A002064 -id:A002064 - cp's OEIS frontend

A050914 a(n) = n*3^n + 1.

1, 4, 19, 82, 325, 1216, 4375, 15310, 52489, 177148, 590491, 1948618, 6377293, 20726200, 66961567, 215233606, 688747537, 2195382772, 6973568803, 22082967874, 69735688021, 219667417264, 690383311399, 2165293113022, 6778308875545, 21182215236076, 66088511536555, 205891132094650

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Cf. A002064, A050915.

Equals A036290(n) + 1.

[ n*3^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n*3^n+1,{n,0,30}] (* or *) LinearRecurrence[{7,-15,9},{1,4,19},30] (* Harvey P. Dale, Nov 07 2012 *)

a(n)=n*3^n+1 \\ Charles R Greathouse IV, Oct 07 2015

From Colin Barker, Oct 14 2012: (Start)
a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3).
G.f.: -(6*x^2 - 3*x + 1)/((x-1)*(3*x-1)^2). (End)
E.g.f.: exp(x)*(3*x*exp(2*x) + 1). - Elmo R. Oliveira, Sep 09 2024

A186947 a(n) = 4^n - n*2^n.

Original entry on oeis.org

1, 2, 8, 40, 192, 864, 3712, 15488, 63488, 257536, 1038336, 4171776, 16728064, 67002368, 268206080, 1073250304, 4293918720, 17177640960, 68714758144, 274867945472, 1099490656256, 4398002470912, 17592093769728, 70368551239680, 281474574057472, 1125899067981824

Offset: 0

Paul Barry, Mar 01 2011

Binomial transform of A186948.

Second binomial transform of A186949.

			G.f. = 1 + 2*x + 8*x^2 + 40*x^3 + 192*x^4 + 864*x^5 + 3712*x^6 + ... - _Michael Somos_, Jul 18 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-20,16).

Cf. A000325, A002064, A186948, A186949.

[4^n - n*2^n: n in [0..30]]; // G. C. Greubel, Aug 14 2018

Table[4^n-n 2^n,{n,0,30}] (* or *) LinearRecurrence[{8,-20,16},{1,2,8},30] (* Harvey P. Dale, Apr 23 2017 *)

{a(n) = 2^n * (2^n - n)}; /* Michael Somos, Jul 18 2018 */

G.f.: (1 - 6*x + 12*x^2)/((1 - 2*x)^2*(1 - 4*x)).
a(n) = 4*a(n-1) + 2^n*(n-2), n >= 1. - Vincenzo Librandi, Mar 13 2011
a(n) = 2^n*A000325(n) = 4^n*A002064(-n) for all n in Z. - Michael Somos, Jul 18 2018
From Elmo R. Oliveira, Sep 15 2024: (Start)
E.g.f.: exp(2*x)*(exp(2*x) - 2*x).
a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) for n > 2. (End)

A229580 Number of defective 3-colorings of an n X 2 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

Original entry on oeis.org

1, 6, 40, 224, 1152, 5632, 26624, 122880, 557056, 2490368, 11010048, 48234496, 209715200, 905969664, 3892314112, 16642998272, 70866960384, 300647710720, 1271310319616, 5360119185408, 22539988369408, 94557999988736

Offset: 1

R. H. Hardin, Sep 26 2013

nonn

			Some solutions for n=3:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 0
  0 0   2 0   0 1   0 2   1 0   2 2   1 2   2 1   0 2   1 2
  1 0   0 2   1 2   1 1   2 1   1 0   0 1   0 0   0 0   0 2

R. H. Hardin, Table of n, a(n) for n = 1..210
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022.

Column 2 of A229586.

Empirical: a(n) = 8*a(n-1) - 16*a(n-2) for n>3.
a(n) = 4*a(n-1) + 4^(n-1) for n > 2. - Gerald Hillier, May 01 2018
a(n) = (2n - 1)*2^(2n - 3) for n > 1 [Gerson W. Barbosa]. - Gerald Hillier, May 02 2018
Empirical g.f.: x*(1 - 2*x + 8*x^2) / (1 - 4*x)^2. - Colin Barker, May 02 2018
a(n) = A002064(2n-2) - A002064(2n-3) for n > 1. - Daniel Forgues, Aug 31 2018
Empirical: a(n) = Integral_{t>0} dt/Beta(n-t,n+t) for n > 1. - Gregory Gerard Wojnar, Feb 10 2024

A367004 a(n) is the smallest prime factor of n*2^n+1.

