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.

User: Benedek Nagy

Benedek Nagy's wiki page.

Benedek Nagy has authored 7 sequences.

A366815 Hyper-Wiener index in diamond nanowires obtained by connecting n unit cells in a sequence.

Original entry on oeis.org

448, 3544, 14294, 40420, 92348, 183208, 328834, 547764, 861240, 1293208, 1870318, 2621924, 3580084, 4779560, 6257818, 8055028, 10214064, 12780504, 15802630, 19331428, 23420588, 28126504, 33508274, 39627700, 46549288, 54340248, 63070494, 72812644
Offset: 1

Views

Author

Benedek Nagy, Oct 24 2023

Keywords

Links

Crossrefs

Cf. A366816.

Programs

  • Magma
    [(338*n^4 + 481*n^3 + 145*n^2 + 416*n - 36)/3 : n in [1..50]]; // Wesley Ivan Hurt, Dec 10 2023
  • Mathematica
    LinearRecurrence[{5, -10, 10, -5, 1}, {448, 3544, 14294, 40420, 92348},50] (* Paolo Xausa, Feb 27 2024 *)
  • PARI
    a(n) = (338*n^4 + 481*n^3 + 145*n^2 + 416*n - 36)/3

Formula

a(n) = (338*n^4 + 481*n^3 + 145*n^2 + 416*n - 36)/3.
G.f.: 2*x*(224 + 652*x + 527*x^2 - 45*x^3 - 6*x^4)/(1 - x)^5. - Stefano Spezia, Oct 24 2023

A366817 Detour index of n body-centered cubic grid unit cells in a row.

Original entry on oeis.org

64, 298, 752, 1476, 2520, 3934, 5768, 8072, 10896, 14290, 18304, 22988, 28392, 34566, 41560, 49424, 58208, 67962, 78736, 90580, 103544, 117678, 133032, 149656, 167600, 186914, 207648, 229852, 253576, 278870, 305784, 334368, 364672, 396746, 430640
Offset: 1

Views

Author

Benedek Nagy, Oct 24 2023

Keywords

Links

Crossrefs

Cf. A273321, A302351, A007290, A032091, A115269, A001386.

Programs

  • Mathematica
    A366817[n_] := (25*n^3 + 180*n^2 - 13*n)/3; Array[A366817, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {64, 298, 752, 1476}, 50] (* Paolo Xausa, May 28 2024 *)
  • PARI
    a(n) = (25*n^3 + 180*n^2 - 13*n)/3 \\ Andrew Howroyd, Oct 24 2023

Formula

a(n) = (25*n^3 + 180*n^2 - 13*n)/3.
From Stefano Spezia, May 28 2024: (Start)
G.f.: 2*x*(32 + 21*x - 28*x^2)/(1 - x)^4.
E.g.f.: exp(x)*x*(192 + 255*x + 25*x^2)/3. (End)

A366816 Wiener index in diamond nanowires obtained by connecting n unit cells in a sequence.

Original entry on oeis.org

232, 1296, 3868, 8624, 16240, 27392, 42756, 63008, 88824, 120880, 159852, 206416, 261248, 325024, 398420, 482112, 576776, 683088, 801724, 933360, 1078672, 1238336, 1413028, 1603424, 1810200, 2034032, 2275596, 2535568, 2814624, 3113440, 3432692, 3773056
Offset: 1

Views

Author

Benedek Nagy, Oct 24 2023

Keywords

Links

Crossrefs

Cf. A366815.

Programs

  • Mathematica
    A366816[n_] := 2/3*((13*n + 9)*13*n + 62)*n; Array[A366816, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {232, 1296, 3868, 8624}, 50] (* Paolo Xausa, Oct 01 2024 *)

Formula

a(n) = (338*n^3 + 234*n^2 + 124*n)/3.

A302351 Hyper-Wiener index of body-centered cubic grid cells in a row.

Original entry on oeis.org

92, 377, 1128, 2700, 5548, 10227, 17392, 27798, 42300, 61853, 87512, 120432, 161868, 213175, 275808, 351322, 441372, 547713, 672200, 816788, 983532, 1174587, 1392208, 1638750, 1916668, 2228517, 2576952, 2964728, 3394700, 3869823, 4393152, 4967842, 5597148
Offset: 1

Views

Author

Benedek Nagy, Jun 09 2018

Keywords

Links

Programs

  • Mathematica
    Table[(25n^4+105n^3+143n^2+171n+108)/6,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{92,377,1128,2700,5548},40] (* Harvey P. Dale, Sep 19 2020 *)
  • PARI
    Vec(x*(92 - 83*x + 163*x^2 - 90*x^3 + 18*x^4 + 50*x^5 - 250*x^6 + 500*x^7 - 500*x^8 + 250*x^9 - 50*x^10) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Jun 11 2018

Formula

a(n) = (25*n^4 + 105*n^3 + 143*n^2 + 171*n + 108)/6 (proven).
From Colin Barker, Jun 11 2018: (Start)
G.f.: x*(92 - 83*x + 163*x^2 - 90*x^3 + 18*x^4 + 50*x^5 - 250*x^6 + 500*x^7 - 500*x^8 + 250*x^9 - 50*x^10) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>11.
(End)

Extensions

Corrected and extended by Harvey P. Dale, Sep 19 2020

A302256 Hyper-Wiener index of rows of unit cells on the face-centered cubic lattice.

