A000602 -id:A000602 - cp's OEIS frontend

A006009 Number of paraffins.

4, 16, 48, 108, 216, 384, 640, 1000, 1500, 2160, 3024, 4116, 5488, 7168, 9216, 11664, 14580, 18000, 22000, 26620, 31944, 38016, 44928, 52728, 61516, 71344, 82320, 94500, 108000, 122880, 139264, 157216, 176868, 198288, 221616, 246924, 274360, 304000, 336000

Offset: 1

N. J. A. Sloane

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

4*A007009.

Cf. A005994, A005997.

a:= n-> (Matrix([[0$4,4,16,48,108]]). Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,-4,-4,10,-4,-4,4,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..40); # Alois P. Heinz, Aug 13 2008

a[n_] := 1/16*(2*n^4+12*n^3+24*n^2+2*(9-(-1)^n)*n-3*(-1)^n+3); Array[a, 40] (* Jean-François Alcover, Mar 17 2014 *)

a(n) = 2*(A005994(n) + binomial(n, 4)).
G.f.: 4*x*(1-x^3) / ((1-x)^4*(1-x^2)^2). - Alois P. Heinz, Aug 13 2008
a(n) = Sum_{i=1..n} i*floor(i^2/2). - Enrique Pérez Herrero, Mar 10 2012

A018212 Alkane (or paraffin) numbers l(11,n).

Original entry on oeis.org

1, 5, 25, 85, 255, 651, 1519, 3235, 6470, 12190, 21942, 37854, 63090, 101850, 160050, 245322, 367983, 541035, 781495, 1110395, 1554553, 2146573, 2927145, 3945045, 5260060, 6942988, 9079292, 11769100, 15131700, 19305540

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

nonn

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
Winston C. Yang (paper in preparation).

N. J. A. Sloane, Classic Sequences
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (5, -6, -10, 29, -9, -36, 36, 9, -29, 10, 6, -5, 1).

Cf. A282011.

LinearRecurrence[{5, -6, -10, 29, -9, -36, 36, 9, -29, 10, 6, -5, 1},{1, 5, 25, 85, 255, 651, 1519, 3235, 6470, 12190, 21942, 37854, 63090},30] (* Ray Chandler, Sep 23 2015 *)

G.f.: (1+6*x^2+x^4)/((1-x)^5*(1-x^2)^4). [ N. J. A. Sloane ]
l(c, r) = 1/2 binomial(c+r-3, r) + 1/2 d(c, r), where d(c, r) is binomial((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, binomial((c + r - 4)/2, r/2) if c is even and r is even, binomial((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
a(n) = (1/(2*8!))*(n+2)*(n+4)*(n+6)*(n+8)*((n+1)*(n+3)*(n+5)*(n+7) + 1*3*5*7) - (1/3)*(1/2^6)*(n^3+(27/2)*n^2+56*n+(279/4))*(1/2)*(1-(-1)^n) [Yosu Yurramendi Jun 23 2013]

A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 4, 4, 2, 0, 0, 3, 6, 10, 6, 3, 0, 0, 3, 9, 16, 16, 9, 3, 0, 0, 4, 12, 28, 32, 28, 12, 4, 0, 0, 4, 16, 40, 60, 60, 40, 16, 4, 0, 0, 5, 20, 60, 100, 126, 100, 60, 20, 5, 0, 0, 5, 25, 80, 160, 226, 226, 160, 80, 25, 5, 0, 0, 6, 30, 110, 240

Offset: 0

N. J. A. Sloane

Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.
Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - Yosu Yurramendi, Aug 08 2008
The first seven columns are A004526, A002620, A006584, A032091, A032092, A032093, A032094. Row sums give essentially A032085. - Yosu Yurramendi, Aug 08 2008
From Álvar Ibeas, Jun 01 2020: (Start)
T(n, k) is the sum of odd-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for A034851(n, k) paths and odd for T(n, k) of them.
For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for A034851(n, k) strings and odd for T(n, k) cases.
(End)

			Triangle begins:
  0;
  0 0;
  0 1 0;
  0 1 1 0;
  0 2 2 2 0;
  0 2 4 4 2 0;
  ...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences

Cf. A007318, A034851, A051159.

