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Search: a213246 -id:a213246
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A083420
a(n) = 2*4^n - 1.
+10
54
1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Sum of divisors of 4^n. - Paul Barry, Oct 13 2005
Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007
If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016
To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017
Largest odd factors of A147590. - César Aguilera, Jan 07 2020
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig 11.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9, 2016.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).
FORMULA
G.f.: (1+2*x)/((1-x)*(1-4*x)).
E.g.f.: 2*exp(4*x)-exp(x).
With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - Paul Barry, May 19 2003
From Benoit Cloitre, Jun 18 2004: (Start)
a(n) = (-16^n/2)*B(2n, 1/4)/B(2n) where B(n, x) is the n-th Bernoulli polynomial and B(k) = B(k, 0) is the k-th Bernoulli number.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)
a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - Reinhard Zumkeller, Feb 07 2006
a(n) = Stirling2(2*(n+1), 2). - Zerinvary Lajos, Dec 06 2006
a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - Vincenzo Librandi, Dec 30 2010
a(n) = A001576(n+1) - 2*A001576(n). - Brad Clardy, Mar 26 2011
a(n) = 6*A002450(n) + 1. - Roderick MacPhee, Jul 06 2012
a(n) = A000203(A000302(n)). - Michel Marcus, Jan 20 2014
a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - Wesley Ivan Hurt, Mar 14 2015
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - Peter Bala, Feb 06 2019
a(n) = A147590(n)/A000079(n). - César Aguilera, Jan 07 2020
a(n) = numerator(zeta_star({2}_(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - Roudy El Haddad, Feb 22 2022
MAPLE
seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011
MATHEMATICA
2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)
PROG
(Magma) [2*4^n-1 : n in [0..30]]; // Wesley Ivan Hurt, Mar 14 2015
(PARI) a(n)=2*4^n-1 \\ Charles R Greathouse IV, Sep 24 2015
(Haskell)
a083420 = subtract 1 . (* 2) . (4 ^) -- Reinhard Zumkeller, Dec 22 2015
CROSSREFS
Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Cf. A000302.
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Apr 29 2003
STATUS
approved
A213243
Number of nonzero elements in GF(2^n) that are cubes.
+10
11
1, 1, 7, 5, 31, 21, 127, 85, 511, 341, 2047, 1365, 8191, 5461, 32767, 21845, 131071, 87381, 524287, 349525, 2097151, 1398101, 8388607, 5592405, 33554431, 22369621, 134217727, 89478485, 536870911, 357913941, 2147483647, 1431655765, 8589934591, 5726623061, 34359738367, 22906492245
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).
FORMULA
a(n) = M / gcd( M, 3 ), where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014, verified by Robert Israel, Apr 22 2016: (Start)
a(n) = (-1)*((-2+(-1)^n)*(-1+2^n))/3.
a(n) = 5*a(n-2) - 4*a(n-4).
G.f.: x*(2*x^2+x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)). (End)
E.g.f.: (-1 + exp(x) - 2*exp(3*x) + 2*exp(4*x))*exp(-2*x)/3. - Ilya Gutkovskiy, Apr 22 2016
MAPLE
A213243:=n->(2^n-1)/gcd(2^n-1, 3): seq(A213243(n), n=1..50); # Wesley Ivan Hurt, Aug 23 2014
MATHEMATICA
Table[(2^n - 1)/GCD[2^n - 1, 3], {n, 50}] (* Vincenzo Librandi, Mar 16 2013 *)
LinearRecurrence[{0, 5, 0, -4}, {1, 1, 7, 5}, 40] (* Harvey P. Dale, Jan 05 2017 *)
PROG
(Magma) [(2^n - 1) / GCD (2^n - 1, 3): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013
(PARI) a(n)=(2^n-1)/gcd(2^n-1, 3) \\ Edward Jiang, Sep 04 2014
CROSSREFS
Cf. A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 07 2012
STATUS
approved
A213245
Number of nonzero elements in GF(2^n) that are 7th powers.
