A096231 -id:A096231 - cp's OEIS frontend

A164001 Spiral of triangles around a hexagon.

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426

Offset: 1

Omar E. Pol, Oct 27 2009

a(n) is the side length of the n-th triangle in a spiral around a hexagon with side length = 1.
Sequence very similar to A134816, but without repeated terms. Records in A134816. Also records in A000931, the Padovan sequence.
Column k=2 of A242464 (with offset 0). - Alois P. Heinz, May 19 2014
a(n) is the number of bitstrings of length n-1 without two consecutive 0's or three consecutive 1's. - Zachary Stier, Mar 16 2021

Alois P. Heinz, Table of n, a(n) for n = 1..1000
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015. See Conjecture 5.8.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A060006.

LinearRecurrence[{0,1,1},{1,2,3,4},50] (* Harvey P. Dale, Jul 08 2017 *)

If n < 5 then a(n) = n, otherwise a(n) = a(n-2) + a(n-3).
G.f.: -x - 1 + (-x^2 - 2*x - 1)/(-1 + x^2 + x^3). a(n) = A000931(n+4) + A000931(n+5) = A000931(n+7), n > 1. - R. J. Mathar, Oct 29 2009
a(n) ~ 1.67873... * 1.32471...^(n-1) where 1.32471... is the real root of x^3 - x - 1 = 0 (see A060006), and 1.67873... is the real root of 23*x^3 - 46*x^2 + 13*x - 1 = 0. - Ricardo Bittencourt, May 14 2023

A078027 Expansion of (1 - x)/(1 - x^2 - x^3).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426

Offset: 0

N. J. A. Sloane, Nov 17 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., 6 (2003), #03.2.3.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

a:=[1,-1,1];; for n in [4..60] do a[n]:=a[n-2]+a[n-3]; od; a; # G. C. Greubel, Aug 04 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1-x^2-x^3) )); // G. C. Greubel, Aug 04 2019

seq(coeff(series((1-x)/(1-x^2-x^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Aug 04 2019

CoefficientList[Series[(1-x)/(1-x^2-x^3), {x,0,60}], x] (* G. C. Greubel, Aug 04 2019 *)
LinearRecurrence[{0,1,1},{1,-1,1},60] (* Harvey P. Dale, Jun 20 2020 *)

Vec((1-x)/(1-x^2-x^3)+O(x^60)) \\ Charles R Greathouse IV, Sep 23 2012

((1-x)/(1-x^2-x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 04 2019

a(n) is asymptotic to r^(n-2) / (2*r+3) where r = 1.3247179572447..., the real root of x^3 = x + 1. For n >= 4, a(n) = a(n-2) + a(n-3). - Philippe Deléham, Jan 13 2004

a(n) = A182097(n) - A182097(n-1). - R. J. Mathar, Jan 27 2018

A124745 Expansion of (1+x)/(1-x^2+x^3).

Original entry on oeis.org

1, 1, 1, 0, 0, -1, 0, -1, 1, -1, 2, -2, 3, -4, 5, -7, 9, -12, 16, -21, 28, -37, 49, -65, 86, -114, 151, -200, 265, -351, 465, -616, 816, -1081, 1432, -1897, 2513, -3329, 4410, -5842, 7739, -10252, 13581, -17991, 23833, -31572, 41824, -55405, 73396, -97229, 128801

Offset: 0

Paul Barry, Nov 06 2006

Paolo Xausa, Table of n, a(n) for n = 0..1000
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,-1).

Row sums of A124744.

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

LinearRecurrence[{0, 1, -1}, {1, 1, 1}, 100] (* Paolo Xausa, Aug 27 2024 *)

a(n) = Sum_{k=0..n} C(floor(k/2),n-k)*(-1)^(n-k) = (-1)^n*A078027(n).

a(n) = a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 1. - Taras Goy, Mar 24 2019

A228361 The number of all possible covers of L-length line segment by 2-length line segments with allowed gaps < 2.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456

Offset: 0

Philipp O. Tsvetkov, Aug 21 2013

Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Second row of A228360.

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

CoefficientList[Series[(1 - x^2 - x^3)^-1 (1 + x)^2 x^2 , {x, 0, 100}], x]

For n>1, a(n) = A134816(n).
G.f.: x^2*(1+x)^2/(1-x^2-x^3).
a(n) = a(n-2) +a(n-3) for n >= 5.
a(n) = A000931(n+5), n>1. - R. J. Mathar, Sep 02 2013

A020720 Pisot sequences E(7,9), P(7,9).

Original entry on oeis.org

7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081, 922111, 1221537

Offset: 0

David W. Wilson

Colin Barker, Table of n, a(n) for n = 0..1000
S. B. Ekhad, N. J. A. Sloane, and D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

A subsequence of A000931.
See A008776 for definitions of Pisot sequences.
The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

LinearRecurrence[{0, 1, 1}, {7, 9, 12}, 50] (* Jean-François Alcover, Aug 31 2018 *)
CoefficientList[Series[(7 + 9 x + 5 x^2)/(1 - x^2 - x^3), {x, 0, 50}], x] (* Stefano Spezia, Aug 31 2018 *)

a(n) = a(n-2) + a(n-3) for n>=3. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016

G.f.: (7+9*x+5*x^2) / (1-x^2-x^3). - Colin Barker, Jun 05 2016

A133034 First differences of Padovan sequence A000931.

Original entry on oeis.org

-1, 0, 1, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396

Offset: 0

Omar E. Pol, Nov 05 2007

Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A002026.

LinearRecurrence[{0,1,1},{-1,0,1},60] (* Harvey P. Dale, Dec 14 2013 *)

a(n+4) = A000931(n).
G.f.: ( 1-2*x^2 ) / ( -1+x^2+x^3 ). - R. J. Mathar, Sep 11 2011
a(n) = a(n-2) + a(n-3) with a(0) = -1, a(1) = 0, a(2) = 1. - Taras Goy, Mar 24 2019

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A164001 Spiral of triangles around a hexagon.

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A078027 Expansion of (1 - x)/(1 - x^2 - x^3).

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A124745 Expansion of (1+x)/(1-x^2+x^3).

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A228361 The number of all possible covers of L-length line segment by 2-length line segments with allowed gaps < 2.

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A020720 Pisot sequences E(7,9), P(7,9).

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A133034 First differences of Padovan sequence A000931.

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