Essentially the same as A034877.

a034852 n k = a034852_tabl !! n !! k
a034852_row n = a034852_tabl !! n
a034852_tabl = zipWith (zipWith (-)) a007318_tabl a034851_tabl
-- Reinhard Zumkeller, Mar 24 2012

nmax = 12; t[n_?EvenQ, k_?EvenQ] := (Binomial[n, k] - Binomial[n/2, k/2])/ 2; t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_?OddQ, k_?EvenQ] := (Binomial[n, k] - Binomial[(n-1)/2, k/2])/2; t[n_?OddQ, k_?OddQ] := (Binomial[n, k] - Binomial[(n-1)/2, (k-1)/2])/2; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 15 2011, after Yosu Yurramendi *)

Equals (A007318-A051159)/2. - Yosu Yurramendi, Aug 08 2008

T(n, k) = T(n - 1, k - 1) + T(n - 1, k); except when n is even and k odd, in which case T(n, k) = A034851(n, k) = T(n - 1, k - 1) + A034841(n - 1, k) = A034841(n - 1, k - 1) + T(n - 1, k) = C(n, k) / 2. - Álvar Ibeas, Jun 01 2020

More terms from James Sellers, May 04 2000

A067608 Number of structural alkanes with combinatorial diameter n.

Original entry on oeis.org

1, 1, 3, 6, 53, 496, 81096, 35292601, 211275732504203, 5013078952131335869356, 4188494841905497365271738826910705731652978, 13998172580873019733546655911268420464183123192214609601699428961

Offset: 1

Peter Freyd (pjf(AT)saul.cis.upenn.edu), Feb 02 2002

nonn

			There are 53 such alkanes where the longest chain of carbon atoms is of length 5.

R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, April 1989, pp. 278-281.

Cf. A000602, A000628, A067609, A067610.

rid[0]=1; rid[r_] := rid[r]=1+Binomial[rid[r-1]+2, 3]; rd[r_] := rid[r]-rid[r-1]; td[1]=1; td[r_] := If[EvenQ[r], Binomial[rd[r/2]+1, 2], Binomial[rid[(r-1)/2]+3, 4]-rd[(r-1)/2]Binomial[rid[(r-3)/2]+2, 3]-Binomial[rid[(r-3)/2]+3, 4]]; td/@Range[12]

Edited by Dean Hickerson, Feb 11 2002

A067609 Number of stereo alkanes with combinatorial diameter n.

Original entry on oeis.org

1, 1, 3, 6, 58, 861, 373141, 525901096, 92709102076260838, 65190291939775823483614581, 1416591403847441323962646602694082865630539057192433

Offset: 1

Peter Freyd (pjf(AT)saul.cis.upenn.edu), Feb 02 2002

nonn

R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, April 1989, pp. 278-281.

Cf. A000602, A000628, A067608, A067610.

f[n_, k_] := Binomial[n+k-1, k]+Binomial[n, k]; rid[0]=1; rid[r_] := rid[r]=1+f[rid[r-1], 3]; rd[r_] := rid[r]-rid[r-1]; td[1]=1; td[r_] := If[EvenQ[r], Binomial[rd[r/2]+1, 2], f[rid[(r-1)/2], 4]-rd[(r-1)/2]f[rid[(r-3)/2], 3]-f[rid[(r-3)/2], 4]]; td/@Range[12]

Edited by Dean Hickerson, Feb 11 2002

A067610 Number of stereo alkanes not containing a certain forbidden substructure with combinatorial diameter n.

Original entry on oeis.org

1, 1, 3, 6, 57, 838, 319924, 35630889

Offset: 1

Peter Freyd (pjf(AT)saul.cis.upenn.edu), Feb 02 2002

nonn

R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, April 1989, pp. 278-281.

Cf. A000602, A000628, A067609, A067608.

Edited by Dean Hickerson, Feb 11 2002

A135106 Number of physical trees of alkane structures with n carbon vertices.

