A158411 -id:A158411 - cp's OEIS frontend

A130130 a(0)=0, a(1)=1, a(n)=2 for n >= 2.

0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Paul Curtz, Aug 01 2007

a(n) is also total number of positive integers below 10^(n+1) requiring 9 positive cubes in their representation as sum of cubes (cf. Dickson, 1939).
A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + a(n) = A002283(n).
a(n) = number of obvious divisors of n. The obvious divisors of n are the numbers 1 and n. - Jaroslav Krizek, Mar 02 2009
Number of colors needed to paint n adjacent segments on a line. - Jaume Oliver Lafont, Mar 20 2009
a(n) = ceiling(n-th nonprimes/n) = ceiling(A018252(n)/A000027(n)) for n >= 1. Numerators of (A018252(n)/A000027(n)) in A171529(n), denominators of (A018252(n)/A000027(n)) in A171530(n). a(n) = A171624(n) + 1 for n >= 5. - Jaroslav Krizek, Dec 13 2009
a(n) is also the continued fraction for sqrt(1/2). - Enrique Pérez Herrero, Jul 12 2010
For n >= 1, a(n) = minimal number of divisors of any n-digit number. See A066150 for maximal number of divisors of any n-digit number. - Jaroslav Krizek, Jul 18 2010
Central terms in the triangle A051010. - Reinhard Zumkeller, Jun 27 2013
Decimal expansion of 11/900. - Elmo R. Oliveira, May 05 2024

Leonard Eugene Dickson, All integers except 23 and 239 are sums of eight cubes, Bulletin of the American Mathematical Society 45 (1939), p. 588-591.
Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000005, A000027, A002283, A018252, A051010, A061439, A066150, A070824, A158411, A171529, A171530, A171624, A181375, A181377, A181379, A181381, A181400, A181402, A181404.

a130130 = min 2
a130130_list = 0 : 1 : repeat 2  -- Reinhard Zumkeller, Jun 27 2013

A130130[0]:=0; A130130[1]:=1; A130130[n_]:=2; (* Enrique Pérez Herrero, Jul 12 2010 *)
A130130[n_]:=ContinuedFraction[Sqrt[1/2],n+1][[n+1]] (* Enrique Pérez Herrero, Jul 12 2010 *)
Join[{0, 1},LinearRecurrence[{1},{2},96]] (* Ray Chandler, Sep 23 2015 *)
PadRight[{0,1},120,{2}] (* Harvey P. Dale, Sep 15 2022 *)

a(n)=min(n,2) \\ Charles R Greathouse IV, Jun 01 2011

G.f.: x*(1+x)/(1-x) = x*(1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 20 2009
a(n) = A000005(n) - A070824(n) for n >= 1.
E.g.f.: 2*exp(x) - x - 2. - Stefano Spezia, May 19 2024

A158799 a(0)=1, a(1)=2, a(n)=3 for n >= 2.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Jaume Oliver Lafont, Mar 27 2009

a(n) = number of neighboring natural numbers of n (i.e., n, n - 1, n + 1). a(n) = number of natural numbers m such that n - 1 <= m <= n + 1. Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2*k + 1 for n >= k + 1. - Jaroslav Krizek, Nov 18 2009
Partial sums of A130716; partial sums give A008486. - Jaroslav Krizek, Dec 06 2009
In atomic spectroscopy, a(n) is the number of P term symbols with spin multiplicity equal to n+1, i.e., there is one singlet-P term (n=0), there are two doublet-P terms (n=1), and there are three P terms for triple multiplicity (n=2) and higher (n>2). - A. Timothy Royappa, Mar 16 2012
a(n+1) is also the domination number of the n-Andrásfai graph. - Eric W. Weisstein, Apr 09 2016
Decimal expansion of 37/300. - Elmo R. Oliveira, May 11 2024
a(n+1) is also the domination number of the n X n rook complement graph. - Eric W. Weisstein, Mar 10 2025

David Applegate, The movie version
Eric Weisstein's World of Mathematics, Andrásfai Graph.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A122553, A130130, A139250, A157532, A158411, A158478, A158515.

Cf. A008486, A130716.

PadRight[{1,2},120,{3}] (* or *) Min[#,3]&/@Range[120] (* Harvey P. Dale, Apr 08 2018 *)

PARI
```
a(n)=if(n>1,3,if(n<0,0,n++))
```

G.f.: (1+x+x^2)/(1-x) = (1-x^3)/(1-x)^2.
a(n) = (n>=0)+(n>=1)+(n>=2).
a(n) = 1 + n for 0 <= n <= 1, a(n) = 3 for n >= 2. a(n) = A157532(n) for n >= 1. - Jaroslav Krizek, Nov 18 2009
E.g.f.: 3*exp(x) - x - 2 = x^2/(2*G(0)) where G(k) = 1 + (k+2)/(x - x*(k+1)/(x + k + 1 - x^4/(x^3 + (k+1)*(k+2)*(k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2012
a(n) = min(n+1,3). - Wesley Ivan Hurt, Apr 16 2014
a(n) = 1 + A130130(n). - Elmo R. Oliveira, May 11 2024

