A158463 -id:A158463 - cp's OEIS frontend

A188386 a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

11, 13, 47, 37, 107, 73, 191, 121, 299, 181, 431, 253, 587, 337, 767, 433, 971, 541, 1199, 661, 1451, 793, 1727, 937, 2027, 1093, 2351, 1261, 2699, 1441, 3071, 1633, 3467, 1837, 3887, 2053, 4331, 2281, 4799, 2521, 5291, 2773, 5807, 3037, 6347, 3313, 6911, 3601

Offset: 1

Gary Detlefs, Mar 29 2011

Denominators are listed in A033931.
A027446 appears to be divisible by a(n).
The sequence lists also the largest odd divisors of 3*m^2-1 (A080663) for m>1. In fact, for m even, the largest odd divisor is 3*m^2-1 itself; for m odd, the largest odd divisor is (3*m^2-1)/2. From this follows the second formula given in Formula field. - Bruno Berselli, Aug 27 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A033931 (denominators), A001008, A001711, A027446, A002805, A003154, A080663, A158463, A213998.

import Data.Ratio ((%), numerator)
a188386 n = a188386_list !! (n-1)
a188386_list = map numerator $ zipWith (-) (drop 3 hs) hs
   where hs = 0 : scanl1 (+) (map (1 %) [1..])
-- Reinhard Zumkeller, Jul 03 2012

[Numerator((3*n^2+6*n+2)/((n*(n+1)*(n+2)))): n in [1..50]]; // Vincenzo Librandi, Mar 30 2011

seq((3-(-1)^n)*(3*n^2+6*n+2)/4, n=1..100);

Table[(3 - (-1)^n)*(3*n^2 + 6*n + 2)/4, {n, 40}] (* Wesley Ivan Hurt, Jan 29 2017 *)
Numerator[#[[4]]-#[[1]]]&/@Partition[HarmonicNumber[Range[0,50]],4,1] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{11,13,47,37,107,73},50] (* Harvey P. Dale, Dec 31 2017 *)

a(n) = numerator((3*n^2+6*n+2)/(n*(n+1)*(n+2))).
a(n) = (3-(-1)^n)*(3*n^2+6*n+2)/4.
a(2n+1) = A158463(n+1), a(2n) = A003154(n+1).
G.f.: -x*(11+13*x+14*x^2-2*x^3-x^4+x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Apr 09 2011
a(n) = numerator of coefficient of x^3 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011
H(n+3) = 3/2 + 2*f(n)/((n+2)*(n+3)), where f(n) = Sum_{k=0..n}((-1)^k*binomial(-3,k)/(n+1-k)). - Gary Detlefs, Jul 17 2011
a(n) = A213998(n+2,2). - Reinhard Zumkeller, Jul 03 2012
Sum_{n>=1} 1/a(n) = c*(tan(c) - cot(c)/2) - 1/2, where c = Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022

A331952 a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.

Original entry on oeis.org

-1, 0, 2, 6, 11, 18, 26, 36, 47, 60, 74, 90, 107, 126, 146, 168, 191, 216, 242, 270, 299, 330, 362, 396, 431, 468, 506, 546, 587, 630, 674, 720, 767, 816, 866, 918, 971, 1026, 1082, 1140, 1199, 1260, 1322, 1386, 1451, 1518, 1586, 1656, 1727, 1800, 1874, 1950, 2027

Offset: 0

Paul Curtz, Feb 02 2020

a(n+1) is once in the hexagonal spiral in A330707. a(n+2) is twice in the same spiral.
a(n) has one odd followed by three evens.
Difference table:
  -1, 0, 2, 6, 11, 18, 26, 36, ... = a(n)
   1, 2, 4, 5,  7,  8, 10, 11, ... = A001651(n+1)
   1, 2, 1, 2,  1,  2,  1,  2, ... = A000034.

			G.f. = -1 + 2*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 26*x^6 + 36*x^7 + 47*x^8 + ... - _Michael Somos_, Sep 08 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for two-way infinite sequences

Cf. A000034, A001651, A158463, 6*A014105, 2*A003154, A152746(n+1), A008585(1+n).

Equals 2 less than A084684, 1 less than A077043, and 1 more than A276382(n-1). - Greg Dresden, Feb 22 2020

a:=[-1,0,2,6]; [n le 4 select a[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..45]]; // Marius A. Burtea, Feb 02 2020

LinearRecurrence[{2, 0, -2, 1}, {-1, 0, 2, 6}, 100] (* Amiram Eldar, Feb 02 2020 *)
a[n_] := Floor[(n^2 - 1)*3/4]; (* Michael Somos, Sep 08 2023 *)

Vec(-(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 03 2020

{a(n) = (n^2 - 1)*3\4}; /* Michael Somos, Sep 08 2023 */

a(-n) = a(n).
a(20+n) - a(n) = 30*(10+n).
a(2+n) = a(n) + 3*(1+n), a(0)=-1 and a(1)=0.
a(4*n) = 12*n^2 - 1, a(1+4*n) = 6*n*(1+2*n), a(2+4*n) = 2 + 12*n*(1+n), a(3+4*n) = 6*(1+n)*(1+2*n) for n>= 0.
From Colin Barker, Feb 02 2020: (Start)
G.f.: -(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3.
a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.
(End)
E.g.f.: (1/8)*(exp(x)*(6*x^2 + 6*x - 7) - exp(-x)). - Stefano Spezia, Feb 02 2020 after Colin Barker
a(n) = floor((n^2 - 1)*3/4). - Michael Somos, Sep 09 2023

a(42)-a(52) from Stefano Spezia, Feb 02 2020

A065532 a(n) = 48*n^2 - 1.