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 686, 4598, 34872, 295044, 2753958, 28103804, 311216626, 3716341042, 47597786154, 650812077852, 9461423560788, 145724617925326, 2370293673319292, 40600119927220706, 730458115445479734

Offset: 1

R. J. Mathar, Feb 12 2008

nonn

Similar to A000602 (alkane trees with n carbon atoms) but keeping track of the history of attaching carbon atoms (methyls) to the backbone, as if these had been labeled.

			Starting with a(1)=1, one C1 methane, we get a(2)=1, the C1-C2 backbone.
The third can be attached to either C1 ending up with C3-C1-C2, or to C2 ending up with C1-C2-C3, yielding a(3)=2 different propanes.
C4 may be attached to any of C1 to C3 in these two propanes, yielding a(4)=6 different butanes, four of which are linear and two of which are stars.

J. V. Knop, W. R. Muller, K. Szymanski et al., Computer enumeration and generation of physical trees, J. Comput. Chem. vol. 8 no. 4 (1987) pp 549-554.

A135106 := proc(n) local numb, stack,istack,N,i ; numb := array(1..n) ; for i from 1 to n do numb[i] := 0 ; od: stack := array(1..7,1..100) ; stack[1,1]:=2 ; stack[2,1]:=0 ; stack[3,1]:=0 ; stack[4,1]:=0 ; stack[5,1]:=1 ; stack[6,1]:=0 ; stack[7,1]:=1 ; istack := 1 ; while istack <> 0 do for i from 1 to 7 do stack[i,istack+1] := stack[i,istack] ; od: if stack[6,istack] = 3 then istack := istack-1 ; else stack[6,istack] := stack[6,istack]+1 ; stack[1,istack+1] := stack[1,istack]+1 ; N := stack[6,istack] ; if stack[N,istack] <> 0 then stack[N,istack+1] := stack[N,istack+1]-1 ; stack[N+1,istack+1] := stack[N+1,istack+1]+1 ; stack[5,istack+1] := stack[N,istack]*stack[5,istack] ; stack[6,istack+1] := 0 ; stack[7,istack+1] := stack[7,istack]+1 ; numb[stack[7,istack+1]]:=numb[stack[7,istack+1]]+stack[5,istack+1] ; if stack[7,istack+1] <> n then istack := istack+1 ; fi ; fi ; fi ; od: numb[n] ; end: for n from 2 do print( A135106(n)) ; end: # R. J. Mathar, Feb 18 2008

a(n) = A248837(n-2). - Georg Fischer, Oct 23 2018

More terms from R. J. Mathar, Feb 18 2008

a(20)-a(21) from Alois P. Heinz, May 27 2013

A005999 Number of paraffins.

Original entry on oeis.org

1, 2, 6, 11, 23, 38, 64, 95, 141, 194, 266, 347, 451, 566, 708, 863, 1049, 1250, 1486, 1739, 2031, 2342, 2696, 3071, 3493, 3938, 4434, 4955, 5531, 6134, 6796, 7487, 8241, 9026, 9878, 10763, 11719, 12710, 13776, 14879, 16061, 17282, 18586, 19931, 21363, 22838, 24404, 26015, 27721, 29474, 31326, 33227, 35231, 37286

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Cf. A005997.

[1+Floor((n-1)/2)+2*(Binomial(n+1,3)-Binomial(Floor((n+1)/2),3)-Binomial(Ceiling((n+1)/2),3))-(n-1)^2 : n in [1..50]]; // Wesley Ivan Hurt, Sep 16 2014

A005999:=n->1+floor((n-1)/2)+2*(binomial(n+1,3)-binomial(floor((n+1)/2),3)-binomial(ceil((n+1)/2),3))-(n-1)^2: seq(A005999(n), n=1..40); # Wesley Ivan Hurt, Sep 16 2014

A005997[n_] := 1 + Floor[(n-1)/2] + 2*(Binomial[n+1,3] -Binomial[Floor[(n+1)/2],3] - Binomial[Ceiling[(n+1)/2],3]); A005999[n_] := A005997[n] - (n-1)^2; Array[A005999, 100] (* Enrique Pérez Herrero, Apr 22 2012 *)

Vec( (x^5+2*x^4+x^3+x^2+1)/(-1+x)^2/(-1+x^2)^2 + O(x^66) ) \\ Joerg Arndt, Sep 16 2014

G.f.: (x^5+2*x^4+x^3+x^2+1)/((-1+x)^2*(-1+x^2)^2).

a(n) = A005997(n) - (n-1)^2. - Enrique Pérez Herrero, Mar 28 2012

A006004 a(n) = C(n+2,3) + C(n,3) + C(n-1,3).

