short Proofs for MIU theorems, Technical Report Series DCC-98-01, University of Porto, 1998.
Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A368946, A369409.
Cf. A000079 (first column and, for n >= 2, row lengths), A062709 (right border, for n >= 2).
Permutation of A001651.

A369414row[n_] := If[n <= 1, {n+1}, Range[2^n, 3+2^(n-2), -3]];
Array[A369414row, 8, 0]

T(n,1) = n + 1 for n < 2.

T(n,k) = 2^n - 3*(k-1) for n >= 2 and 1 <= k <= 2^(n-2).

A067390 Number of distinct prime factors in 2^n + 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 1, 1, 3, 1, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 4, 1, 3, 2, 4, 3, 2, 3, 3, 2, 2, 2, 3, 2, 4, 4, 2, 2, 3, 2, 3, 3, 4, 2, 5, 2, 1, 3, 3, 3, 4, 3, 2, 2, 4, 2, 5, 2, 1, 5, 3, 3, 3, 3, 3, 4, 4, 2, 4, 3, 4, 5, 4, 4, 5, 1, 4, 4, 2, 3, 5, 4, 4, 4, 4, 5, 4, 4, 6, 3, 2, 2

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

			a(5) = 2 since 2^5 + 3 = 35 = 5 * 7 has 2 distinct prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..699

Cf. A001221, A062709.

[#PrimeDivisors(2^n+3):n in [1..100]]; // Marius A. Burtea, Feb 06 2020

PrimeNu @ Table[2^n + 3, {n, 1, 50}] (* Amiram Eldar, Feb 06 2020 *)

a(n) = A001221(A062709(n)). - Amiram Eldar, Feb 06 2020

A228132 First differences of A014311.

Original entry on oeis.org

4, 2, 1, 5, 2, 1, 3, 1, 2, 7, 2, 1, 3, 1, 2, 5, 1, 2, 4, 11, 2, 1, 3, 1, 2, 5, 1, 2, 4, 9, 1, 2, 4, 8, 19, 2, 1, 3, 1, 2, 5, 1, 2, 4, 9, 1, 2, 4, 8, 17, 1, 2, 4, 8, 16, 35, 2, 1, 3, 1, 2, 5, 1, 2, 4, 9, 1, 2, 4, 8, 17, 1, 2, 4, 8, 16, 33, 1, 2, 4, 8, 16, 32

Offset: 1

Jon Perry, Nov 02 2013

The records are: 4, 5, 7, 11, 19, 35, 67, ... and they occur at these indices of A014311: 11, 19, 35, 67, ... (for both, see A062709). - Michel Marcus, Jun 11 2015

The record (maximum) among the first 1000 terms is 65539. - Harvey P. Dale, May 29 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A062709 (2^n+3), A014311 (numbers with exactly 3 ones in binary expansion).

Cf. A145057.

oo=0;
for (i=1;i<500;i++) {
s=i.toString(2);
o=0;
for (j=0;j

Differences[Select[Range[500],DigitCount[#,2,1]==3&]] (* Harvey P. Dale, May 29 2018 *)

lista(nn) = {my(last = 0); for (n=1, nn, if (hammingweight(n)==3, if (last, print1(n-last,", ")); last = n;););} \\ Michel Marcus, Jun 10 2015

from math import isqrt, comb
from sympy import integer_nthroot
def A228132(n): return (1<<(r:=n-comb((m:=integer_nthroot(6*n+6,3)[0])+(t:=(n>=comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n+1-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+(1<comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))-(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))-(1<Chai Wah Wu, Apr 07 2025

A377610 a(n) is the number of iterations of x -> 2*x - 3 until (# composites reached) = (# primes reached), starting with prime(n+2).

Original entry on oeis.org

13, 9, 7, 21, 7, 1, 15, 1, 5, 23, 5, 13, 1, 3, 1, 1, 3, 19, 1, 1, 11, 1, 7, 9, 1, 19, 1, 17, 7, 1, 3, 1, 1, 1, 11, 1, 5, 1, 1, 11, 3, 5, 1, 1, 15, 15, 1, 1, 3, 1, 5, 5, 1, 5, 1, 1, 1, 1, 13, 1, 1, 9, 1, 5, 3, 1, 3, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 9, 3

Offset: 1

Clark Kimberling, Nov 05 2024

nonn

For a guide to related sequences, see A377609.

			Starting with prime(3) = 5, we have 2*5-3 = 7, then 2*7-3 = 11, etc., resulting in a chain 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027, 2051, 4099, 8195, 16387 having 7 primes and 7 composites. Since every initial subchain has fewer composites than primes, a(1) = 14-1 = 13. (For more terms from the mapping x -> 2x-3, see A062709.)

Cf. A062709, A377609.

chain[{start_, u_, v_}] := If[CoprimeQ[u, v] && start*u + v != start,
   NestWhile[Append[#, u*Last[#] + v] &, {start}, !
      Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &], {}];
chain[{Prime[3], 2, -3}]
Map[Length[chain[{Prime[#], 2, -3}]] &, Range[3, 100]] - 1
(* Peter J. C. Moses, Oct 31 2024 *)

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

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

A347512 Number of minimal dominating sets in the n-book graph.

Original entry on oeis.org

6, 7, 11, 19, 35, 67, 131, 259, 515, 1027, 2051, 4099, 8195, 16387, 32771, 65539, 131075, 262147, 524291, 1048579, 2097155, 4194307, 8388611, 16777219, 33554435, 67108867, 134217731, 268435459, 536870915, 1073741827, 2147483651, 4294967299, 8589934595

Offset: 1

Eric W. Weisstein, Sep 04 2021

Eric Weisstein's World of Mathematics, Book Graph
Eric Weisstein's World of Mathematics, Minimal Dominating Set
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A062709 (essentially the same).

Join[{6}, 2^Range[2, 20] + 3]
CoefficientList[Series[(6 - 11 x + 2 x^2)/((-1 + x) (-1 + 2 x)), {x, 0, 20}], x]

a(n) = A062709(n) = 2^n + 3 for n > 1.
G.f.: x*(6 - 11*x + 2*x^2)/((-1 + x)*(-1 + 2*x)).
E.g.f.: exp(x)*(3 + exp(x)) - 4 + x. - Stefano Spezia, Sep 04 2021

A376322 (1/4) times obverse convolution (2)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 5, 35, 385, 7315, 256025, 17153675, 2247131425, 582007039075, 299733625123625, 307826433001962875, 631352014087025856625, 2587911905742718986305875, 21207938067561582092776645625, 347534481113131645754330891856875, 11389052480558437163015177657041650625

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

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

Cf. A000051, A000079, A007395, A010701, A374848, A375054, A139030.

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

a(n) = a(n-1)*A062709(n) for n>=1.

a(n) = (1/4)((3)**(2^n)) = (1/4)(A010701(n)**A000079(n)) for n>=0.

cp's OEIS Frontend

A254464 a(n) = 21*2^n + 10*4^n + 15*3^n + 3*6^n + 6*5^n + 7^n + 28.

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A369414 Irregular triangle read by rows: row n lists the values of the vertices at the n-th level of the MI graph (see comments).

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A067390 Number of distinct prime factors in 2^n + 3.

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A228132 First differences of A014311.

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A377610 a(n) is the number of iterations of x -> 2*x - 3 until (# composites reached) = (# primes reached), starting with prime(n+2).

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

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

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A347512 Number of minimal dominating sets in the n-book graph.

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A254464 a(n) = 212^n + 104^n + 153^n + 36^n + 6*5^n + 7^n + 28.