A176271 -id:A176271 - cp's OEIS frontend

A027694 a(n) = n^2 + n + 9.

9, 11, 15, 21, 29, 39, 51, 65, 81, 99, 119, 141, 165, 191, 219, 249, 281, 315, 351, 389, 429, 471, 515, 561, 609, 659, 711, 765, 821, 879, 939, 1001, 1065, 1131, 1199, 1269, 1341, 1415, 1491, 1569, 1649, 1731, 1815, 1901, 1989, 2079, 2171, 2265, 2361, 2459, 2559

Offset: 0

Patrick De Geest

Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A176271.

with (combinat):seq(fibonacci(3, n)+n+8, n=0..46); # Zerinvary Lajos, Jun 07 2008

Table[n^2+n+9,{n,0,50}]  (* Harvey P. Dale, Feb 07 2011 *)

a(n)=n^2+n+9 \\ Charles R Greathouse IV, Oct 07 2015

For n > 3, a(n) = A176271(n+1,5). - Reinhard Zumkeller, Apr 13 2010
G.f.: (-9 + 16*x - 9*x^2)/(x-1)^3. - R. J. Mathar, Feb 07 2011
a(0) = 9, a(n) = a(n-1) + 2*n. - Vincenzo Librandi, Feb 07 2011
Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(35)/2)/sqrt(35). - Amiram Eldar, Jan 17 2021
From Elmo R. Oliveira, Oct 28 2024: (Start)
E.g.f.: exp(x)*(9 + 2*x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A108309 Consider the triangle of odd numbers where the n-th row has the next n odd numbers. The sequence is the number of primes in the n-th row.

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 3, 4, 4, 5, 3, 4, 6, 4, 6, 6, 4, 6, 7, 6, 8, 7, 5, 8, 9, 8, 7, 8, 9, 8, 9, 10, 10, 8, 10, 12, 5, 12, 12, 13, 9, 11, 11, 9, 13, 14, 9, 14, 14, 10, 10, 19, 14, 12, 12, 12, 12, 16, 15, 16, 15, 13, 18, 16, 16, 12, 16, 17, 15, 16, 18, 14, 15, 20, 18, 19, 14, 19, 20, 18, 16

Offset: 1

Giovanni Teofilatto, Jul 25 2005

Except for the initial term, a(n)>=2 because in the interval 2n-1 of odd numbers there are always at least two primes.
For n>2, this is the same as the number of primes between n^2-n and n^2+n, which is the sum of A089610 and A094189. - T. D. Noe, Sep 16 2008
a(n) = SUM(A010051(A176271(n,k)): 1<=k<=n). - Reinhard Zumkeller, Apr 13 2010
From Pierre CAMI, Sep 03 2014: (Start)
For n>1 a(n)~floor(1/2 + n/log(n)).
The number of primes < n^2 is ~ n^2/2/log(n) by the prime number theorem, as a(n) ~ floor(1/2 + n/log(n)) we have:
n^2/2/log(n) ~ 1 + floor(1/2 + 2/log(2)) + floor(1/2 + 3/log(3)) + floor(1/2 + 4/log(4)) + ... + floor(1/2 + (n-1)/log(n-1)) + floor(1/2 + n/log(n)).
For n=16000 the number of primes < n^2 is 13991985, the sum: 1 + floor(1/2 + 2/log(2)) + floor(1/2 + 3/log(3)) + floor(1/2 + 4/log(4))+ ... + floor(1/2 + (n-1)/log(n-1)) + floor(1/2 + n/log(n)) is 13991101 and (n^2)/(2*log(n)) is 13222671.
So between n^2+n and n^2+3*n there are n odd numbers and ~floor(1/2 + n/log(n)) prime numbers.
The twin primes are of the form T1=n^2+n-1 and T2=n^2+n+1, or n^2+n+T1 and n^2+n+T2 with T1<=2*n-1, or n^2+n+P and n^2+n+P(-2 or +2) with P prime <=2*n-1.
(End)

			Triangle begins:
1: 1 -> 0 primes,
2: 3,5 -> 2 primes,
3: 7,9,11 -> 2 primes,
4: 13,15,17,19 -> 3 primes.

T. D. Noe, Table of n, a(n) for n=1..10000
Pierre CAMI, Table of n, a(n) and Floor(1/2+n/log(n))for n=1..10000

Cf. A014085, A088485.

a108309 = sum . (map a010051) . a176271_row
-- Reinhard Zumkeller, May 24 2012

seq(numtheory:-pi(n^2+n-1)-numtheory:-pi(n^2-n),n=1..100); # Robert Israel, Sep 03 2014

f[n_] := PrimePi[n^2 + n - 1] - PrimePi[n^2 - n]; Table[f[n], {n, 81}] (* Ray Chandler, Jul 26 2005 *)

Edited and extended by Ray Chandler, Jul 26 2005

A065599 If n odd, a(n) = n^2 else a(n) = n.

Original entry on oeis.org

0, 1, 2, 9, 4, 25, 6, 49, 8, 81, 10, 121, 12, 169, 14, 225, 16, 289, 18, 361, 20, 441, 22, 529, 24, 625, 26, 729, 28, 841, 30, 961, 32, 1089, 34, 1225, 36, 1369, 38, 1521, 40, 1681, 42, 1849, 44, 2025, 46, 2209, 48, 2401, 50, 2601, 52, 2809, 54, 3025, 56, 3249

Offset: 0

George E. Antoniou, Dec 01 2001

a(n) = ABS(alternating sum of n-th row of the triangle in A176271), n>0. [Reinhard Zumkeller, Apr 13 2010]

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

Cf. A000217, A176271.

Table[ n^(Mod[n, 2] + 1), {n, 1, 60} ]
LinearRecurrence[{0,3,0,-3,0,1},{0,1,2,9,4,25},80] (* Harvey P. Dale, Sep 10 2017 *)

a(n) = { if (n%2, n^2, n) } \\ Harry J. Smith, Oct 23 2009

a(n) = n^( n (mod 2) + 1 ).
O.g.f.: (x + 2*x^2 + 6*x^3 - 2*x^4 + x^5)/(1 - x^2)^3. - Len Smiley, Dec 04 2001
a(n) = A000217(n)-(-1)^n*A000217(n-1) with A000217(-1)=0. - Bruno Berselli, Jun 07 2013
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - Wesley Ivan Hurt, Apr 26 2021
a(n) = n*((n+1)-(n-1)*(-1)^n)/2. - Aaron J Grech, Sep 03 2024
E.g.f.: x*(cosh(x) + (1 + x)*sinh(x)). - Stefano Spezia, Sep 26 2024

A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.

Original entry on oeis.org

1, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Robert Price, Jun 12 2013

nonn

Appears to be essentially the same as A176271, A140139, A130773, A062545. - R. J. Mathar, Aug 23 2024

Robert Price, Table of n, a(n) for n = 0..100

Cf. A226357, A226359, A222945, A222947, A223133.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

cp's OEIS Frontend

A027694 a(n) = n^2 + n + 9.

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A108309 Consider the triangle of odd numbers where the n-th row has the next n odd numbers. The sequence is the number of primes in the n-th row.

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A065599 If n odd, a(n) = n^2 else a(n) = n.

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A223134 Number of distinct sums i+j+k with i,j,k >= 0, i*j*k <= n.

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A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.