A047259 -id:A047259 - cp's OEIS frontend

A080341 Sum of the first n terms that are congruent to 1, 4 or 5 mod 6 (A047259).

1, 5, 10, 17, 27, 38, 51, 67, 84, 103, 125, 148, 173, 201, 230, 261, 295, 330, 367, 407, 448, 491, 537, 584, 633, 685, 738, 793, 851, 910, 971, 1035, 1100, 1167, 1237, 1308, 1381, 1457, 1534, 1613, 1695, 1778, 1863, 1951, 2040, 2131, 2225, 2320, 2417

Offset: 1

Christian Mercat (Integer.Sequence(AT)entrelacs.net), Mar 20 2003

Number of edges needed in a sector of a hexagon of size n paved by rhombi coming from triangular/hexagonal lattices.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A047259.

Accumulate[Select[Range[100],MemberQ[{1,4,5},Mod[#,6]]&]] (* Harvey P. Dale, Aug 16 2012 *)

a(n) = n^2+(n+1)/3 with integer division, that is n mod 3 = 0 : n^2+n/3 n mod 3 = 1 : n^2+(n-1)/3 n mod 3 = 2 : n^2+(n+1)/3.

G.f.: x*(1+3*x+x^2+x^3)/(1-x)^3/(1+x+x^2). [Colin Barker, Feb 12 2012]

A301451 Numbers congruent to {1, 7} mod 9.

Original entry on oeis.org

1, 7, 10, 16, 19, 25, 28, 34, 37, 43, 46, 52, 55, 61, 64, 70, 73, 79, 82, 88, 91, 97, 100, 106, 109, 115, 118, 124, 127, 133, 136, 142, 145, 151, 154, 160, 163, 169, 172, 178, 181, 187, 190, 196, 199, 205, 208, 214, 217, 223, 226, 232, 235, 241, 244, 250, 253, 259, 262, 268

Offset: 1

Bruno Berselli, Mar 21 2018

First bisection of A056991, second bisection of A242660.

The squares of the terms of A174396 are the squares of this sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A274406: numbers congruent to {0, 8} mod 9.
Cf. A193910: numbers congruent to {2, 6} mod 9.
Subsequence of A016777, A026225, A029739, A033627, A047236, A047259, A055047, A055054, A056991, A153053, A187318, A242660.

a := [1,7,10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;

Magma
```
&cat [[9*n+1, 9*n+7]: n in [0..40]];
```

Table[2 (2 n - 1) + (2 n - 3 (1 - (-1)^n))/4, {n, 1, 60}]
{#+1,#+7}&/@(9*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,7,10},60] (* Harvey P. Dale, Nov 08 2020 *)

Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018

[2*(2*n-1)+(2*n-3*(1-(-1)**n))/4 for n in range(1,70)]

[n for n in (1..300) if n % 9 in (1,7)]

O.g.f.: x*(1 + 6*x + 2*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (3 + 8*exp(x) - 11*exp(2*x) + 18*x*exp(2*x))*exp(-x)/4.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 2*(2*n - 1) + (2*n - 3*(1 - (-1)^n))/4. Therefore, for n even a(n) = (9*n - 4)/2, otherwise a(n) = (9*n - 7)/2.
a(2n+1) = A017173(n). a(2n) = A017245(n-1). - R. J. Mathar, Feb 28 2019

A144430 a(n) = 1 + A144429(n).

Original entry on oeis.org

2, 3, 4, 5, 7, 10, 11, 13, 16, 17, 19, 22, 23, 25, 28, 29, 31, 34, 35, 37, 40, 41, 43, 46, 47, 49, 52, 53, 55, 58, 59, 61, 64, 65, 67, 70, 71, 73, 76, 77, 79, 82, 83, 85, 88, 89, 91, 94, 95, 97, 100, 101, 103, 106, 107, 109, 112, 113, 115, 118, 119, 121, 124, 125, 127, 130

Offset: 1

Milton L. Brown (miltbrown(AT)earthlink.net), Oct 13 2008

If you cross out every 3rd term starting at 4 and cross out multiples of 5, the reduced sequence is 2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, ... which starts with 14 primes.

Essentially the same as A047259. - Georg Fischer, Oct 07 2018

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Cf. A144429.

a:=[2,3,4,5,7];; for n in [6..70] do a[n]:=a[n-3]+6; od; a; # Muniru A Asiru, Oct 07 2018

a(n) = a(n-3) + 6, n > 5.

Edited and extended by R. J. Mathar, Oct 15 2008

cp's OEIS Frontend

A080341 Sum of the first n terms that are congruent to 1, 4 or 5 mod 6 (A047259).

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A301451 Numbers congruent to {1, 7} mod 9.

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A144430 a(n) = 1 + A144429(n).

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