A029578 -id:A029578 - cp's OEIS frontend

A294178 a(2n) = 2n + 1, a(2n+1) = 6n + 3.

1, 3, 3, 9, 5, 15, 7, 21, 9, 27, 11, 33, 13, 39, 15, 45, 17, 51, 19, 57, 21, 63, 23, 69, 25, 75, 27, 81, 29, 87, 31, 93, 33, 99, 35, 105, 37, 111, 39, 117, 41, 123, 43, 129, 45, 135, 47, 141, 49, 147, 51, 153

Offset: 0

Paul Curtz, Jun 28 2018

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

Cf. A029578, A289296.

LinearRecurrence[{0,2,0,-1},{1,3,3,9},100] (* Paolo Xausa, Nov 13 2023 *)

Vec((1 + 3*x)*(1 + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^40)) \\ Colin Barker, Jun 29 2018

a(n) = 2*a(n-2) - a(n-4).
a(n) = A289296(n+1) - A289296(n).
G.f.: (1 + 3*x)*(1 + x^2) / ((1 - x)^2*(1 + x)^2). - Colin Barker, Jun 29 2018

A329482 Interleave 1 - n + 3n^2, 1 + 3n*(1+n) for n >= 0.

Original entry on oeis.org

1, 1, 3, 7, 11, 19, 25, 37, 45, 61, 71, 91, 103, 127, 141, 169, 185, 217, 235, 271, 291, 331, 353, 397, 421, 469, 495, 547, 575, 631, 661, 721, 753, 817, 851, 919, 955, 1027, 1065, 1141, 1181, 1261

Offset: 0

Paul Curtz, Nov 14 2019

a(n+1) - 2*a(n) = -1, 1, 1, -3, -3, -13, -13, -29, -29, ...
Hexagonal spiral for A000265:
.
                   17--35---9--37
                   /
                 33  17---9--19---5
                 /   /             \
                1   1   3---7---1  21
               /   /   /         \   \
             31  15   5   1---1   9  11
               \   \   \     /   /   /
               15   7   1---3   5  23
                 \   \         /   /
                 29  13---3--11   3
                   \             /
                    7--27--13--25
.
The two sequences are perpendicular.
a(n+1) - a(n) = 0, 2, 4, 4, 8, 6, 12, ... = 2*A029578(n+2).
A003215 is a bisection of 1, 1, 13, 7, 49, 19, 109, 37, ... .

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. A003215, A029578, A056106, A056108.

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 1, 3, 7, 11}, 42] (* Amiram Eldar, Nov 23 2019 *)
Module[{nn=20,a,b},a=Table[1-n+3 n^2,{n,0,nn}];b=Table[1+3n(1+n),{n,0,nn}];Riffle[a,b]] (* Harvey P. Dale, Apr 30 2023 *)

Vec((1 + 4*x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, Nov 15 2019

From Colin Barker, Nov 14 2019: (Start)
G.f.: (1 + 4*x^3 + x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
a(n) = (5 + 3*(-1)^n - 2*(1 + (-1)^n)*n + 6*n^2) / 8.
(End)
E.g.f.: (1/8)*exp(-x)*(3 + 2*x + exp(2*x)*(5 + 4*x + 6*x^2)). - Stefano Spezia, Nov 14 2019 after Colin Barker
a(-n) = 1, 1, 5, 7, 15, 19, ... = interleave 1 + n + 3*n^2, 1 + 3*n*(1+n), both in the spiral.

cp's OEIS Frontend

A294178 a(2n) = 2n + 1, a(2n+1) = 6n + 3.

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A329482 Interleave 1 - n + 3n^2, 1 + 3n*(1+n) for n >= 0.

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cp's OEIS Frontend

A294178 a(2n) = 2*n + 1, a(2n+1) = 6*n + 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A329482 Interleave 1 - n + 3*n^2, 1 + 3*n*(1+n) for n >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294178 a(2n) = 2n + 1, a(2n+1) = 6n + 3.

A329482 Interleave 1 - n + 3n^2, 1 + 3n*(1+n) for n >= 0.