A203286 -id:A203286 - cp's OEIS frontend

A005994 Alkane (or paraffin) numbers l(7,n).

1, 3, 9, 19, 38, 66, 110, 170, 255, 365, 511, 693, 924, 1204, 1548, 1956, 2445, 3015, 3685, 4455, 5346, 6358, 7514, 8814, 10283, 11921, 13755, 15785, 18040, 20520, 23256, 26248, 29529, 33099, 36993, 41211, 45790, 50730, 56070, 61810, 67991

Offset: 0

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

Equals A000217 (1, 3, 6, 10, 15, ...) convolved with A193356 (1, 0, 3, 0, 5, ...). - Gary W. Adamson, Feb 16 2009
F(1,4,n) is the number of bracelets with 1 blue, 4 red and n black beads. If F(1,4,1)=3 and F(1,4,2)=9 taken as a base;
F(1,4,n) = n(n+1)(n+2)/6+F(1,2,n) + F(1,4,n-2). [F(1,2,n) is the number of bracelets with 1 blue, 2 red and n black beads. If F(1,2,1)=2 and F(1,2,2)=4 taken as a base F(1,2,n)=n+1+F(1,2,n-2)]. - Ata Aydin Uslu and Hamdi G. Ozmenekse, Jan 11 2012
a(A254338(n)) = 6 for n > 0. - Reinhard Zumkeller, Feb 27 2015

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 = 0..1000
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
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
Ata Aydin Uslu and Hamdi G. Ozmenekse, Number of bracelets made with 1 blue, 4 red and n black beads.
Ata Aydin Uslu and Hamdi G. Ozmenekse, Number of bracelets made with 1 blue, 2 red and n black beads.
Index entries for linear recurrences with constant coefficients, signature (3, -1, -5, 5, 1, -3, 1).

Cf. A006009, A005997, A005993 (first differences).

Cf. A000217, A005408, A282011.

--  Following Gary W. Adamson.
import Data.List (inits, intersperse)
a005994 n = a005994_list !! n
a005994_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
                   tail $ inits $ tail a000217_list
-- Reinhard Zumkeller, Feb 27 2015

a:= n -> (Matrix([[1, 0$4, 1, 3]]). Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [3, -1, -5, 5, 1, -3, 1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..40); # Alois P. Heinz, Jul 31 2008

LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,3,9,19,38,66,110},50] (* or *) CoefficientList[Series[(1+x^2)/((1-x)^3(1-x^2)^2),{x,0,50}],x] (* Harvey P. Dale, May 02 2011 *)
nn=45;With[{a=Accumulate[Range[nn]],b=Riffle[Range[1,nn,2],0]}, Flatten[ Table[ListConvolve[Take[a,n],Take[b,n]],{n,nn}]]] (* Harvey P. Dale, Nov 11 2011 *)

{a(n)=if(n<-4, n=-5-n); polcoeff( (1+x^2)/((1-x)^3*(1-x^2)^2)+x*O(x^n), n)} /* Michael Somos, Mar 08 2007 */

G.f.: (1+x^2)/((1-x)^3*(1-x^2)^2) = (1+x^2)/((1-x)^5*(1+x)^2).
l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
a(-5-n)=a(n). - Michael Somos, Mar 08 2007
Euler transform of length 4 sequence [3, 3, 0, -1]. - Michael Somos, Mar 08 2007
a(n) = 3a(n-1) - a(n-2) - 5a(n-3) + 5a(n-4) + a(n-5) - 3a(n-6) + a(n-7), with a(0)=1, a(1)=3, a(2)=9, a(4)=19, a(5)=38, a(6)=66, a(7)=110. - Harvey P. Dale, May 02 2011
a(n) = A006009(n)/2 - A000332(n+4) = ((1/2)*Sum_{i=1..n+1} (i+1)*floor((i+1)^2/2)) - binomial(n+4,4). - Enrique Pérez Herrero, May 11 2012
a(n) = (1/48)*(n+1)*(n+3)*((n+2)*(n+4)+3)+1/32*(2*n+5)*(1+(-1)^n). - Yosu Yurramendi, Jun 20 2013
Conjecture: a(n)+a(n+1) = A203286(n+1). - R. J. Mathar, Mar 08 2025

A115264 Diagonal sums of correlation triangle for floor((n+2)/2).

