A134816 -id:A134816 - cp's OEIS frontend

A152545 Padovan-Fibonacci triangle, read by rows, where the first column equals the Padovan spiral numbers (A134816), while the row sums equal the Fibonacci numbers (A000045).

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 1, 5, 4, 4, 3, 3, 1, 1, 7, 5, 5, 5, 4, 3, 3, 1, 1, 9, 7, 7, 7, 5, 5, 5, 4, 3, 1, 1, 1, 12, 9, 9, 9, 8, 7, 7, 7, 5, 4, 4, 4, 1, 1, 1, 1, 16, 12, 12, 12, 12, 9, 9, 9, 8, 8, 8, 7, 5, 4, 4, 4, 1, 1, 1, 1, 1, 21, 16, 16, 16, 16, 13, 12, 12, 12, 12, 12, 11, 9, 8

Offset: 0

Paul D. Hanna, Dec 13 2008

The number of terms in each row equal the Padovan spiral numbers (A134816, with offset).

			Triangle begins:
[1],
[1],
[1,1],
[2,1],
[2,2,1],
[3,2,2,1],
[4,3,3,2,1],
[5,4,4,3,3,1,1],
[7,5,5,5,4,3,3,1,1],
[9,7,7,7,5,5,5,4,3,1,1,1],
[12,9,9,9,8,7,7,7,5,4,4,4,1,1,1,1],
[16,12,12,12,12,9,9,9,8,8,8,7,5,4,4,4,1,1,1,1,1],
[21,16,16,16,16,13,12,12,12,12,12,11,9,8,8,8,5,5,5,5,4,1,1,1,1,1,1,1],
[28,21,21,21,21,20,16,16,16,16,16,16,13,13,12,12,12,12,11,11,8,6,5,5,5,5,5,5,1,1,1,1,1,1,1,1,1],
[37,28,28,28,28,28,22,21,21,21,21,21,20,20,20,20,18,16,16,16,14,13,12,12,12,12,12,11,6,6,6,6,6,5,5,5,5,1,1,1,1,1,1,1,1,1,1,1,1],
[49,37,37,37,37,37,33,28,28,28,28,28,28,28,28,28,27,22,22,21,21,21,20,20,20,20,20,18,17,17,16,16,14,12,12,12,12,7,6,6,6,6,6,6,6,6,6,6,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1],
...
ILLUSTRATION OF RECURRENCE.
Start out with row 0 and row 1 consisting of a single '1'.
To obtain any given row of this irregular triangle, first
sum the prior two rows term-by-term; for rows 7 and 8 we get:
[5,4,4,3,3,1,1] + [7,5,5,5,4,3,3,1,1] = [12,9,9,8,7,4,4,1,1].
Place markers in an array so that the number of contiguous markers
in each row correspond to the term-by-term sums like so:
--------------------------
12:o o o o o o o o o o o o
9: o o o o o o o o o - - -
9: o o o o o o o o o - - -
8: o o o o o o o o - - - -
7: o o o o o o o - - - - -
4: o o o o - - - - - - - -
4: o o o o - - - - - - - -
1: o - - - - - - - - - - -
1: o - - - - - - - - - - -
--------------------------
Then count the markers by columns to obtain the desired row;
here, the number of markers in each column yields row 9:
[9,7,7,7,5,5,5,4,3,1,1,1].
Continuing in this way generates all the rows of this triangle.

Paul D. Hanna, Table of n, a(n) for n = 0..261

Cf. A134816, A000045, A000931; A152546 (row squared sums).

{T(n,k)=local(G000931=(1-x^2)/(1-x^2-x^3+x*O(x^(n+6))));if(n<0,0,if(n<2&k==0,1, polcoeff(sum(j=0,polcoeff(G000931,n+5)-1,(x^(T(n-1,j)+T(n-2,j)) - 1)/(x-1)),k) ))};
/* To print, use Padovan g.f. to get the number of terms in row n: */
for(n=0,10,for(k=0,polcoeff((1-x^2)/(1-x^2-x^3+x*O(x^(n+6))),n+6)-1,print1(T(n,k),","));print(""))

G.f. for row n: Sum_{k=0..A000931(n+5)-1} (x^{T(n-1,k)+T(n-2,k)} - 1)/(x-1) = Sum_{k=0..A000931(n+6)-1} T(n,k)*x^k for n>1 with T(0,0)=T(1,0)=1, where A000931 is the Padovan sequence.

A285551 Volume of each square prism building the next 3-dimensional box in A100538 where side lengths form the Padovan spiral number sequence (A134816), starting with 1 X 1 X 1, 1 X 1 X 2, 2 X 2 X 2, 2 X 2 X 3, 4 X 4 X 5, ...

Original entry on oeis.org

1, 2, 8, 12, 36, 80, 175, 441, 972, 2304, 5376, 12348, 29008, 67081, 156065, 363350, 843144, 1962396, 4560200, 10600000, 24648975, 57288465, 133194600, 309636096, 719790336, 1673379352, 3890033728, 9043304417, 21023197601, 48872682810, 113615800200, 264124052396

Offset: 1

Peter M. Chema, Apr 25 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Robert Dickau, Padovan's Spiral Numbers, Wolfram Demonstrations Project Published: August 19, 2010.
Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2, 4,-4,-1,-1,0,-1).

Cf. A000931, A100538, A134816.

A[n_]:=Sum[Binomial[k, n - 2k], {k, 0, Floor[n/2]}]; a000931[n_]:=If[n==0, 1, If[n<3, 0, A[n  - 3]]]; a[n_]:=a000931[n + 5]^2*a000931[n + 6]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Apr 26 2017 *)
LinearRecurrence[{1, 2, 3, -2, 4, -4, -1, -1, 0, -1}, {1, 2, 8, 12, 36, 80, 175, 441, 972, 2304}, 40] (* Vincenzo Librandi, Jul 19 2017 *)

A(n) = sum(k=0, n\2, binomial(k, n - 2*k));
a000931(n) = if(n==0, 1, if(n<3, 0, A(n - 3)));
a(n) = a000931(n + 5)^2*a000931(n + 6); \\ Indranil Ghosh, Apr 26 2017

from sympy import binomial
def A(n): return sum([binomial(k, n - 2*k) for k in range(int(n/2) + 1)])
def a000931(n): return 1 if n==0 else 0 if n<3 else A(n - 3)
def a(n): return a000931(n + 5)**2*a000931(n + 6) # Indranil Ghosh, Apr 26 2017

a(n) = A000931(n+5)^2*A000931(n+6).

a(n) = A100538(n+1) - A100538(n).