Original entry on oeis.org

213, 942, 2956, 7326, 15447, 29038, 50142, 81126, 124681, 183822, 261888, 362542, 489771, 647886, 841522, 1075638, 1355517, 1686766, 2075316, 2527422, 3049663, 3648942, 4332486, 5107846, 5982897, 6965838, 8065192, 9289806, 10648851, 12151822
Offset: 1

Views

Author

Benedek Nagy, Apr 04 2018

Keywords

Comments

This sequence is related to the Wiener-index of the FCC grid (cf. A273322). Now the second order distances are also counted (see definition of Hyper-Wiener index).

Links

Crossrefs

Cf. A273322.

Programs

  • Mathematica
    Table[(81*n^4 + 261*n^3 + 264*n^2 + 540*n + 132)/6, {n, 30}] (* Wesley Ivan Hurt, Jan 20 2024 *)
  • PARI
    a(n) = (81*n^4+261*n^3+264*n^2+540*n+132)/6; \\ Altug Alkan, Apr 04 2018
  • PARI
    Vec(x*(213 - 123*x + 376*x^2 - 164*x^3 + 22*x^4) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Jun 11 2018

Formula

a(n) = (81*n^4+261*n^3+264*n^2+540*n+132)/6. Proved in the Hamzeh Mujahed - Benedek Nagy paper.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5); with a(1)=213, a(2)=942, a(3)=2956, a(4)=7326 and a(5)=15447.
G.f.: x*(213 - 123*x + 376*x^2 - 164*x^3 + 22*x^4) / (1 - x)^5. - Colin Barker, Jun 11 2018

Extensions

a(5) corrected by Altug Alkan, Apr 04 2018

A273322 Wiener index of graphs of f.c.c. unit cells in a line = Sum of distances in face-centered cubic grid unit cells connected in a row.

Original entry on oeis.org

150, 536, 1336, 2712, 4826, 7840, 11916, 17216, 23902, 32136, 42080, 53896, 67746, 83792, 102196, 123120, 146726, 173176, 202632, 235256, 271210, 310656, 353756, 400672, 451566, 506600, 565936, 629736, 698162, 771376, 849540, 932816, 1021366
Offset: 1

Views

Author

Benedek Nagy, May 20 2016

Keywords

Links

Programs

  • Mathematica
    Table[27 n^3 + 45 n^2 + 62 n + 16, {n, 33}] (* or *)
Rest@ CoefficientList[Series[2 x (75 - 32 x + 46 x^2 - 8 x^3)/(1 - x)^4, {x, 0, 33}], x] (* Michael De Vlieger, May 20 2016 *)
LinearRecurrence[{4,-6,4,-1},{150,536,1336,2712},40] (* Harvey P. Dale, Dec 04 2018 *)
  • PARI
    Vec(2*x*(75-32*x+46*x^2-8*x^3)/(1-x)^4 + O(x^50)) \\ Colin Barker, May 20 2016

Formula

a(n) = 27*n^3 + 45*n^2 + 62*n + 16.
From Colin Barker, May 20 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
G.f.: 2*x*(75 - 32*x + 46*x^2 - 8*x^3) / (1-x)^4.
(End)

A273321 Wiener index of graph of b.c.c. unit cells in a line = Sum of distances in a b.c.c. row graph.

Original entry on oeis.org

64, 206, 488, 960, 1672, 2674, 4016, 5748, 7920, 10582, 13784, 17576, 22008, 27130, 32992, 39644, 47136, 55518, 64840, 75152, 86504, 98946, 112528, 127300, 143312, 160614, 179256, 199288, 220760, 243722, 268224, 294316, 322048, 351470, 382632, 415584, 450376, 487058, 525680, 566292
Offset: 1

Views

Author

Benedek Nagy, May 20 2016

Keywords

Links

Programs

  • Mathematica
    Table[(25/3) n^3 + 20 n^2 + (71/3) n + 12, {n, 40}] (* or *)
Rest@ CoefficientList[Series[2 x (32 - 25 x + 24 x^2 - 6 x^3)/(1 - x)^4, {x, 0, 40}], x] (* Michael De Vlieger, May 20 2016 *)
  • PARI
    Vec(2*x*(32-25*x+24*x^2-6*x^3)/(1-x)^4 + O(x^50)) \\ Colin Barker, May 20 2016

Formula

a(n) = (25/3)*n^3 + 20*n^2 + (71/3)*n + 12.
From Colin Barker, May 20 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
O.g.f.: 2*x*(32 - 25*x + 24*x^2 - 6*x^3) / (1 - x)^4. (End)
E.g.f.: (12 + 52*x + 45*x^2 + (25/3)*x^3)*exp(x) - 12. - Benedict W. J. Irwin, May 27 2016

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