Corrected by Jaroslav Krizek, Dec 17 2009

A158478 Number of colors needed to paint n sectors of a circle.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2

Offset: 0

Jaume Oliver Lafont, Mar 20 2009

nonn

No pair of adjacent sectors can have the same color.
For n > 0: smallest prime factor of A098548(n).
Decimal expansion of 61/4950. - Elmo R. Oliveira, May 05 2024

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010693, A020639, A098548, A158411.

a158478 n = if n < 4 then n else 2 + mod n 2
a158478_list = [0..3] ++ cycle [2,3]
-- Reinhard Zumkeller, Nov 30 2014

Join[Range[0, 1], ConstantArray[{2, 3}, 20]] // Flatten (* Robert Price, Jun 15 2020 *)

PARI
```
a(n)=if(n<4,n,2+n%2)
```

G.f.: x*(1+2*x+2*x^2)/(1-x^2).
E.g.f.: 2*cosh(x) + 3*sinh(x) - 2*(1 + x). - Stefano Spezia, Mar 24 2020
a(n) = a(n-2) for n > 3. - Elmo R. Oliveira, May 05 2024
a(n) = (n mod 2) + (2 mod (n+1)). - Aaron J Grech, Sep 02 2024

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

A278317 Number of neighbors of each new term in a right triangle read by rows.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 4, 4, 3, 2, 2, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2

Offset: 1

Omar E. Pol, Nov 18 2016

To evaluate T(n,k) consider only the neighbors of T(n,k) that are present in the triangle when T(n,k) should be a new term in the triangle.
Apart from the first column and the first two diagonals the rest of the elements are 4's.
For the same idea but for an isosceles triangle see A275015; for a square array see A278290, for a square spiral see A278354; and for a hexagonal spiral see A047931.

			Triangle begins:
0;
1, 2;
2, 3, 2;
2, 4, 3, 2;
2, 4, 4, 3, 2;
2, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 4, 4, 3, 2;
...

Apart from the initial zero, row sums give A004767.
Column 1 is A130130.
Columns > 1 give the terms greater than 1 of A158411.
Right border gives 0 together with A007395, also twice A057427.
Second right border gives A122553.
Cf. A047931, A274912, A274913, A275015, A278290, A278317, A278354,

A158515 Number of colors needed to paint a wheel graph on n nodes.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4

Offset: 0

Jaume Oliver Lafont, Mar 20 2009

Adjacent nodes are not allowed to have the same color.

Antti Karttunen, Table of n, a(n) for n = 0..1001
Eric Weisstein's World of Mathematics, Wheel Graph
Wikipedia, Wheel graph
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A158411, A158478.

PadRight[{0, 1, 2}, 100, {4, 3}] (* Paolo Xausa, Apr 22 2024 *)

PARI
```
a(n)=if(n<4,n,4-n%2)
```

(define (A158515 n) (if (< n 4) n (- 4 (modulo n 2)))) ;; Antti Karttunen, Sep 14 2017

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

A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 05 2017

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
The partial sum of this sequence is A184985.
a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

			For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
V. Manturov, Knot Theory, CRC Press, 2004.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.
J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
A. Banerjee, Knot theory [Foil knot family].
D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0,1,2},120,{1}] (* Harvey P. Dale, Feb 20 2023 *)

makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

PARI
```
a(n) = if(n>2, 1, n);
```

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
A005803(a(n)) = A005096(a(n)) = A000007(n).
A107583(a(n)) = A103775(n+5).
a(n+1) = 2^A185012(n+1), with a(0) = 0.
a(n) = A163985(n) mod A004278(n+1).
a(n) = A157928(n) + A171386(n+1).
a(n) = A063524(n) + A157928(n) + A185012(n).
a(n) = A010701(n) - A141044(n) - A179184(n).
G.f.: (x + x^2 - x^3)/(1 - x).
E.g.f.: (2*exp(x) - 2 + x^2)/2.

A168092 a(n) = number of natural numbers m such that n - 2 <= m <= n + 2.