A000931 Padovan sequence (or Padovan numbers): a(n) = a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 0.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts congruent to 2 mod 3 (offset -1). - Vladeta Jovovic, Feb 09 2005
a(n) is the number of compositions of n into parts that are odd and >= 3. Example: a(10)=3 counts 3+7, 5+5, 7+3. - David Callan, Jul 14 2006
Referred to as N0102 in R. K. Guy's "Anyone for Twopins?" - Rainer Rosenthal, Dec 05 2006
Zagier conjectures that a(n+3) is the maximum number of multiple zeta values of weight n > 1 which are linearly independent over the rationals. - Jonathan Sondow and Sergey Zlobin (sirg_zlobin(AT)mail.ru), Dec 20 2006
Starting with offset 6: (1, 1, 2, 2, 3, 4, 5, ...) = INVERT transform of A106510: (1, 1, -1, 0, 1, -1, 0, 1, -1, ...). - Gary W. Adamson, Oct 10 2008
Starting with offset 7, the sequence 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, ... is called the Fibonacci quilt sequence by Catral et al., in Fib. Q. 2017. - N. J. A. Sloane, Dec 24 2021
Triangle A145462: right border = A000931 starting with offset 6. Row sums = Padovan sequence starting with offset 7. - Gary W. Adamson, Oct 10 2008
Starting with offset 3 = row sums of triangle A146973 and INVERT transform of [1, -1, 2, -2, 3, -3, ...]. - Gary W. Adamson, Nov 03 2008
a(n+5) corresponds to the diagonal sums of "triangle": 1; 1; 1,1; 1,1; 1,2,1; 1,2,1; 1,3,3,1; 1,3,3,1; 1,4,6,4,1; ..., rows of Pascal's triangle (A007318) repeated. - Philippe Deléham, Dec 12 2008
With offset 3: (1, 0, 1, 1, 1, 2, 2, ...) convolved with the tribonacci numbers prefaced with a "1": (1, 1, 1, 2, 4, 7, 13, ...) = the tribonacci numbers, A000073. (Cf. triangle A153462.) - Gary W. Adamson, Dec 27 2008
a(n) is also the number of strings of length (n-8) from an alphabet {A, B} with no more than one A or 2 B's consecutively. (E.g., n = 4: {ABAB,ABBA,BABA,BABB,BBAB} and a(4+8) = 5.) - Toby Gottfried, Mar 02 2010
p(n):=A000931(n+3), n >= 1, is the number of partitions of the numbers {1,2,3,...,n} into lists of length two or three containing neighboring numbers. The 'or' is inclusive. For n=0 one takes p(0)=1. For details see the W. Lang link. There the explicit formula for p(n) (analog of the Binet-de Moivre formula for Fibonacci numbers) is also given. Padovan sequences with different inputs are also considered there. - Wolfdieter Lang, Jun 15 2010
Equals the INVERTi transform of Fibonacci numbers prefaced with three 1's, i.e., (1 + x + x^2 + x^3 + x^4 + 2x^5 + 3x^6 + 5x^7 + 8x^8 + 13x^9 + ...). - Gary W. Adamson, Apr 01 2011
When run backwards gives (-1)^n*A050935(n).
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [0, 0, 1; 1, 0, 1; 0, 1, 0] or of the 3 X 3 matrix [0, 1, 0; 0, 0, 1; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
Figure 4 of Brauchart et al., 2014, shows a way to "visualize the Padovan sequence as cuboid spirals, where the dimensions of each cuboid made up by the previous ones are given by three consecutive numbers in the sequence". - N. J. A. Sloane, Mar 26 2014
a(n) is the number of closed walks from a vertex of a unidirectional triangle containing an opposing directed edge (arc) between the second and third vertices. Equivalently the (1,1) entry of A^n where the adjacency matrix of digraph is A=(0,1,0;0,0,1;1,1,0). - David Neil McGrath, Dec 19 2014
Number of compositions of n-3 (n >= 4) into 2's and 3's. Example: a(12)=5 because we have 333, 3222, 2322, 2232, and 2223. - Emeric Deutsch, Dec 28 2014
The Hoffman (2015) paper "offers significant evidence that the number of quantities needed to generate the weight-n multiple harmonic sums mod p is" a(n). - N. J. A. Sloane, Jun 24 2016
a(n) gives the number of compositions of n-5 into odd parts where the order of the 1's does not matter. For example, a(11)=4 counts the following compositions of 6: (5,1)=(1,5), (3,3), (3,1,1,1)=(1,3,1,1)=(1,1,3,1)=(1,1,1,3), (1,1,1,1,1,1). - Gregory L. Simay, Aug 04 2016
For n > 6, a(n) is the number of maximal matchings in the (n-5)-path graph, maximal independent vertex sets and minimal vertex covers in the (n-6)-path graph, and minimal edge covers in the (n-5)-pan graph and (n-3)-path graphs. - Eric W. Weisstein, Mar 30, Aug 03, and Aug 07 2017
From James Mitchell and Wilf A. Wilson, Jul 21 2017: (Start)
a(2n + 5) + 2n - 4, n > 2, is the number of maximal subsemigroups of the monoid of order-preserving mappings on a set with n elements.
a(n + 6) + n - 3, n > 3, is the number of maximal subsemigroups of the monoid of order-preserving or reversing mappings on a set with n elements.
(End)
Has the property that the largest of any four consecutive terms equals the sum of the two smallest. - N. J. A. Sloane, Aug 29 2017 [David Nacin points out that there are many sequences with this property, such as 1,1,1,2,1,1,1,2,1,1,1,2,... or 2,3,4,5,2,3,4,5,2,3,4,5,... or 2,2,1,3,3,   4,1,4,    5,5,1,6,6,   7,1,7,   8,8,1,9,9,   10,1,10, ...  (spaces added for clarity), and a conjecture I made here in 2017 was simply wrong. I have deleted it. - N. J. A. Sloane, Oct 23 2018]
a(n) is also the number of maximal cliques in the (n+6)-path complement graph. - Eric W. Weisstein, Apr 12 2018
a(n+8) is the number of solus bitstrings of length n with no runs of 3 zeros. - Steven Finch, Mar 25 2020
Named after the architect Richard Padovan (b. 1935). - Amiram Eldar, Jun 08 2021
Shannon et al. (2006) credit a French architecture student Gérard Cordonnier with the discovery of these numbers.
For n >= 3, a(n) is the number of sequences of 0s and 1s of length (n-2) that begin with a 0, end with a 0, contain no two consecutive 0s, and contain no three consecutive 1s. - Yifan Xie, Oct 20 2022
For n >= 2, a(n+5) is the number of ways to tile the 1xn board with dominoes and squares (ie. size 1x1) such that are either none or one squares between dominoes, none or one squares at both ends of the board, and there is at least one domino. For example, for n=6, a(11)=4 since the tilings are |2|2, |22|, 2|2| and 222 (where 2 represents a domino and | a square). - Enrique Navarrete, Aug 31 2024

			G.f. = 1 + x^3 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 4*x^11 + ...

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Mariana Nagy, Simon R. Cowell and Valeriu Beiu, Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight, arXiv:1902.05944 [math.HO], 2019.
Wilbert Osmond, Growing Trees in Padovan Sequence For The Enhancement of L-System Algorithm, 2014.
Richard Padovan, Dom Hans Van Der Laan And The Plastic Number, pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.
Richard Padovan, Dom Hans van der Laan and the Plastic Number, Chapter 74, pp. 407-419, Volume II of K. Williams and M. J. Ostwald (eds.), Architecture and Mathematics from Antiquity to the Future, DOI 10.1007/978-3-319-00143-2_27, Springer International Publishing Switzerland, 2015.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, Integers (2024) Vol. 24, Art. No. A73. See p. 11.
Salah Eddine Rihane, Chèfiath Awero Adegbindin and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
Shingo Saito, Tatsushi Tanaka and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq., Vol. 14 (2011), Article 11.2.4, Conjecture 2.
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.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Steven J. Tedford, Combinatorial identities for the Padovan numbers, Fib. Q., 57:4 (2019), 291-298.
Michel Waldschmidt, Lectures on Multiple Zeta Values (IMSC 2011).
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set.
Eric Weisstein's World of Mathematics, Minimal Edge Cover.
Eric Weisstein's World of Mathematics, Minimal Vertex Cover.
Eric Weisstein's World of Mathematics, Padovan Sequence.
Eric Weisstein's World of Mathematics, Pan Graph.
Eric Weisstein's World of Mathematics, Path Complement Graph.
Eric Weisstein's World of Mathematics, Path Graph.
Erv Wilson, The Scales of Mt. Meru.
Iwona Włoch, Urszula Bednarz, Dorota Bród, Andrzej Włoch and Małgorzata Wołowiec-Musiał, On a new type of distance Fibonacci numbers, Discrete Applied Math., Vol. 161, No. 16-17 (November 2013) pp. 2695-2701.
Richard Yanco, Letter and Email to N. J. A. Sloane, 1994.
Richard Yanco and Ansuman Bagchi, K-th order maximal independent sets in path and cycle graphs, Unpublished manuscript, 1994. (Annotated scanned copy)
Diyar O. Mustafa Zangana and Ahmet Öteleş, Padovan Numbers by the Permanents of a Certain Complex Pentadiagonal Matrix, J. of Garmian Univ., Vol. 5, No. 2 (2018), pp. 330-338.
Sergey Zlobin, A note on arithmetic properties of multiple zeta values, arXiv:math/0601151 [math.NT], 2006.
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.
Closely related to A001608.
Cf. A000073, A005682-A005691, A103372-A103380, A106510, A145462, A146973, A153462.
Doubling every term gives A291289.

