A001068 -id:A001068 - cp's OEIS frontend

A087099 Partial sums of A063914.

1, 3, 6, 11, 16, 24, 31, 42, 51, 65, 76, 93, 106, 126, 141, 164, 181, 207, 226, 255, 276, 308, 331, 366, 391, 429, 456, 497, 526, 570, 601, 648, 681, 731, 766, 819, 856, 912, 951, 1010, 1051, 1113, 1156, 1221, 1266, 1334, 1381, 1452, 1501, 1575, 1626

Offset: 1

Jeremy Gardiner, Aug 10 2003

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

Cf. A086848, A087098, A001068.

With[{nn=50},Accumulate[Riffle[Range[1,2*nn,2],3*Range[0,nn]+2]]] (* Harvey P. Dale, Jul 25 2013 *)

From Chai Wah Wu, Feb 02 2021: (Start)
a(n) = 2*a(n-2) - a(n-4) for n > 4.
G.f.: x*(x^3 + x^2 + 2*x + 1)/((x - 1)^2*(x + 1)^2). (End)

A364414 Numbers k with the property that the second part of the symmetric representation of sigma(k) is an octagon of width 1 and one of its vertices is also the central vertex of the first valley of the largest Dyck path of the diagram.

Original entry on oeis.org

21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 147, 153, 159, 171, 177, 183, 189, 201, 207, 213, 219, 231, 237, 243, 249, 261, 267, 273, 279, 291, 297, 303, 309, 321, 327, 333, 339, 351, 357

Offset: 1

Omar E. Pol, Jul 23 2023

nonn

Conjecture 1: These are the numbers > 9 that are congruent to {3, 9, 21, 27} mod 30.
Conjecture 2: These are the terms > 9 of A016945 except the terms ending in 5.
Conjecture 3: The polygon mentioned in the definition is an "S"-shaped concave octagon.
Conjecture 4: Every term of this sequence has as nearest neighbor a term of A091999.
Conjecture 5: The terms of A091999 greater than 2 are the numbers k with the property that the first part of the symmetric representation of sigma(k) is an octagon.
Conjecture 6: The octagon mentioned in the definition shares at least an edge with the octagon mentioned in conjecture 5.
Also the row numbers of the triangle A364639 where the rows start with [0, 0, 1, 0, -1]. - Omar E. Pol, Aug 23 2023

			The symmetric representation of sigma(21) in the first quadrant looks like this:
   _ _ _ _ _ _ _ _ _ _ _
  |_ _ _ _ _ _ _ _ _ _ _|
                        |
                        |
                        |_ _ _
                        |_ _  |_
                            |_ _|_
                                | |_
                                |_  |
                                  | |
                                  |_|_ _ _ _
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          |_|
.
Its second part is an octagon of width 1 and one of its vertices is also the central vertex of the first valley of the largest Dyck path of the structure, so 21 is in the sequence.
Note that 10 is not in the sequence because the second part of the symmetric representation of sigma(10) is an octagon of width 1 in accordance with the definition but none of its vertices is the central vertex of the first valley of the largest Dyck path of the diagram.

Subsequence of A016945.
Cf. A000203, A001068, A045572, A091999, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A244145, A245092, A249223, A249351
(widths), A262626, A362866, A364639, A365081.

A187230 Rank transform of the sequence floor(5n/4); complement of A187231.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 11, 14, 15, 17, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 59, 62, 63, 65, 66, 69, 70, 72, 73, 75, 77, 79, 80, 83, 84, 85, 87, 89, 91, 93, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 112, 113, 114, 117, 119, 120, 122, 124, 125, 127, 128, 131, 132, 134, 135, 138, 139, 141, 142, 144, 146, 148, 149, 151, 153, 154, 156, 159, 160, 161, 163, 165

Offset: 1

Clark Kimberling, Mar 07 2011

nonn

See A187224.

Cf. A187224, A187231.

seqA=Table[Floor[5n/4],{n,1,220}] (*A001068*)
seqB=Table[n,{n,1,220}];(*A000027*)
jointRank[{seqA_,seqB_}]:={Flatten@Position[#1,{,1}],Flatten@Position[#1,{,2}]}&[Sort@Flatten[{{#1,1}&/@seqA,{#1,2}&/@seqB},1]];
limseqU=FixedPoint[jointRank[{seqA,#1[[1]]}]&,jointRank[{seqA,seqB}]][[1]] (*A187230*)
Complement[Range[Length[seqA]],limseqU] (*A187231*)
(*by Peter J. C. Moses, Mar 07 2011*)

A047421 Floor(8n/7).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75

Offset: 0

N. J. A. Sloane

Up to the offset identical to A004777, cf formula. - M. F. Hasler, Oct 06 2014

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A032766, A004773, A001068, A047226, A047368, A004777.

Table[Floor[8 n/7], {n, 0, 80}] (* Bruno Berselli, Oct 06 2014 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,8},70] (* Harvey P. Dale, Mar 06 2016 *)

PARI
```
a(n)=n\7+n \\ M. F. Hasler, Oct 06 2014
```

a(n) = A004777(n+1). - M. F. Hasler, Oct 06 2014
G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6) / (1 - x - x^7 + x^8). [Bruno Berselli, Oct 06 2014]
a(n) = n + floor(n/7) = a(n-1) + a(n-7) - a(n-8). [Bruno Berselli, Oct 06 2014]

More terms from Ray Chandler, Sep 05 2004

Restored to version of early 2008 by M. F. Hasler, Oct 06 2014

cp's OEIS Frontend

A087099 Partial sums of A063914.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A364414 Numbers k with the property that the second part of the symmetric representation of sigma(k) is an octagon of width 1 and one of its vertices is also the central vertex of the first valley of the largest Dyck path of the diagram.

Views

Author

Keywords

Comments

Examples

Crossrefs

A187230 Rank transform of the sequence floor(5n/4); complement of A187231.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A047421 Floor(8n/7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions