A005997 -id:A005997 - 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

A123610 Triangle read by rows, where T(n,k) = (1/n)Sum_{d|(n,k)} phi(d) binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 39, 68, 39, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 100, 392, 618, 392, 100, 8, 1, 1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1, 1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1, 1, 11, 275, 2475, 9900

Offset: 0

Paul D. Hanna, Oct 03 2006

A variant of the triangle A047996 of circular binomial coefficients.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,  10,    4,    1;
  1,  5,  20,   20,    5,    1;
  1,  6,  39,   68,   39,    6,    1;
  1,  7,  63,  175,  175,   63,    7,    1;
  1,  8, 100,  392,  618,  392,  100,    8,   1;
  1,  9, 144,  786, 1764, 1764,  786,  144,   9,  1;
  1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1;
  ...
Example of column g.f.s are:
column 1: 1/(1 - x)^2;
column 2: Ser([1, 1, 3, 1]) / ((1 - x)^2*(1 - x^2)^2) = g.f. of A005997;
column 3: Ser([1, 2, 11, 26, 30, 26, 17, 6, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 -x^3)^2);
column 4: Ser([1, 3, 28, 94, 240, 440, 679, 839, 887, 757, 550, 314, 148, 48, 11, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2);
where Ser() denotes a polynomial in x with the given coefficients, as in Ser([1, 1, 3, 1]) = (1 + x + 3*x^2 + x^3).

Paul D. Hanna, Rows n = 0..45, flattened.
Petros Hadjicostas, Proofs of some formulae for g.f.'s of this sequence.

Cf. Columns: A005997, A123613, A123614, A123615, A123616.
Cf. A123611 (row sums), A123612 (antidiagonal sums), A123617 (central terms).
Cf. A123618, A123619, A047996 (variant), A128545.

T[, 0] = 1; T[n, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]* Binomial[n/#, k/#]^2, 0]&]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

{T(n,k)=if(k==0,1,(1/n)*sumdiv(n,d,if(gcd(k,d)==d, eulerphi(d)*binomial(n/d,k/d)^2,0)))}

T(2*n+1, n) = (2*n + 1)*A000108(n)^2 = (2*n + 1)*((2*n)!/(n!(n+1)!))^2 = A000891(n) for n >= 0.
Row sums are 2*A047996(2*n,n) = 2*A003239(n) for n > 0.
Row sums equal the row sums of triangle A128545.
For n >= 1, the g.f. of column n has the form: P_n(x)/(Product_{m=1..n} (1 - x^m)^2), where P_n(x) is a polynomial with n^2 coefficients such that the sum of the coefficients is P_n(1) = (2*n - 1)!.
From Petros Hadjicostas, Oct 24 2017: (Start)
Proofs of the following formulae can be found in the links.
G.f.: Sum_{n>=1, k>=0} T(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(f(x^s,y^s)), where phi(s) is Euler's totient function at s, f(x,y) = (sqrt(g(x,y)) + 1 -(1 + y)*x)/2, and g(x,y) = 1 - 2*(1 + y)*x + (1 - y)^2*x^2. (Term T(0,0) is not used in this g.f.)
Row g.f.: Sum_{k>=0} T(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*R(n/d, y^d), where R(m, y) = [z^m] (1 + (1 + y)*z + y*z^2)^m. (End)

A123613 Column 3 of triangle A123610.

Original entry on oeis.org

1, 4, 20, 68, 175, 392, 786, 1440, 2475, 4036, 6292, 9464, 13805, 19600, 27200, 36996, 49419, 64980, 84238, 107800, 136367, 170696, 211600, 260000, 316881, 383292, 460404, 549460, 651775, 768800, 902066, 1053184, 1223915, 1416108, 1631700, 1872792, 2141581

Offset: 0

Paul D. Hanna, Oct 03 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,6,-9,12,-9,6,-6,4,-1).

Cf. A123610 (triangle); columns: A005997, A123614, A123615, A123616.

CoefficientList[Series[(1 + 2*x + 11*x^2 + 26*x^3 + 30*x^4 + 26*x^5 + 17*x^6 + 6*x^7 + x^8)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 16 2017 *)
LinearRecurrence[{4,-6,6,-9,12,-9,6,-6,4,-1},{1,4,20,68,175,392,786,1440,2475,4036},40] (* Harvey P. Dale, Apr 22 2019 *)

{a(n)=polcoeff(truncate(Ser([1,2,11,26,30,26,17,6,1]))/((1-x)^2*(1-x^2)^2*(1-x^3)^2 +x*O(x^n)),n)}

G.f.: P_3(x) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2), with P_3(1) = 5!, where P_3(x) = (1+2*x+11*x^2+26*x^3+30*x^4+26*x^5+17*x^6+6*x^7+x^8).