Original entry on oeis.org

3, 3, 5, 5, 7, 5, 3, 3, 11, 7, 13, 13, 3, 3, 17, 17, 5, 11, 3, 3, 23, 13, 5, 5, 3, 3, 7, 29, 31, 17, 3, 3, 47, 19, 37, 37, 3, 3, 41, 41, 13, 23, 3, 3, 11, 5, 7, 7, 3, 3, 53, 7, 5591, 29, 3, 3, 5, 31, 37, 61, 3, 3, 5, 5, 67, 5, 3, 3, 7, 37, 11, 41, 3, 3, 149

Offset: 1

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..914 (terms 1..350 from Robert Israel)

Cf. A002064, A005849, A020639, A049479, A367002, A367003, A367005.

seq(min(numtheory:-factorset(n*2^n+1)), n=1..100); # Robert Israel, Nov 09 2023

Table[FactorInteger[n 2^n+1][[1,1]],{n,80}] (* Harvey P. Dale, Aug 14 2024 *)

a(n) = A020639(A002064(n)).

A029544 Near Cullen numbers: k such that (k+1)*2^k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 5, 6, 13, 26, 65, 66, 86, 114, 133, 186, 294, 445, 866, 1325, 1478, 1823, 2765, 7553, 7943, 10178, 20960, 20964, 21337, 26562, 85374, 96749, 247038

Offset: 1

Henri Lifchitz

Primes in the sequence are 2, 5, 13, 1823, 96749, ... - R. J. Mathar, Oct 15 2011

We can write (k+1)*2^k + 1 = {(k+1)/2}*4^{(k+1)/2} + 1, and when k is odd, this takes the form of a generalized Cullen prime (base 4). These are listed in A007646. In other words, {2*A007646 - 1} gives all the odd terms of this sequence. - Jeppe Stig Nielsen, Oct 16 2019

Steven Harvey, NearCullen and NearWoodall Primes

Cf. A002064, A007646, A230769.

isok(n) = isprime((n+1)*2^n+1); \\ Michel Marcus, Nov 09 2013

Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008

A050915 a(n) = n*4^n + 1.

Original entry on oeis.org

1, 5, 33, 193, 1025, 5121, 24577, 114689, 524289, 2359297, 10485761, 46137345, 201326593, 872415233, 3758096385, 16106127361, 68719476737, 292057776129, 1236950581249, 5222680231937, 21990232555521, 92358976733185

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (9,-24,16).

Cf. A002064.

[ n*4^n+1: n in [0..30]]; // Vincenzo Librandi, Sep 16 2011

CoefficientList[Series[-(12 x^2 - 4 x + 1)/((x - 1) (4 x - 1)^2), {x, 0, 21}], x] (* Michael De Vlieger, Jan 04 2020 *)
Table[n*4^n+1,{n,0,30}] (* or *) LinearRecurrence[{9,-24,16},{1,5,33},30] (* Harvey P. Dale, Sep 18 2024 *)

From Colin Barker, Oct 14 2012: (Start)
a(n) = 9*a(n-1) - 24*a(n-2) + 16*a(n-3).
G.f.: -(12*x^2 - 4*x + 1)/((x-1)*(4*x-1)^2). (End)
E.g.f.: exp(x)*(1 + 4*exp(3*x)*x). - Stefano Spezia, Jan 05 2020

A134081 Triangle T(n, k) = binomial(n, k)((2k+1)*(n-k) +k+1)/(k+1), read by rows.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 12, 8, 1, 5, 22, 26, 11, 1, 6, 35, 60, 45, 14, 1, 7, 51, 115, 125, 69, 17, 1, 8, 70, 196, 280, 224, 98, 20, 1, 9, 92, 308, 546, 574, 364, 132, 23, 1, 10, 117, 456, 966, 1260, 1050, 552, 171, 26, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

			First few rows of the triangle are:
  1;
  2,  1;
  3,  5,   1;
  4, 12,   8,   1;
  5, 22,  26,  11,  1;
  6, 35,  60,  45, 14,  1;
  7, 51, 115, 125, 69, 17, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A002064, A007318, A112295.