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 17, 21, 32, 39, 55, 66, 89, 105, 136, 159, 200, 231, 284, 325, 392, 445, 528, 595, 697, 780, 903, 1005, 1152, 1275, 1449, 1596, 1800, 1974, 2211, 2415, 2689, 2926, 3240, 3514, 3872, 4186, 4592, 4950, 5408, 5814, 6328, 6786, 7361

Offset: 0

Paul Barry, Jan 18 2006

Diagonal sums of A115263.
This is associated with the root system F4, and can be described using the additive function on the affine F4 diagram:
2--4--3--2--1
a(n-4) seems to be the number of face-magic cubes or order 2 with magic sum n, which means the sum of the 4 numbers at the 4 corners of each of the 6 faces equals n. (The 8 integers at the corners do not need to be distinct; copies by the 48 operations of rotations and flips are counted only once, cf. A203286, A381589. All 8 integers are positive.). E.g., 1=a(4-4) is the cube with magic sum 4, placing 1 at each corner. 1 =a(5-4) is the number of cubes with magic sum 5 obtained by placing 1 at 6 of the 8 corners but 2 at two corners opposite along a space diagonal. - R. J. Mathar, Mar 11 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1,-2,0,2,1,1,-2,-1,1).

For G2, the corresponding sequence is A001399.
For E6, the corresponding sequence is A164680.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210631 for a very similar sequence.

R:=PowerSeriesRing(Integers(), 0); Coefficients(R!( 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Jan 13 2020

CoefficientList[Series[1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)), {x,0,50}], x] (* G. C. Greubel, Jan 13 2020 *)

A115264(n) := block( A099837(n+3)/27 + A056594(n)/16+(-1)^n*(2*n^2+24*n+63)/256 +(6*n^4 +144*n^3+1194*n^2+3960*n+4267)/6912 )$ /* R. J. Mathar, Mar 19 2012 */

my(x='x+O('x^50)); Vec(1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4))) \\ G. C. Greubel, Jan 13 2020

x=PowerSeriesRing(QQ,'x').gen(); 1/((1-x)*(1-x**2)**2*(1-x**3)*(1-x**4))

G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)*(1-x^4)).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} [j<=k]*floor((k-j+2)/2)*[j<=n-2k]*floor((n-2k-j+2)/2).
a(n) = A099837(n+3)/27 + A056594(n)/16 + (-1)^n*(2*n^2 +24*n +63)/256 +(6*n^4 +144*n^3 +1194*n^2 +3960*n +4267)/6912 . - R. J. Mathar, Mar 19 2012

A381589 The number of face-magic cubes with magic sum n and distinct positive integers at the vertices including 1.

Original entry on oeis.org

3, 2, 6, 6, 16, 13, 21, 28, 38, 40, 57, 58, 81, 92, 108, 118, 150, 158, 188, 213, 242, 257, 309, 324, 373, 408, 448, 483, 551, 578, 643, 695, 759, 804, 894, 935, 1023, 1097, 1177, 1243, 1360, 1416, 1528, 1625, 1731, 1816, 1959, 2041, 2181, 2300, 2430, 2541, 2721, 2822, 2992, 3141, 3300, 3441, 3650, 3781, 3985, 4163, 4358, 4526, 4777, 4934

Offset: 18

R. J. Mathar, Mar 12 2025

nonn

The face-magic cubes counted here have 8 distinct positive integers (including 1) at the vertices, and each sum over the 4 vertices of the 6 faces is the same. Cubes obtained by rotations or mirrors of the octahedral point group are counted only once.

			The 3 face-magic cubes with sum 18 are 1 4 5 8 - 6 7 2 3, 1 4 5 8 - 7 6 3 2 and 1 6 3 8 - 7 4 5 2, values at the base and values at the top face separated by a dash.

R. J. Mathar, Illustrations, conjectured g.f.

Cf. A203286, A115264.

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A005994 Alkane (or paraffin) numbers l(7,n).

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A115264 Diagonal sums of correlation triangle for floor((n+2)/2).

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A381589 The number of face-magic cubes with magic sum n and distinct positive integers at the vertices including 1.

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