Original entry on oeis.org

-1, 47, 191, 431, 767, 1199, 1727, 2351, 3071, 3887, 4799, 5807, 6911, 8111, 9407, 10799, 12287, 13871, 15551, 17327, 19199, 21167, 23231, 25391, 27647, 29999, 32447, 34991, 37631, 40367, 43199, 46127, 49151, 52271, 55487, 58799, 62207, 65711, 69311, 73007, 76799

Offset: 0

Labos Elemer, Nov 28 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A158463.

[48*n^2 - 1: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[(1-50*x-47*x^2)/(x-1)^3,{x,0,40}],x] (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3,-3,1},{-1,47,191},40] (* Harvey P. Dale, Dec 13 2017 *)

PARI
```
A065532(n)=48*n^2-1
```

From Vincenzo Librandi, Jul 08 2012: (Start)
G.f.: (1 - 50*x - 47*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 19 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) - 1)/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(48*x^2 + 48*x - 1).
a(n) = A158463(2*n). (End)

Better description from Randall L Rathbun, Jan 19 2002

Offset changed from 1 to 0 by Harry J. Smith, Oct 21 2009

A158543 a(n) = 144*n^2 - 12.

Original entry on oeis.org

132, 564, 1284, 2292, 3588, 5172, 7044, 9204, 11652, 14388, 17412, 20724, 24324, 28212, 32388, 36852, 41604, 46644, 51972, 57588, 63492, 69684, 76164, 82932, 89988, 97332, 104964, 112884, 121092, 129588, 138372, 147444, 156804, 166452, 176388, 186612, 197124, 207924

Offset: 1

Vincenzo Librandi, Mar 21 2009

The identity (24*n^2 - 1)^2 - (144*n^2 - 12)*(2*n)^2 = 1 can be written as A158544(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158463, A158544.

I:=[132, 564, 1284]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012

LinearRecurrence[{3, -3, 1}, {132, 564, 1284}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
144*Range[40]^2-12 (* Harvey P. Dale, Oct 20 2012 *)

for(n=1, 40, print1(144*n^2 - 12", ")); \\ Vincenzo Librandi, Feb 14 2012

From Vincenzo Librandi, Feb 14 2012: (Start)
G.f.: -x*(132 + 168*x - 12*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 06 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)))/24.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)) - 1)/24. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 12*(exp(x)*(12*x^2 + 12*x - 1) + 1).
a(n) = 12*A158463(n). (End)

Comment rewritten by R. J. Mathar, Oct 16 2009

A158462 a(n) = 36*n^2 - 6.

Original entry on oeis.org

30, 138, 318, 570, 894, 1290, 1758, 2298, 2910, 3594, 4350, 5178, 6078, 7050, 8094, 9210, 10398, 11658, 12990, 14394, 15870, 17418, 19038, 20730, 22494, 24330, 26238, 28218, 30270, 32394, 34590, 36858, 39198, 41610, 44094, 46650, 49278, 51978, 54750, 57594, 60510

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (12*n^2 - 1)^2 - (36*n^2 - 6)*(2*n)^2 = 1 can be written as A158463(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A140811, A158463.

I:=[30, 138, 318]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 12 2012

LinearRecurrence[{3, -3, 1}, {30, 138, 318}, 50] (* Vincenzo Librandi, Feb 12 2012 *)
36Range[50]^2-6 (* Harvey P. Dale, Jul 19 2025 *)

for(n=1, 40, print1(36*n^2-6", ")); \\ Vincenzo Librandi, Feb 12 2012

G.f.: 6*x*(5 + 8*x - x^2)/(1-x)^3. - Bruno Berselli, Aug 27 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 12 2012
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(6))*Pi/sqrt(6))/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 6*(exp(x)*(6*x^2 + 6*x - 1) + 1).
a(n) = 6*A140811(n). (End)

A193051 Primes p such that 12p^2-1 and 16p^3-1 are also primes.

Original entry on oeis.org

2, 3, 17, 29, 107, 167, 173, 599, 1667, 1889, 2129, 3407, 3539, 3797, 3863, 5189, 6779, 6983, 7529, 8849, 11399, 11519, 11657, 12227, 12437, 12809, 13217, 14153, 15227, 16223, 16607, 17609, 21683, 21863, 22193, 23789, 25127

Offset: 1

Juri-Stepan Gerasimov, Jul 15 2011

Primes p such that 3*(2p)^2-1 (see A089681) and 2*(2p)^3-1 are primes.

			For p=2, 2 is a prime number, 12*2^2-1=47 is a prime number and 16*2^3-1=127 is a prime number.
For p=3, 3 is a prime number, 12*3^2-1=109 is a prime number and 16*3^3-1=431 is a prime number.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A158463.

[p: p in PrimesUpTo(26000)|IsPrime(12*p^2-1) and IsPrime(16*p^3-1)]; // Vincenzo Librandi, Apr 10 2013

fQ[n_] := PrimeQ[12 n^2 - 1] && PrimeQ[16 n^3 - 1]; Select[ Prime@ Range@ 3000, fQ] (* Robert G. Wilson v, Aug 08 2011 *)
Select[Prime[Range[5000]], PrimeQ[12 #^2 - 1] && PrimeQ[16 #^3 - 1]&] (* Vincenzo Librandi, Apr 10 2013 *)

cp's OEIS Frontend

A188386 a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

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A331952 a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.

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A065532 a(n) = 48*n^2 - 1.

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A158543 a(n) = 144*n^2 - 12.

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A158462 a(n) = 36*n^2 - 6.

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A193051 Primes p such that 12*p^2-1 and 16*p^3-1 are also primes.

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Author

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Comments

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Programs

Magma

Mathematica

A193051 Primes p such that 12p^2-1 and 16p^3-1 are also primes.