+10
8
1, 3, 1, 15, 31, 9, 127, 255, 73, 1023, 2047, 585, 8191, 16383, 4681, 65535, 131071, 37449, 524287, 1048575, 299593, 4194303, 8388607, 2396745, 33554431, 67108863, 19173961, 268435455, 536870911, 153391689, 2147483647, 4294967295, 1227133513, 17179869183, 34359738367, 9817068105
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,9,0,0,-8).
FORMULA
a(n) = M / gcd( M, 7 ), where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014, verified by Robert Israel, Nov 20 2016: (Start)
a(n) = 9*a(n-3)-8*a(n-6).
G.f.: x*(4*x^4+6*x^3+x^2+3*x+1) / ( (x-1)*(2*x-1)*(x^2+x+1)*(4*x^2+2*x+1) ). (End)
MAPLE
A213245:=n->(2^n-1)/gcd(2^n-1, 7): seq(A213245(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014
MATHEMATICA
Table[(2^n - 1)/GCD[2^n - 1, 7], {n, 60}] (* Vincenzo Librandi, Mar 16 2013 *)
PROG
(Magma) [(2^n - 1) / GCD (2^n - 1, 7): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013
(PARI) a(n)=(2^n-1)/gcd(2^n-1, 7) \\ Edward Jiang, Sep 04 2014
CROSSREFS
Cf. A213243 (cubes), A213244 (5th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).
KEYWORD
nonn,easy
AUTHOR
Joerg Arndt, Jun 07 2012
STATUS
approved
A213247
Number of nonzero elements in GF(2^n) that are 11th powers.
+10
8
1, 3, 7, 15, 31, 63, 127, 255, 511, 93, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 95325, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 97612893, 2147483647, 4294967295, 8589934591, 17179869183, 34359738367, 68719476735
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = M / GCD( M, 11 ) where M=2^n-1.
From Colin Barker, Aug 24 2014: (Start)
a(n) = 1025*a(n-10)-1024*a(n-20).
G.f.: x*(512*x^18 +768*x^17 +896*x^16 +960*x^15 +992*x^14 +1008*x^13 +1016*x^12 +1020*x^11 +1022*x^10 +93*x^9 +511*x^8 +255*x^7 +127*x^6 +63*x^5 +31*x^4 +15*x^3 +7*x^2 +3*x +1) / (1024*x^20 -1025*x^10 +1).
(End)
a(n) = (2^n - 1)/11 if n is divisible by 10, 2^n - 1 otherwise. - Robert Israel, Aug 24 2014
MAPLE
A213247:=n->(2^n-1)/igcd(2^n-1, 11): seq(A213247(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014
MATHEMATICA
Table[(2^n - 1)/GCD[2^n - 1, 11], {n, 50}] (* Vincenzo Librandi, Mar 16 2013 *)
PROG
(Magma) [(2^n - 1) / GCD (2^n - 1, 11): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013
(PARI) { for(n=1, 36, if(n%10, a=2^n-1, a=(2^n-1)/11); print1(a, ", ")) } \\ K. Spage, Aug 23 2014
CROSSREFS
Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213248 (13th powers).
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 07 2012
STATUS
approved
A213244
Number of nonzero elements in GF(2^n) that are 5th powers.
+10
7
1, 3, 7, 3, 31, 63, 127, 51, 511, 1023, 2047, 819, 8191, 16383, 32767, 13107, 131071, 262143, 524287, 209715, 2097151, 4194303, 8388607, 3355443, 33554431, 67108863, 134217727
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = M / GCD( M, 5 ) where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014: (Start)
a(n) = 17*a(n-4)-16*a(n-8).
G.f.: x*(8*x^6+12*x^5+14*x^4+3*x^3+7*x^2+3*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(x^2+1)*(4*x^2+1)).