Original entry on oeis.org

2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Jaroslav Krizek, Nov 18 2009

nonn

Generalization: If a(n,k) = number of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = a(n-1,k) + 1 = n + k for 0 <= n <= k, a(n,k) = a(n-1,k) = 2k + 1 for n >= k + 1 (see, e.g., A158799). a(n) = (2 + n) for 0 <= n <= 2, a(n) = 5 for n >= 3.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A158411. - Jaume Oliver Lafont, Nov 29 2009

G.f.: (2+x+x^2+x^3)/(1-x) = 1/(1-x)+(1-x^4)/(1-x)^2. - Jaume Oliver Lafont, Nov 29 2009

A373005 Array read by ascending antidiagonals: A(n,k) is the maximum possible cardinality of a set of points of diameter at most k-1 in {0,1}^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 2, 3, 2, 1, 0, 1, 2, 4, 4, 2, 2, 0, 1, 2, 5, 6, 4, 2, 1, 0, 1, 2, 6, 8, 7, 4, 2, 0, 0, 1, 2, 7, 10, 11, 8, 4, 2, 1, 0, 1, 2, 8, 12, 16, 14, 8, 4, 2, 2, 0, 1, 2, 9, 14, 22, 22, 15, 8, 4, 2, 1, 0, 1, 2, 10, 16, 29, 32, 26, 16, 8, 4, 2, 0

Offset: 0

Stefano Spezia, May 19 2024

A(n,k) is also the size of the Hamming ball in {0,1}^n of radius (k-1)/2 if k is odd and of the union of two Hamming balls in {0,1}^n of radius k/2-1 whose centers are of Hamming distance 1 if k is even.

			The array begins:
  1, 1, 2, 1,  0,  1,  2,  1, ...
  0, 1, 2, 2,  2,  2,  2,  2, ...
  0, 1, 2, 3,  4,  4,  4,  4, ...
  0, 1, 2, 4,  6,  7,  8,  8, ...
  0, 1, 2, 5,  8, 11, 14, 15, ...
  0, 1, 2, 6, 10, 16, 22, 26, ...
  0, 1, 2, 7, 12, 22, 32, 42, ...
  0, 1, 2, 8, 14, 29, 44, 64, ...
  ...

Noga Alon, Zhihan Jin, and Benny Sudakov, The Helly number of Hamming balls and related problems, arXiv:2405.10275 [math.CO], 2024. See p. 3.
S. L. Bezrukov, Specification of all maximal subsets of the unit cube with respect to given diameter. Problemy Peredachi Informatsii, pages 106-109, 1987. On ResearchGate.
G. Katona, Intersection theorems for systems of finite sets, Acta Mathematica Academiae Scientiarum Hungaricae 15, 329-337 (1964).
Daniel J. Kleitman, On a combinatorial conjecture of Erdös, Journal of Combinatorial Theory, Series A 1, 209-214, (1966).

Cf. A000007 (k=0), A000012 (k=1), A000124 (k=5), A000125 (k=7), A005843 (k=4), A006261 (k=11), A007395 (k=2), A008859 (k=13), A011782 (main diagonal), A014206, A046127 (k=8), A059173, A059174, A130130 (n=1), A158411 (n=2), A373006 (antidiagonal sums).

A[n_,k_]:=If[OddQ[k],Sum[Binomial[n,i],{i,0,(k-1)/2}], Binomial[n-1,k/2-1]+Sum[Binomial[n,i],{i,0,k/2-1}]]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(n,k) = Sum_{i=0..(k-1)/2} binomial(n,i) if k is odd;
A(n,k) = binomial(n-1,k/2-1) + Sum_{i=0..k/2-1} binomial(n,i) if k is even.
A(n,3) = n+1.
A(n,6) = A014206(n-1).
A(n,9) = A000127(n+1).
A(n,10) = A059173(n) for n > 0.
A(n,12) = A059174(n) for n > 0.
A(0,k) = A007877(k) for k > 0.

A210032 a(n)=n for n=1,2,3 and 4; a(n)=5 for n >= 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

A. Timothy Royappa, Mar 16 2012

In atomic spectroscopy, a(n) is the number of D term symbols with spin multiplicity equal to n, i.e., there is one singlet-D term (n=1), and there are two doublet-D terms (n=2), three triple-D terms (n=3), four quartet-D terms (n=4) and five terms for every other D term of multiplicity 5 or higher (n >= 5).

Decimal expansion of 11111/9000. - Arkadiusz Wesolowski, Mar 29 2012

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010716, A101272, A158411, A168092, A128999.

Join[Range@4, Table[5, {83}]] (* Arkadiusz Wesolowski, Mar 29 2012 *)
PadRight[{1,2,3,4},120,{5}] (* Harvey P. Dale, Sep 23 2017 *)

a(n) = min(n,5). - Wesley Ivan Hurt, Apr 16 2014
From Elmo R. Oliveira, Jun 26 2024: (Start)
G.f.: x*(1+x+x^2+x^3+x^4)/(1-x) = x*(1-x^5)/(1-x)^2.
a(n) = 1 + A158411(n-1) = A101272(n+1) - 1 = A168093(n-1) - 2. (End)

cp's OEIS Frontend

A130130 a(0)=0, a(1)=1, a(n)=2 for n >= 2.

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A158799 a(0)=1, a(1)=2, a(n)=3 for n >= 2.

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A158478 Number of colors needed to paint n sectors of a circle.

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

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A278317 Number of neighbors of each new term in a right triangle read by rows.

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A158515 Number of colors needed to paint a wheel graph on n nodes.

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A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

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