Original entry on oeis.org

1, 4, 11, 25, 49, 86, 139, 211, 305, 424, 571, 749, 961, 1210, 1499, 1831, 2209, 2636, 3115, 3649, 4241, 4894, 5611, 6395, 7249, 8176, 9179, 10261, 11425, 12674, 14011, 15439, 16961, 18580, 20299, 22121, 24049, 26086, 28235, 30499, 32881, 35384, 38011, 40765

Offset: 1

N. J. A. Sloane

Equals binomial transform of [1, 3, 4, 3, 0, 0, 0, ...]. Example: a(4) = 25 = (1, 3, 3, 1) dot (1, 3, 4, 3) = (1 + 9 + 12 + 3). - Gary W. Adamson, Jul 25 2008

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

A006004:=n->(n^3 - 2*n^2 + 5*n - 2)/2; seq(A006004(n), n=1..50); # Wesley Ivan Hurt, Feb 09 2014

Table[Binomial[n+2,3]+Binomial[n,3]+Binomial[n-1,3],{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,11,25},50] (* Harvey P. Dale, Jun 15 2011 *)

a(n) = (n^3 - 2*n^2 + 5*n - 2)/2 \\ Charles R Greathouse IV, Feb 10 2017

a(n) = (n^3 - 2n^2 + 5n - 2)/2.
G.f.: (x^3+x^2+1)/(x-1)^4. - Harvey P. Dale, Jun 15 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=4, a(2)=11, a(3)=25. - Harvey P. Dale, Jun 15 2011

Terms added by Wesley Ivan Hurt, Feb 09 2014

A005998 Number of paraffins.

Original entry on oeis.org

1, 2, 7, 14, 29, 48, 79, 116, 169, 230, 311, 402, 517, 644, 799, 968, 1169, 1386, 1639, 1910, 2221, 2552, 2927, 3324, 3769, 4238, 4759, 5306, 5909, 6540, 7231, 7952, 8737, 9554, 10439, 11358, 12349, 13376, 14479, 15620, 16841, 18102, 19447, 20834, 22309

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

[1/8*(2*n^3-2*n^2+5*n-(-1)^n*(n+1)+1): n in [1..50]]; // Vincenzo Librandi, Mar 15 2014

a:= n-> (Matrix([[0, -1, -4, -11, -22, -41]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1,1]:
seq(a(n), n=1..38); # Alois P. Heinz, Jul 31 2008

a[n_] := 1/8*(2*n^3-2*n^2+5*n-(-1)^n*(n+1)+1); Array[a, 40] (* Jean-François Alcover, Mar 13 2014 *)
CoefficientList[Series[(x^4 + 2 x^3 + 2 x^2 + 1)/(-1 + x)^2/(-1 + x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,2,7,14,29,48},50] (* Harvey P. Dale, Oct 13 2024 *)

G.f.: x*(x^4+2*x^3+2*x^2+1)/(-1+x)^2/(-1+x^2)^2.

More terms from Vincenzo Librandi, Mar 15 2014

cp's OEIS Frontend

A006009 Number of paraffins.

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A018212 Alkane (or paraffin) numbers l(11,n).

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A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

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A067608 Number of structural alkanes with combinatorial diameter n.

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Mathematica

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A067609 Number of stereo alkanes with combinatorial diameter n.

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Mathematica

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A067610 Number of stereo alkanes not containing a certain forbidden substructure with combinatorial diameter n.

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A135106 Number of physical trees of alkane structures with n carbon vertices.

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Maple

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A005999 Number of paraffins.

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