(End)
MAPLE
A213244:=n->(2^n-1)/gcd(2^n-1, 5): seq(A213244(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014
MATHEMATICA
Table[(2^n - 1)/GCD[2^n - 1, 5], {n, 50}] (* Vincenzo Librandi, Mar 16 2013 *)
PROG
(Magma) [(2^n - 1) / GCD (2^n - 1, 5): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013
(PARI) a(n)=(2^n-1)/gcd(2^n-1, 5) \\ Edward Jiang, Sep 04 2014
CROSSREFS
Cf. A213243 (cubes), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 07 2012
STATUS
approved
A213248
Number of nonzero elements in GF(2^n) that are 13th powers.
+10
7
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 315, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 1290555, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591, 17179869183, 34359738367, 5286113595
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = M / gcd( M, 13 ) where M=2^n-1.
Conjectures from Colin Barker, Aug 24 2014: (Start)
a(n) = 4097*a(n-12)-4096*a(n-24).
G.f.: x*(2048*x^22 +3072*x^21 +3584*x^20 +3840*x^19 +3968*x^18 +4032*x^17 +4064*x^16 +4080*x^15 +4088*x^14 +4092*x^13 +4094*x^12 +315*x^11 +2047*x^10 +1023*x^9 +511*x^8 +255*x^7 +127*x^6 +63*x^5 +31*x^4 +15*x^3 +7*x^2 +3*x +1) / (4096*x^24 -4097*x^12 +1). (End)
MAPLE
A213248:=n->(2^n-1)/gcd(2^n-1, 13): seq(A213248(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014
MATHEMATICA
Table[(2^n - 1)/GCD[2^n - 1, 13], {n, 40}] (* Vincenzo Librandi, Mar 17 2013 *)
PROG
(Magma) [(2^n - 1) / GCD (2^n - 1, 13): n in [1..40]]; // Vincenzo Librandi, Mar 17 2013
(PARI) a(n)=(2^n-1)/gcd(2^n-1, 13) \\ Edward Jiang, Sep 04 2014
CROSSREFS
Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers).
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 07 2012
STATUS
approved
A228147
Triangle T(n,k), read by rows: T(n,k) is the denominator of (1+2^(n-k+1))/(1-2^(k+1)).
+10
2
1, 1, 1, 1, 3, 7, 1, 1, 7, 5, 1, 3, 7, 3, 31, 1, 1, 7, 5, 31, 21, 1, 3, 7, 15, 31, 63, 127, 1, 1, 7, 5, 31, 7, 127, 85, 1, 3, 7, 3, 31, 63, 127, 51, 511, 1, 1, 7, 5, 31, 21, 127, 85, 511, 341, 1, 3, 7, 15, 31, 63, 127, 15, 511, 1023, 2047, 1, 1, 7, 5, 31
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The numerators are given in A228146.
The first diagonal is A213243, the second diagonal is A213244, the third diagonal is A213246, the fourth diagonal is A213247.
LINKS
Vincenzo Librandi, Rows n = 0..100, flattened
EXAMPLE
Triangle begins:
1;
1,1;
1,3,7;
1,1,7,5;
1,3,7,3,31;
1,1,7,5,31,21;
1,3,7,15,31,63,127;
1,1,7,5,31,7,127,85;
1,3,7,3,31,63,127,51,511;
1,1,7,5,31,21,127,85,511,341;
1,3,7,15,31,63,127,15,511,1023,2047;
1,1,7,5,31,21,127,85,511,341,2047,1365; etc.
MATHEMATICA
a[n_, k_] := Denominator[(1 + 2^(n - k + 1))/(1 - 2^(k + 1))]; Table[a[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
PROG
(Magma) [Denominator((1+2^(n-k+1))/(1-2^(k+1))): k in [0..n], n in [0..11]];
CROSSREFS
Cf. A213243, A213244, A213246, A213247.
KEYWORD
nonn,tabl
AUTHOR
Vincenzo Librandi, Aug 15 2013
STATUS
approved
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