cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A255844 a(n) = 2*n^2 + 6.

Original entry on oeis.org

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614
Offset: 0

Views

Author

Avi Friedlich, Mar 08 2015

Keywords

Comments

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
Equivalently, numbers m such that 2*m-12 is a square.
For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

Links

Crossrefs

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
Cf. A152811: nonnegative numbers of the form 2*m^2-6.
Subsequence of A000378.
Cf. similar sequences listed in A255843.

Programs

  • Magma
    [2*n^2+6: n in [0..50]];
  • Mathematica
    Table[2 n^2 + 6, {n, 0, 50}]
  • PARI
    vector(50, n, n--; 2*n^2+6)
  • Sage
    [2*n^2+6 for n in (0..50)]

Formula

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117950(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Extensions

Corrected and extended by Bruno Berselli, Mar 11 2015

A294774 a(n) = 2*n^2 + 2*n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905
Offset: 0

Views

Author

Bruno Berselli, Nov 08 2017

Keywords

Comments

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:
2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Links

Crossrefs

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

Programs

  • Maple
    seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017
  • Mathematica
    Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)
  • PARI
    Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

Formula

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A154669 Averages k of twin prime pairs such that 2*k^3 + 12*k^2 is a square.

Original entry on oeis.org

12, 282, 642, 1452, 12162, 17292, 34842, 98562, 194682, 233922, 347772, 383682, 410412, 544962, 749082, 1071642, 1302492, 1421292, 1503372, 1685442, 2794242, 3011052, 3235962, 3312732, 3792252, 3875322, 4755522
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

Keywords

Comments

(11,13) is a twin prime pair with average 12; sqrt(2*12^3 + 12*12^2) = 72.

Crossrefs

Cf. A152811.

Programs

  • Maple
    a := proc (n) if isprime(n-1) = true and isprime(n+1) = true and type(sqrt(2*n^3+12*n^2), integer) = true then n else end if end proc: seq(a(n), n = 3 .. 5000000); # Emeric Deutsch, Jan 20 2009
  • Mathematica
    a[n_]:=Sqrt[2*n^3+12*n^2];lst={};Do[If[Floor[a[n]]==a[n],If[PrimeQ[n-1]&&PrimeQ[n+1],AppendTo[lst,n]]],{n,9!}];lst
Select[Mean/@Select[Partition[Prime[Range[400000]],2,1],#[[2]]-#[[1]] == 2&],IntegerQ[Sqrt[2#^3+12#^2]]&] (* Harvey P. Dale, Sep 05 2017 *)

Extensions

Extended by Emeric Deutsch, Jan 20 2009
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