A010729 -id:A010729 - cp's OEIS frontend

A171472 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 7, a(1) = 30.

7, 30, 124, 504, 2032, 8160, 32704, 130944, 524032, 2096640, 8387584, 33552384, 134213632, 536862720, 2147467264, 8589901824, 34359672832, 137438822400, 549755551744, 2199022731264, 8796091973632, 35184369991680

Offset: 0

Klaus Brockhaus, Dec 09 2009

Related to Reverse and Add trajectory of 22 in base 2: A061561(4*n+2) = 12*a(n).
Third binomial transform of A010729.
a(n) in base 2 is n+3 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A061561, A010729 (repeat 7, 9), A171470, A171471, A171473, A171499.

[8*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, May 31 2011

LinearRecurrence[{6,-8},{7,30},30] (* Harvey P. Dale, Sep 01 2016 *)

{m=22; v=concat([7, 30], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 8*4^n-2^n.
G.f.: (7-12*x)/((1-2*x)*(1-4*x)).
a(n) = A171499(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: exp(2*x)*(8*exp(2*x) - 1). - Stefano Spezia, Sep 27 2023

A306289 The smallest prime factor of numbers greater than 1 and coprime to 6.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 5, 29, 31, 5, 37, 41, 43, 47, 7, 53, 5, 59, 61, 5, 67, 71, 73, 7, 79, 83, 5, 89, 7, 5, 97, 101, 103, 107, 109, 113, 5, 7, 11, 5, 127, 131, 7, 137, 139, 11, 5, 149, 151, 5, 157, 7, 163, 167, 13, 173, 5, 179, 181, 5, 11, 191, 193

Offset: 1

Davis Smith, Feb 03 2019

nonn

a(n) is the least prime factor of the n-th number that is greater than 1 and congruent to 1 or 5 (mod 6).
a(n) = 5 when n is congruent to {1, 8} (mod 10) (n is a term in A017281, A017365, or A306277). a(n) = 7 when n is congruent to {2, 11} (mod 14) but not {1, 8} (mod 10). a(n) = 11 when n is congruent to {3, 18} (mod 22) but not a case where it equals 5 or 7. a(n) = 13 when n is congruent to {4, 21} (mod 26) (n is a term in A306285) but not a case where it equals 5, 7, or 11. a(n) = 17 when n is congruent to {5, 28} (mod 34) but not a case where it equals 5, 7, 11, or 13. a(n) = 19 when n is congruent to {6, 31} (mod 38) (n is a term in A306331) but not a case where it equals 5, 7, 11, 13, or 17.
Conjecture: This pattern continues indefinitely. a(n) = A007310(m + 1) when n is congruent to {m, A306277(m + 1)} (mod A091999(m + 1)) but not congruent to {k, A306277(k + 1)} (mod A091999(k + 1)), m > k >= 1. The indices of the first appearance of a number in this sequence supports this conjecture in that they are never, for m > 0, congruent to A306277(m + 1) mod A091999(m + 1).

			a(n) is the least term, other than 0, in n-th row of the array A(m,n), where A(m,n) is A007310(m + 1) when A007310(n + 1) mod A007310(m + 1) is congruent to 0, otherwise 0.
Table begins
  \m  1 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 ...
  n\
   1| 5 0  0  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   2| 0 7  0  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   3| 0 0 11  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   4| 0 0  0 13  0  0  0  0  0   0   0   0   0   0   0   0 ...
   5| 0 0  0  0 17  0  0  0  0   0   0   0   0   0   0   0 ...
   6| 0 0  0  0  0 19  0  0  0   0   0   0   0   0   0   0 ...
   7| 0 0  0  0  0  0 23  0  0   0   0   0   0   0   0   0 ...
   8| 5 0  0  0  0  0  0 25  0   0   0   0   0   0   0   0 ...
   9| 0 0  0  0  0  0  0  0 29   0   0   0   0   0   0   0 ...
  10| 0 0  0  0  0  0  0  0  0  31   0   0   0   0   0   0 ...
  11| 5 7  0  0  0  0  0  0  0   0  35   0   0   0   0   0 ...
  12| 0 0  0  0  0  0  0  0  0   0   0  37   0   0   0   0 ...
  13| 0 0  0  0  0  0  0  0  0   0   0   0  41   0   0   0 ...
  14| 0 0  0  0  0  0  0  0  0   0   0   0   0  43   0   0 ...
  15| 0 0  0  0  0  0  0  0  0   0   0   0   0   0  47   0 ...
  16| 0 7  0  0  0  0  0  0  0   0   0   0   0   0   0  49 ...
For the n-th row of this square array, the leftmost terms, other than 0, are the factors of A(n,n). A(n,n) = A007310(n + 1). If for every m, m < n, A(m,n) = 0, then a(n) = A007310(n + 1) and A007310(n + 1) is prime.

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 2, Section 2, Problems 96 and 105.

Davis Smith, Table of n, a(n) for n = 1..1000

Cf. A000034, A007310, A010729, A017281, A017365, A020639, A091999, A273669, A306277, A306285, A306331.

Cf. A107744, A111863 (bisections).

seq(min(op(numtheory[factorset] (6*ceil(n/2)+(-1)^n))), n=1..64) ;

FactorInteger[Rest@ Flatten@ Array[6 # + {1, 5} &, 33, 0]][[All, 1, 1]] (* Michael De Vlieger, Feb 15 2019 *)
FactorInteger[#][[1,1]]&/@Select[Range[2,200],CoprimeQ[#,6]&] (* Harvey P. Dale, Jul 10 2020 *)

for(n=2, 211, if((n%6==1)||(n%6==5), print1(factor(n)[1,1], ", ")))

vector(64,n,factor(6*ceil(n/2)+(-1)^n)[1,1])

a(n) = n++; factor(n\2*6-(-1)^n)[1,1]; \\ Michel Marcus, Feb 06 2019

a(n) = A020639(A007310(n + 1)).
a(n) = A020639(3n + A000034(n + 1)).
a(n) = A020639(6*ceiling(n/2) + (-1)^n).
a(floor(prime(n + 2)/3)) = prime(n + 2).

A173512 a(n) = 8*n + 4 + n mod 2.

Original entry on oeis.org

4, 13, 20, 29, 36, 45, 52, 61, 68, 77, 84, 93, 100, 109, 116, 125, 132, 141, 148, 157, 164, 173, 180, 189, 196, 205, 212, 221, 228, 237, 244, 253, 260, 269, 276, 285, 292, 301, 308, 317, 324, 333, 340, 349, 356, 365, 372, 381, 388, 397, 404, 413, 420, 429, 436

Offset: 0

Reinhard Zumkeller, Feb 20 2010

nonn

First differences of A173511;

a(n+1) - a(n) = A010729(n+1).

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

[8*n+4+n mod 2: n in [0..60]]; // Vincenzo Librandi, Nov 29 2016

LinearRecurrence[{1, 1, -1}, {4, 13, 20}, 60] (* Vincenzo Librandi, Nov 29 2016 *)

G.f.: (4 + 9*x + 3*x^2)/((1 + x)*(1 - x)^2). - Philippe Deléham, Nov 29 2016

A134965 a(1)=3, a(n) = a(n-1) + 7 + 2*mod(n-1, 2) for n>=2.

Original entry on oeis.org

3, 12, 19, 28, 35, 44, 51, 60, 67, 76, 83, 92, 99, 108, 115, 124, 131, 140, 147, 156, 163, 172, 179, 188, 195, 204, 211, 220, 227, 236, 243, 252, 259, 268, 275, 284, 291, 300, 307, 316, 323, 332, 339, 348, 355, 364, 371, 380, 387, 396, 403, 412, 419, 428

Offset: 1

Roger L. Bagula, Jan 31 2008

Starting weights for pyramid game.
Numbers n such that the equation m(m + 1)/2 + 1 - n == 0 mod m has a solution.
Numbers congruent to {3, 12} mod 16. - Philippe Deléham, Nov 28 2016