Columns: A000027, A000326, A002413, A051740, A051879.

A134081:= func< n,k | Binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1) >;
[A134081(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

T[n_, k_]:= Binomial[n, k]*((2*k+1)*(n-k) +k+1)/(k+1);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

def A134081(n,k): return binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1)
flatten([[A134081(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

Binomial transform of A112295(unsigned).
From G. C. Greubel, Feb 17 2021: (Start)
T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1).
Sum_{k=0..n} T(n, k) = 2^n *n + 1 = A002064(n). (End)

A367007 Number of distinct prime factors of n*2^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 31 2023

nonn

The numbers n*2^n+1 are called Cullen numbers.

Amiram Eldar, Table of n, a(n) for n = 0..865

Cf. A002064, A001221, A367004, A367005, A367006.

Table[PrimeNu[n*2^n + 1], {n, 0, 100}] (* Amiram Eldar, Jan 06 2024 *)

a(n) = omega(n*2^n + 1); \\ Amiram Eldar, Jan 06 2024

a(n) = omega(n*2^n + 1) = A001221(A002064(n)).

A050916 a(n) = n*5^n + 1.

Original entry on oeis.org

1, 6, 51, 376, 2501, 15626, 93751, 546876, 3125001, 17578126, 97656251, 537109376, 2929687501, 15869140626, 85449218751, 457763671876, 2441406250001, 12969970703126, 68664550781251, 362396240234376, 1907348632812501, 10013580322265626, 52452087402343751

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-35,25).

Cf. A002064, A036291.

[ n*5^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n 5^n+1,{n,0,20}] (* or *) LinearRecurrence[{11,-35,25},{1,6,51},20] (* Harvey P. Dale, Sep 15 2011 *)

a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3); a(0)=1, a(1)=6, a(2)=51. - Harvey P. Dale, Sep 15 2011
G.f.: -(20*x^2-5*x+1)/((x-1)*(5*x-1)^2). - Colin Barker, Oct 14 2012
From Elmo R. Oliveira, May 03 2025: (Start)
E.g.f.: exp(x)*(1 + 5*x*exp(4*x)).
a(n) = A036291(n) + 1. (End)

A050917 a(n) = n*6^n + 1.

Original entry on oeis.org

1, 7, 73, 649, 5185, 38881, 279937, 1959553, 13436929, 90699265, 604661761, 3990767617, 26121388033, 169789022209, 1097098297345, 7052774768641, 45137758519297, 287753210560513, 1828079220031489, 11577835060199425, 73123168801259521, 460675963447934977

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-48,36).

Cf. A002064, A036292.

[ n*6^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n 6^n+1,{n,0,40}] (* or *) LinearRecurrence[{13,-48,36},{1,7,73},40] (* Harvey P. Dale, Feb 20 2013 *)

vector(20, n, n--; n*6^n + 1) \\ Michel Marcus, Jun 11 2015

G.f.: -(30*x^2-6*x+1)/((x-1)*(6*x-1)^2). - Colin Barker, Oct 14 2012
a(n) = 13*a(n-1) - 48*a(n-2) + 36*a(n-3); a(0)=1, a(1)=7, a(2)=73. - Harvey P. Dale, Feb 20 2013
From Elmo R. Oliveira, May 03 2025: (Start)
E.g.f.: exp(x)*(1 + 6*x*exp(5*x)).
a(n) = A036292(n) + 1. (End)

cp's OEIS Frontend

A050914 a(n) = n*3^n + 1.

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A186947 a(n) = 4^n - n*2^n.

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A229580 Number of defective 3-colorings of an n X 2 0..2 array connected horizontally and antidiagonally with exactly one mistake, and colors introduced in row-major 0..2 order.

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A367004 a(n) is the smallest prime factor of n*2^n+1.

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A029544 Near Cullen numbers: k such that (k+1)*2^k + 1 is prime.

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PARI

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A050915 a(n) = n*4^n + 1.

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Magma

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A134081 Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.

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A367007 Number of distinct prime factors of n*2^n + 1.

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A134081 Triangle T(n, k) = binomial(n, k)((2k+1)*(n-k) +k+1)/(k+1), read by rows.