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.

Showing 1-4 of 4 results.

A085636 Erroneous version of A046142.

Original entry on oeis.org

1, 7, 33, 185, 553, 1233, 2321, 3913, 6105, 8993, 12673, 17241, 22793, 29425, 37233, 46313, 56761, 68673, 82145, 97273, 114153, 132881, 153553, 176265, 201113, 228193, 257601, 289433, 323785, 360753, 400433, 442921, 488313, 536705
Offset: 1

Views

Author

Keywords

A085601 a(n) = 2 * (4^n + 2^n) + 1.

Original entry on oeis.org

5, 13, 41, 145, 545, 2113, 8321, 33025, 131585, 525313, 2099201, 8392705, 33562625, 134234113, 536903681, 2147549185, 8590065665, 34360000513, 137439477761, 549756862465, 2199025352705, 8796097216513, 35184380477441
Offset: 0

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jul 07 2003

Keywords

Comments

1. Begin with a square tile.
2. Place square tiles on each edge to form a diamond shape.
3. Count the tiles: a(0) = 5.
4. Add tiles to fill the enclosing square.
5. Go to step 2.

Links

Crossrefs

Cf. A028403, A046142, A056640, A064412.
Cf. A343175 (essentially the same).

Programs

  • Mathematica
    Table[2(4^n+2^n)+1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{5,13,41},30] (* Harvey P. Dale, Dec 30 2017 *)
  • PARI
    first(n) = Vec((5 - 22*x + 20*x^2)/(1 - 7*x + 14*x^2 - 8*x^3) + O(x^n)) \\ Iain Fox, Dec 30 2017

Formula

From R. J. Mathar, Apr 20 2009: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: -(5 - 22*x + 20*x^2)/((x - 1)*(2*x - 1)*(4*x - 1)).
(End)
E.g.f.: e^x + 2*(e^(2*x) + e^(4*x)). - Iain Fox, Dec 30 2017

Extensions

Edited by Franklin T. Adams-Watters and Don Reble, Aug 15 2006

A254473 24-hedral numbers: a(n) = (2*n + 1)*(8*n^2 + 14*n + 7).

Original entry on oeis.org

7, 87, 335, 847, 1719, 3047, 4927, 7455, 10727, 14839, 19887, 25967, 33175, 41607, 51359, 62527, 75207, 89495, 105487, 123279, 142967, 164647, 188415, 214367, 242599, 273207, 306287, 341935, 380247, 421319, 465247, 512127, 562055, 615127, 671439, 731087
Offset: 0

Views

Author

Luciano Ancora, Mar 26 2015

Keywords

Comments

This sequence is very close to the A046142 sequence: a(n) is asymptotic to A046142(n) as n tends to infinity.
The formula for A046142, the Haüy rhombic dodecahedral number, is remarkably similar, (2*n-1)*(8*n^2-14*n+7), where the first factor of the dodecahedral formula has "+1" versus "-1" in the 24-hedral formula, and the second factor of the former has "-14n" versus the latter of "+14n". Note that the rhombic dodecahedron has 24 faces, further explaining the relationship. The difference of these sequences is diff(n)=72*n^2 + 14. - Peter M. Chema, Jan 09 2016

Links

Crossrefs

Cf. A000447, A016755, A046142.

Programs

  • Magma
    [(2*n+1)*(8*n^2+14*n+7): n in [0..40]]; // Bruno Berselli, Mar 27 2015
  • Maple
    seq((2*n + 1)*(8*n^2 + 14*n + 7), n=0..100); # Robert Israel, Jan 11 2016
  • Mathematica
    Table[(2 n + 1) (8 n^2 + 14 n + 7), {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {7, 87, 335, 847}, 40]
  • PARI
    vector(40, n, n--; (2*n+1)*(8*n^2+14*n+7)) \\ Bruno Berselli, Mar 27 2015
  • Sage
    [(2*n+1)*(8*n^2+14*n+7) for n in (0..40)] # Bruno Berselli, Mar 27 2015

Formula

G.f.: (7 + 59*x + 29*x^2 + x^3)/(1 - x)^4.
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) - a(n-4).
a(n) = -A046142(-n) with A046142(0) = -7.
a(n) = 6*Sum_{k=0..n} (2*k+1)^2 + (2*n+1)^3. - Robert FERREOL, Nov 13 2023

A102250 Indices of semiprime Haüy rhombic dodecahedral numbers.

Original entry on oeis.org

2, 3, 4, 6, 12, 15, 16, 22, 34, 36, 51, 66, 87, 99, 100, 106, 117, 139, 141, 159, 166, 169, 174, 177, 180, 192, 201, 205, 232, 274, 282, 307, 337, 339, 342, 367, 370, 372, 379, 381, 411, 412, 429, 430, 432, 439, 444, 454, 460, 471, 477, 507, 510, 517, 555, 577
Offset: 1

Views

Author

Jonathan Vos Post, Feb 18 2005

Keywords

Comments

Because the Haüy rhombic dodecahedral numbers are A046142(n) = (2*n-1)(8*n^2-14*n+7) no Haüy rhombic dodecahedral number can be prime.
Integers n such that both (2*n-1) and (8*n^2-14*n+7) are primes.
Integers n such that (2*n-1)*(8*n^2-14*n+7) is an element in the intersection of A046142 and A001358.

Examples

			a(3) = 4 because the 3rd Haüy rhombic dodecahedral number is A046142(3) = (2*4-1)(8*4^2-14*4+7) = 553 and because 553 = 7 * 79 is a semiprime.

References

  • R.-J. Haüy, Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées, 1784.
  • H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.

Links

Crossrefs

Cf. A001358, A046142, A085601, A102130.

Programs

  • Magma
    [n: n in [0..600] | IsPrime(2*n-1) and IsPrime(8*n^2-14*n+7)]; // Vincenzo Librandi, Sep 22 2012
  • Mathematica
    Select[ Range[1000], PrimeQ[2# - 1] && PrimeQ[8#^2 - 14# + 7] &]
Select[Range[1000],AllTrue[{2#-1,8#^2-14#+7},PrimeQ]&] (* Harvey P. Dale, Apr 13 2025 *)
Showing 1-4 of 4 results.

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