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

Flatten[Table[If[ IntegerQ[2*Sqrt[ -7 + 8*n]] && Mod[n - 7, 8] == 0, f[n], {}], {n, 1, 10000}]]
LinearRecurrence[{1,1,-1},{3,12,19},60] (* Harvey P. Dale, Oct 05 2017 *)

Vec(x*(3 + 9*x + 4*x^2) / ((1 - x)^2 * (1 + x)) + O(x^100)) \\ Colin Barker, Nov 29 2016

a(n)=8*n - 4 - n%2 \\ Charles R Greathouse IV, Nov 29 2016

From R. J. Mathar, Feb 05 2008: (Start)
G.f.: (3+9*x+4*x^2)/((1-x)^2*(x+1)).
a(n) - a(n-1) = A010729(n).
(End)
From Colin Barker, Nov 29 2016: (Start)
a(n) = 8*n - 4 for n even.
a(n) = 8*n - 5 for n odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
(End)
E.g.f.: 4 + ((16*x - 9)*exp(x) + exp(-x))/2. - David Lovler, Aug 21 2022

Definition adapted to offset by Georg Fischer, Jun 19 2021

A176442 Decimal expansion of (21+sqrt(469))/6.

Original entry on oeis.org

7, 1, 0, 9, 4, 0, 1, 3, 0, 4, 6, 1, 7, 9, 5, 2, 5, 3, 3, 6, 3, 1, 0, 3, 3, 5, 1, 4, 4, 6, 5, 1, 9, 6, 2, 1, 6, 2, 4, 1, 5, 5, 2, 9, 2, 2, 7, 9, 7, 7, 6, 2, 9, 7, 7, 4, 9, 4, 0, 2, 1, 2, 7, 3, 4, 8, 4, 8, 7, 9, 4, 5, 6, 3, 0, 7, 0, 3, 7, 8, 2, 6, 6, 2, 9, 8, 7, 3, 3, 0, 9, 6, 9, 6, 2, 1, 7, 4, 9, 3, 3, 7, 7, 1, 7

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (21+sqrt(469))/6 is A010729.

			(21+sqrt(469))/6 = 7.10940130461795253363...

Cf. A176443 (decimal expansion of sqrt(469)), A010729 (repeat 7, 9).

A176520 Decimal expansion of (63+3*sqrt(469))/14.

Original entry on oeis.org

9, 1, 4, 0, 6, 5, 8, 8, 2, 0, 2, 2, 3, 0, 8, 1, 8, 2, 8, 9, 5, 4, 1, 8, 5, 9, 4, 7, 1, 6, 9, 5, 3, 7, 9, 9, 2, 3, 1, 0, 5, 6, 8, 0, 4, 3, 5, 9, 7, 1, 2, 3, 8, 2, 8, 2, 0, 6, 5, 9, 8, 7, 8, 0, 1, 9, 4, 8, 4, 5, 0, 1, 5, 2, 5, 1, 9, 0, 5, 7, 7, 7, 0, 9, 5, 5, 5, 1, 3, 9, 8, 1, 8, 0, 8, 5, 1, 0, 6, 2, 9, 1, 3, 5, 0

Offset: 1

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of (63+3*sqrt(469))/14 is A010729 preceded by 9.

			(63+3*sqrt(469))/14 = 9.14065882022308182895...

Cf. A176443 (decimal expansion of sqrt(469)), A010729 (repeat 7, 9).

RealDigits[(63+3*Sqrt[469])/14,10,120][[1]] (* Harvey P. Dale, Jan 20 2013 *)

(3*sqrt(469)+63)/14 \\ Charles R Greathouse IV, Apr 25 2016

cp's OEIS Frontend

A171472 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 7, a(1) = 30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A306289 The smallest prime factor of numbers greater than 1 and coprime to 6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A173512 a(n) = 8*n + 4 + n mod 2.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

Formula

A134965 a(1)=3, a(n) = a(n-1) + 7 + 2*mod(n-1, 2) for n>=2.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

PARI

Formula

Extensions

A176442 Decimal expansion of (21+sqrt(469))/6.

Views

Author

Keywords

Comments

Examples

Crossrefs

A176520 Decimal expansion of (63+3*sqrt(469))/14.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A171472 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 7, a(1) = 30.