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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A036672 Number of stereoisomers of acyclic hydrocarbons with n carbon atoms.

Original entry on oeis.org

1, 3, 4, 13, 31, 109, 372, 1446, 5714, 23791, 100827, 438019, 1931818, 8648820, 39178079, 179383748, 828905252, 3861958783, 18125392905, 85631735301, 406977645228, 1944737525915, 9338989516911, 45051405221284, 218236995129380, 1061256971559421
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Comment from Sean A. Irvine, edited by Natan Arie Consigli, Dec 26 2016 : (Start)
This is the counting series for the hypothetical stereo-isomers of all acyclic hydrocarbons that satisfy the octet rule.
A036673 is the variant with triple bonds excluded.
A002986 doesn't count stereoisomers.
The reference gives a three-variable generating function and cycle-index over A4 which can produce both these sequences. There are also dependencies on earlier generating functions.
(End)
Read has incorrect a(10)=27100. - Sean A. Irvine, Nov 20 2020

Examples

			From _M. F. Hasler_, Dec 26 2016: (Start)
For n = 1, there is only a(1) = 1 possibility, CH4.
For n = 2, one has C2H6 (ethane, H3C-CH3), C2H4 (ethylene, H2C=CH2 with a double bond), C2H2 (ethyne, HC≡CH, triple bond), whence a(2) = 3.
For n = 3, one has C3H8 (H3C-CH2-CH3), C3H6 (H2C=CH-CH3, propene), and two C3H4 (H2C=C=CH2, propadiene, and HC≡C-CH3: methylacetylene), thus a(3) = 4. Cyclic molecules like cyclopropane C3H6 and cyclopropropene C3H4 are excluded. (End)
From _Natan Arie Consigli_, Dec 26 2016: (Start)
For n = 4, we have butane, isobutane, 1-butene, cis/trans-2-butene, buta-1,2-diene, buta-1,3-diene, butatriene, isobutylene, but-1-yne, but-2-yne, diacetylene, but-1-en-3-yne.
For n = 5 we have:
- 3 alkanes: pentane, methylbutane and neopentane.
- 17 alkenes: 1-pentene, (E/Z)-2-pentene, 1,2-pentadiene, (E/Z)-1,3-pentadiene, 1,4-pentadiene, 1,2,3-petatriene, penta-1,2,4-triene, pentatetraene, 2-methylbut-1-ene, 2-methylbut-2-ene, 3-methylbut-1-ene, isoprene, 3-methylbuta-1,2-diene, (R/S)-penta-2,3-diene.
-11 alkynes: 1-pentyne, 2-pentyne, pent-1-en-4-yne, (E/Z)-pent-3-en-1-yne, penta-1,2-dien-4-yne, penta-1,4-diyne, penta-1,3-diyne, pent-1-en-3-yne, 3-methylbut-1-yne, 2-methylbut-1-en-3-yne. (End)

Links

Crossrefs

Cf. A000598-A000628, A002986, A010372, ..., A086194, ... A134818, ..., A173289, ..., A261336.
Cf. A107984, A036673.

Extensions

a(10) corrected and more terms from Sean A. Irvine, Nov 20 2020

A354968 Triangle read by rows: T(n, k) = n*k*(n+k)*(n-k)/6.

Original entry on oeis.org

1, 4, 5, 10, 16, 14, 20, 35, 40, 30, 35, 64, 81, 80, 55, 56, 105, 140, 154, 140, 91, 84, 160, 220, 256, 260, 224, 140, 120, 231, 324, 390, 420, 405, 336, 204, 165, 320, 455, 560, 625, 640, 595, 480, 285, 220, 429, 616, 770, 880, 935, 924, 836, 660, 385, 286, 560, 810, 1024
Offset: 2

Views

Author

Ali Sada and Yifan Xie, Jun 14 2022

Keywords

Comments

Given a Pythagorean triple (a,b,c), define S = c^4 - a^4 - b^4. Using Euclid's parameterization (a = 2*n*k, b = n^2 - k^2, c = n^2 + k^2), substituting to get S in terms of n and k gives S = 8*n^2*k^2*((n^2 - k^2))^2, which is a multiple of 288; T(n, k) = sqrt(S/288) = n*k*(n^2 - k^2)/6 = n*k*(n+k)*(n-k)/6.

Examples

			Triangle begins:
  n/k   1    2    3    4    5    6    7
  2     1;
  3     4,   5;
  4    10,  16,  14;
  5    20,  35,  40,  30;
  6    35,  64,  81,  80,  55;
  7    56, 105, 140, 154, 140,  91;
  8    84, 160, 220, 256, 260, 224, 140;
  ...
For n = 3, k = 2, a = 5, b = 12, c = 13. T(3, 2) = sqrt((13^4 - 5^4 - 12^4)/288) = 5.

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 72.

Links

Crossrefs

Cf. A120070 (b leg), A055096 (c hypotenuse).
Cf. A006414 (row sums), A000292 (column 1), A077414 (column 2), A000330 (diagonal), A107984 (transpose), A210440 (diagonal which begins with 4).

Programs

  • Mathematica
    T[n_,k_]:=n*k(n^2-k^2)/6; Table[T[n,k],{n,2,11},{k,n-1}]//Flatten (* Stefano Spezia, Jul 11 2025 *)
  • PARI
    apply( {A354968(n, k=0)=k|| k=n-1-(1-n=ceil(sqrt(8*n-7)/2+.5))*(2-n)\2; k*(n-k)*n*(n+k)\6}, [2..66]) \\ M. F. Hasler, May 08 2025

Formula

G.f.: x^2*y*(1 + x*y - 4*x^2*y + x^3*y + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 11 2025
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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