A073760 -id:A073760 - cp's OEIS frontend

A073762 a(n) = 24*n - 12.

12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 852, 876, 900, 924, 948, 972, 996, 1020, 1044, 1068, 1092, 1116, 1140, 1164, 1188, 1212

Offset: 1

Labos Elemer, Aug 08 2002

Previous name: "Smallest unrelated number belonging to a term of this sequence equals 8."
This is also the list of numbers k such that A259748(k)/k = 5/12. - José María Grau Ribas, Jul 12 2015.
Also the total number of line segments creating a stellated octahedron, where the length of each stellated edge equals n-1, and where the octahedron has 12 edges, each fixed at unit length. - Peter M. Chema, Apr 28 2016

			URSet[12] = {8,9,10} so 12 is here.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016825, A045763, A073758, A073759, A073760, A259748.

[24*n-12: n in [1..60]]; // Vincenzo Librandi, Jun 15 2011

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Do[s=Min[unr[n]]; If[Equal[s, 8], Print[n]], {n, 1, 1000}]
Range[12, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)

a(n)=24*n-12 \\ Charles R Greathouse IV, Jun 14 2011

x='x+O('x^100); Vec(12*(1+x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015

Min{URS[m]} = 8, where UNR[m] = Complement[RRS[m], Divisors[m]].
a(n) = 24*n - 12. - Max Alekseyev, Mar 03 2007
a(n) = 12*A005408(n-1). - Danny Rorabaugh, Oct 22 2015
G.f.: 12*x*(1 + x)/(1 - x)^2. - Ilya Gutkovskiy, Apr 28 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/48. - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 12*(exp(x)*(2*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A113127 Expansion of (1 + x + x^2 + x^3)/(1-x)^2.

Original entry on oeis.org

1, 3, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 0

Paul Barry, Oct 14 2005

Row sums of number triangle A113126.
Equals binomial transform of [1, 2, 1, 0, -1, 2, -3, 4, -5, ...]. - Gary W. Adamson, Feb 14 2009
From 6 on the same as A016825. - R. J. Mathar, Jul 21 2013
The size of a maximal 4-degenerate graph of order n-2 (this class includes 4-trees). - Allan Bickle, Nov 14 2021
Maximum size of an apex graph of order n-2 (an apex graph can be made planar by deleting a single vertex). - Allan Bickle, Nov 14 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016825, A073760, A111284, A131034.

a(n) - a(n-1) = A158411(n+1). - Jaume Oliver Lafont, Mar 27 2009

[4*n-2+2*Binomial(0, n)+Binomial(1, n): n in [0..80]]; // Vincenzo Librandi, Nov 03 2018

CoefficientList[Series[(1 + x + x^2 + x^3) / (1 - x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Nov 03 2018 *)
LinearRecurrence[{2,-1},{1,3,6,10},60] (* Harvey P. Dale, Jul 08 2019 *)

x='x+O('x^66); Vec((1+x+x^2+x^3)/(1-x)^2) \\ Joerg Arndt, May 06 2013

a(n) = 4*n - 2 + 2*binomial(0, n) + binomial(1, n);
a(n) = binomial(n+1, n) + binomial(n, n-1) + binomial(n-1, n-2) + binomial(n-2, n-3).
Row sums of triangle A131034. - Gary W. Adamson, Jun 10 2007
G.f.: (x^2-1)/Q(0), where Q(k)= 4*x - 1 + x*k - x*(x-1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 05 2013
a(n) = A111284(n+1) for n >= 2. - Georg Fischer, Nov 02 2018
a(n) = 4*(n+2) - 10 for n >= 2. - Allan Bickle, Nov 14 2021

A133653 A007318^(-1) * A003261.

Original entry on oeis.org

1, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154

Offset: 1

Gary W. Adamson, Sep 19 2007

It appears this sequence gives the positive integers m such that the sum of the first m Fibonacci numbers divides their product. For example, if n=2 and m=a(2)=6, we have the sum 1+1+2+3+5+8=20 which clearly divides the corresponding product 480. See A175553 for the analogous sequence when using the triangular numbers. Sum_{k=1..n} Fibonacci(k) divides Product_{k=1..n} Fibonacci(k). - John W. Layman, Jul 10 2010

			a(4) = 14 = (1, 3, 3, 1) dot (1, 5, -1, 1) = (1, 15, -3, 1).

Cf. A003261, A007318, A175553.

Essentially the same as A130824, A113127, A111284, A073760, A016825.

Inverse binomial transform of A003261: (1, 7, 23, 63, 159, 383, ...).
Binomial transform of [1, 5, -1, 1, -1, 1, ...].
"1" followed by 2 * [3, 5, 7, 9, 11, ...].
O.g.f.: x*(1+4x-x^2)/(1-x)^2. a(n) = 4n-2, n > 1. - R. J. Mathar, Jun 08 2008
1/(1+1/(6+1/(10+1/(14+1/(...(continued fraction)))))) = (e-1)/2 with e = 2.718281...- Philippe Deléham, Mar 09 2013

More terms from R. J. Mathar, Jun 08 2008

A190035 Number of nondecreasing arrangements of n+2 numbers in 0..3 with the last equal to 3 and each after the second equal to the sum of one or two of the preceding three.

Original entry on oeis.org

5, 7, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230, 234, 238

Offset: 1

R. H. Hardin, May 04 2011

Column 3 of A190041.

			All solutions for n=4:
..1....1....0....3....2....0....1....0....1....1....0....1....1....1
..1....1....3....3....3....1....1....1....2....1....1....2....3....1
..2....1....3....3....3....1....2....1....3....1....1....2....3....1
..2....1....3....3....3....2....3....2....3....2....1....3....3....2
..3....2....3....3....3....2....3....3....3....3....2....3....3....2
..3....3....3....3....3....3....3....3....3....3....3....3....3....3

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A190041, A073760, A016825.

Empirical: a(n) = 4*n - 2 for n>2.
Conjectures from Colin Barker, May 03 2018: (Start)
G.f.: x*(5 - 3*x + x^2 + x^3) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>4.
(End)

cp's OEIS Frontend

A073762 a(n) = 24*n - 12.

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A113127 Expansion of (1 + x + x^2 + x^3)/(1-x)^2.

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A133653 A007318^(-1) * A003261.

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A190035 Number of nondecreasing arrangements of n+2 numbers in 0..3 with the last equal to 3 and each after the second equal to the sum of one or two of the preceding three.

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