A382154 -id:A382154 - cp's OEIS frontend

A319384 a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

1, 5, 9, 21, 29, 49, 61, 89, 105, 141, 161, 205, 229, 281, 309, 369, 401, 469, 505, 581, 621, 705, 749, 841, 889, 989, 1041, 1149, 1205, 1321, 1381, 1505, 1569, 1701, 1769, 1909, 1981, 2129, 2205, 2361, 2441, 2605, 2689, 2861, 2949, 3129, 3221, 3409, 3505, 3701, 3801, 4005, 4109, 4321, 4429, 4649, 4761, 4989, 5105, 5341, 5461

Offset: 0

Paul Curtz, Sep 18 2018

The two bisections A136392(n+1)=1,9,29,61, ... and A201279(n)=5,21,49, ... are in the hexagonal spiral based on 2*n+1:
.
                    67--65--63--61
                    /             \
                  69  33--31--29  59
                  /   /         \   \
                71  35  11---9  27  57
                /   /   /     \   \   \
              73  37  13   1   7  25  55
                  /   /   /   /   /   /
                39  15   3---5  23  53
                  \   \         /   /
                  41  17--19--21  51
                    \             /
                    43--45--47--49
.
A201279(n) - A136892(n) = 20*n.

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

Cf. A001844, A032766, A104585.
In the spiral: A003154(n+1), A080859, A126587, A136392, A201279, A227776.
Partial sums of A382154.

[(6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jan 19 2021

Table[(6 n^2 + 6 n + 5 - (2 n + 1)*(-1)^n)/4, {n, 0, 80}] (* Wesley Ivan Hurt, Jan 07 2021 *)

Vec((1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019

def A319384(n): return (n*(3*n+4)+3 if n&1 else n*(3*n+2)+2)>>1 # Chai Wah Wu, Mar 25 2025

a(2*n) = A136392(n+1), a(2*n+1) = A201279(n).
a(-n) = a(n).
a(2*n) + a(2*n+1) = 6*A001844(n).
a(n) = (6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4. - Wesley Ivan Hurt, Oct 04 2018
G.f.: (1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019
a(n) = A104585(n) + A032766(n+1). - Alex W. Nowak, Jan 08 2021

More terms from N. J. A. Sloane, Mar 23 2025

A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4n if n even or 2n if n odd.

Original entry on oeis.org

1, 2, 6, 6, 16, 10, 24, 14, 32, 18, 40, 22, 48, 26, 56, 30, 64, 34, 72, 38, 80, 42, 88, 46, 96, 50, 104, 54, 112, 58, 120, 62, 128, 66, 136, 70, 144, 74, 152, 78, 160, 82, 168, 86, 176, 90, 184, 94, 192, 98, 200, 102, 208, 106, 216, 110, 224, 114, 232, 118, 240, 122, 248, 126, 256, 130, 264, 134, 272, 138, 280, 142, 288, 146, 296

Offset: 0

N. J. A. Sloane, Mar 23 2025

nonn

Let G denote the 2-dimensional grid obtained from the square grid Z X Z by deleting the vertices with both coordinates odd and the four edges at each of those vertices (see link). G has vertices with valency either 2 (one coordinate even and one odd, indicated by X) or 4 (both coordinates even, indicated by O). The present sequence is the coordination sequence of G with respect to a vertex of valency 2.

See A382154 for further information.

Paolo Xausa, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, The grid G (Each X-node is joined to two O-nodes, and each O-node to four X-nodes.)
N. J. A. Sloane, Illustrates the initial terms of the coordination sequence of G with respect to a vertex of degree 2. E.g. the 6 red vertices labeled 3 correspond to a(3) = 6, and the 16 blue vertices labeled 4 correspond to a(4) = 16.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Partial sums give A382156.

Cf. A382154, A319384.

Join[{1, 2, 6}, Riffle[4*# + 2, 8*(# + 1)]] & [Range[50]] (* Paolo Xausa, Mar 24 2025 *)

def A382155(n): return (1,2,6)[n] if n<3 else n<<(2>>(n&1)) # Chai Wah Wu, Mar 24 2025

G.f.: (-2*x^6+5*x^4+2*x^3+4*x^2+2*x+1)/(1-x^2)^2.

A382156 Partial sums of A382155.

Original entry on oeis.org

1, 3, 9, 15, 31, 41, 65, 79, 111, 129, 169, 191, 239, 265, 321, 351, 415, 449, 521, 559, 639, 681, 769, 815, 911, 961, 1065, 1119, 1231, 1289, 1409, 1471, 1599, 1665, 1801, 1871, 2015, 2089, 2241, 2319, 2479, 2561, 2729, 2815, 2991, 3081, 3265, 3359, 3551, 3649, 3849, 3951, 4159, 4265, 4481, 4591, 4815, 4929, 5161, 5279, 5519

Offset: 0

N. J. A. Sloane, Mar 23 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A319384, A382154, A382155.

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 3, 9, 15, 31, 41, 65}, 100] (* Paolo Xausa, Mar 24 2025 *)

cp's OEIS Frontend

A319384 a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

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A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4n if n even or 2n if n odd.

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A382156 Partial sums of A382155.

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cp's OEIS Frontend

A319384 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

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Magma

Mathematica

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A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4*n if n even or 2*n if n odd.

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Python

Formula

A382156 Partial sums of A382155.

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Mathematica

A319384 a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

A382155 a(n) = (n+1)! if n <= 2; thereafter a(n) = 4n if n even or 2n if n odd.