A123614 Column 4 of triangle A123610.

Original entry on oeis.org

1, 5, 39, 175, 618, 1764, 4420, 9900, 20439, 39325, 71603, 124215, 207076, 333200, 520272, 790704, 1173805, 1705725, 2432375, 3409175, 4704846, 6400900, 8596484, 11407500, 14972643, 19452069, 25034835, 31936975, 40410504, 50740800, 63257408, 78330560

Offset: 0

Paul D. Hanna, Oct 03 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,12,-12,4,12,-22,12,4,-12,12,-4,-4,4,-1).

Cf. A123610 (triangle); columns: A005997, A123613, A123615, A123616.

CoefficientList[Series[(1 + 3*x + 28*x^2 + 94*x^3 + 240*x^4 + 440*x^5 + 679*x^6 + 839*x^7 + 887*x^8 + 757*x^9 + 550*x^10 + 314*x^11 + 148*x^12 + 48*x^13 + 11*x^14 + x^15)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 16 2017 *)
LinearRecurrence[{4,-4,-4,12,-12,4,12,-22,12,4,-12,12,-4,-4,4,-1},{1,5,39,175,618,1764,4420,9900,20439,39325,71603,124215,207076,333200,520272,790704},40] (* Harvey P. Dale, Feb 04 2023 *)

{a(n)=polcoeff(truncate(Ser([1,3,28,94,240,440,679,839,887,757,550,314,148,48,11,1]))/ ((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2 +x*O(x^n)),n)}

G.f.: P_4(x) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2), with P_4(1) = 7!, where P_4(x) = (1+3*x+28*x^2+94*x^3+240*x^4+440*x^5+679*x^6+839*x^7+ 887*x^8+757*x^9+550*x^10+314*x^11+148*x^12+48*x^13+11*x^14+x^15).

A123615 Column 5 of triangle A123610.

Original entry on oeis.org

1, 6, 63, 392, 1764, 6352, 19404, 52272, 127413, 286286, 601203, 1192464, 2252432, 4078368, 7116336, 12018704, 19718181, 31521798, 49228487, 75274584, 112911880, 166423400, 241382700, 344962800, 486301725, 676932006, 931282191, 1267259168

Offset: 0

Paul D. Hanna, Oct 03 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A123610 (triangle); columns: A005997, A123613, A123614, A123616.

CoefficientList[Series[(1 + 4*x + 50*x^2 + 262*x^3 + 930*x^4 + 2566*x^5 + 5795*x^6 + 11156*x^7 + 18699*x^8 + 27712*x^9 + 36699*x^10 + 43696*x^11 + 46988*x^12 + 45696*x^13 + 40167*x^14 + 31828*x^15 + 22603*x^16 + 14268*x^17 + 7899*x^18 + 3762*x^19 + 1498*x^20 + 474*x^21 + 110*x^22 + 16*x^23 + x^24)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2*(1 - x^5)^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 16 2017 *)

{a(n)=polcoeff(truncate(Ser([1,4,50,262,930,2566,5795,11156,18699,27712, 36699,43696,46988,45696,40167,31828,22603,14268,7899,3762,1498,474,110,16,1])) /((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)^2 +x*O(x^n)),n)}

G.f.: P_5(x) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)^2), with P_5(1) = 9!, where P_5(x) = (1+4*x+50*x^2+262*x^3+930*x^4+2566*x^5+5795*x^6+11156*x^7+ 18699*x^8+27712*x^9+36699*x^10+43696*x^11+46988*x^12+45696*x^13+ 40167*x^14+31828*x^15+22603*x^16+14268*x^17+7899*x^18+3762*x^19+ 1498*x^20+474*x^21+110*x^22+16*x^23+x^24).

A123616 Column 6 of triangle A123610.

Original entry on oeis.org

1, 7, 100, 786, 4420, 19404, 71188, 226512, 644231, 1670015, 4008200, 9009728, 19146090, 38744496, 75117600, 140218218, 253051227, 443056383, 754838884, 1254576400, 2038689796, 3245256396, 5069041432, 7780827600, 11752298725

Offset: 0

Paul D. Hanna, Oct 03 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A123610 (triangle); columns: A005997, A123613, A123614, A123615.

{a(n)=polcoeff(truncate(Ser([1,5,85,581,2763,9987,29644,74546,164629, 324255,579250,946960,1429875,2003713,2620218,3205496,3679773,3967701, 4024087,3837087,3440204,2894878,2283089,1681653,1153208,731684,427027, 226843,108486,45806,16737,5073,1221,211,23,1])) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)^2*(1-x^6)^2 +x*O(x^n)),n)}

G.f.: P_6(x) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))^2, with P_6(1) = 11!, where P_6(x) = (1+5*x+85*x^2+581*x^3+2763*x^4+9987*x^5+29644*x^6+74546*x^7+ 164629*x^8+324255*x^9+579250*x^10+946960*x^11+1429875*x^12+2003713*x^13+ 2620218*x^14+3205496*x^15+3679773*x^16+3967701*x^17+4024087*x^18+ 3837087*x^19+3440204*x^20+2894878*x^21+2283089*x^22+1681653*x^23+ 1153208*x^24+731684*x^25+427027*x^26+226843*x^27+108486*x^28+45806*x^29+ 16737*x^30+5073*x^31+1221*x^32+211*x^33+23*x^34+x^35).

A323724 a(n) = n(2(n - 2)*n + (-1)^n + 3)/4.

Original entry on oeis.org

0, 0, 2, 6, 20, 40, 78, 126, 200, 288, 410, 550, 732, 936, 1190, 1470, 1808, 2176, 2610, 3078, 3620, 4200, 4862, 5566, 6360, 7200, 8138, 9126, 10220, 11368, 12630, 13950, 15392, 16896, 18530, 20230, 22068, 23976, 26030, 28158, 30440, 32800, 35322, 37926, 40700

Offset: 0

Stefano Spezia, Jan 25 2019

For n > 1, a(n) is the superdiagonal sum of the matrix M(n) whose permanent is A322277(n).

All the terms of this sequence are even numbers (A005843), but do not end with 4.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000290, A000982, A004526, A005843, A005997, A317614, A322277, A323723, A325516.

Flat(List([0..50], n->(1/2)*(-1 + n)^2*n - (-1 + n)*Int(n/2) + 2*(Int(n/2))^2));

[(1/2)*(-1 + n)^2*n - (-1 + n)*Floor(n/2) + 2*(Floor(n/2))^2: n in [0..50]];

a:=n->(1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*(floor(n/2))^2: seq(a(n), n=0..50);

a[n_] := 1/2 (-1 + n)^2 n - (-1 + n) Floor[n/2] + 2 Floor[n/2]^2; Array[a, 50, 0];
Table[n (2 (n - 2) n + (-1)^n + 3)/4, {n, 0, 50}] (* Bruno Berselli, Feb 06 2019 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,0,2,6,20,40},50] (* Harvey P. Dale, Jan 13 2024 *)

makelist((1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*(floor(n/2))^2, n, 0, 50);

a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*(floor(n/2))^2;

T(i,j,n) = if (i %2, j + n*(i-1), n*i - j + 1);
a(n) = sum(k=1, n-1, T(k,k+1,n)); \\ Michel Marcus, Feb 06 2019

[int((1/2)*(-1 + n)**2*n - (-1 + n)*int(n/2) + 2*(int(n/2))**2) for n in range(0,50)]

O.g.f.: 2*x^2*(1 + x + 3*x^2 + x^3)/((1 - x)^4*(1 + x)^2).
E.g.f.: (1/2)*x*(exp(x)*x*(1 + x) + sinh(x)).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.
a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*floor(n/2)^2.
a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*A004526(n) + 2*A000290(A004526(n)).
a(n) = (n/2)*((n - 1)^2 + 1) for even n; a(n) = (n/2)*(n - 1)^2 otherwise. - Bruno Berselli, Feb 06 2019
a(n) = 2*A004526(n*A000982(n-1)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021
a(n) = 2*A005997(n-1) for n >= 2. - Hugo Pfoertner, Dec 13 2021

Definition by Bruno Berselli, Feb 06 2019

A006009 Number of paraffins.

Original entry on oeis.org

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

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

cp's OEIS Frontend

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

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A123610 Triangle read by rows, where T(n,k) = (1/n)*Sum_{d|(n,k)} phi(d) * binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

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A123613 Column 3 of triangle A123610.

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A123614 Column 4 of triangle A123610.

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A123615 Column 5 of triangle A123610.

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A123616 Column 6 of triangle A123610.

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A323724 a(n) = n*(2*(n - 2)*n + (-1)^n + 3)/4.

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A123610 Triangle read by rows, where T(n,k) = (1/n)Sum_{d|(n,k)} phi(d) binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

A323724 a(n) = n(2(n - 2)*n + (-1)^n + 3)/4.