a:=[1,0,0];; for n in [4..50] do a[n]:=a[n-2]+a[n-3]; od; a; # G. C. Greubel, Dec 30 2019

a000931 n = a000931_list !! n
a000931_list = 1 : 0 : 0 : zipWith (+) a000931_list (tail a000931_list)
-- Reinhard Zumkeller, Feb 10 2011

I:=[1,0,0]; [n le 3 select I[n] else Self(n-2) + Self(n-3): n in [1..60]]; // Vincenzo Librandi, Jul 21 2015

A000931 := proc(n) option remember; if n = 0 then 1 elif n <= 2 then 0 else procname(n-2)+procname(n-3); fi; end;
A000931:=-(1+z)/(-1+z^2+z^3); # Simon Plouffe in his 1992 dissertation; gives sequence without five leading terms
a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to 50 do a[n]:=a[n-2]+a[n-3]; end do; # Francesco Daddi, Aug 04 2011

CoefficientList[Series[(1-x^2)/(1-x^2-x^3), {x, 0, 50}], x]
a[0]=1; a[1]=a[2]=0; a[n_]:= a[n]= a[n-2] + a[n-3]; Table[a[n], {n, 0, 50}] (* Robert G. Wilson v, May 04 2006 *)
LinearRecurrence[{0,1,1}, {1,0,0}, 50] (* Harvey P. Dale, Jan 10 2012 *)
Table[RootSum[-1 -# +#^3 &, 5#^n -6#^(n+1) +4#^(n+2) &]/23, {n,0,50}] (* Eric W. Weisstein, Nov 09 2017 *)

Vec((1-x^2)/(1-x^2-x^3) + O(x^50)) \\ Charles R Greathouse IV, Feb 11 2011

{a(n) = if( n<0, polcoeff(1/(1+x-x^3) + x * O(x^-n), -n), polcoeff( (1 - x^2)/(1-x^2-x^3) + x * O(x^n), n))}; /* Michael Somos, Sep 18 2012 */

def aupton(nn):
    alst = [1, 0, 0]
    for n in range(3, nn+1): alst.append(alst[n-2]+alst[n-3])
    return alst
print(aupton(49)) # Michael S. Branicky, Mar 28 2022

def A000931_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)/(1-x^2-x^3) ).list()
A000931_list(50) # G. C. Greubel, Dec 30 2019

G.f.: (1-x^2)/(1-x^2-x^3).
a(n) is asymptotic to r^n / (2*r+3) where r = 1.3247179572447... = A060006, the real root of x^3 = x + 1. - Philippe Deléham, Jan 13 2004
a(n)^2 + a(n+2)^2 + a(n+6)^2 = a(n+1)^2 + a(n+3)^2 + a(n+4)^2 + a(n+5)^2 (Barniville, Question 16884, Ed. Times 1911).
a(n+5) = a(0) + a(1) + ... + a(n).
a(n) = central and lower right terms in the (n-3)-th power of the 3 X 3 matrix M = [0 1 0 / 0 0 1 / 1 1 0]. E.g., a(13) = 7. M^10 = [3 5 4 / 4 7 5 / 5 9 7]. - Gary W. Adamson, Feb 01 2004
G.f.: 1/(1 - x^3 - x^5 - x^7 - x^9 - ...). - Jon Perry, Jul 04 2004
a(n+4) = Sum_{k=0..floor((n-1)/2)} binomial(floor((n+k-2)/3), k). - Paul Barry, Jul 06 2004
a(n+3) = Sum_{k=0..floor(n/2)} binomial(k, n-2k). - Paul Barry, Sep 17 2004, corrected by Greg Dresden and Zi Ye, Jul 06 2021
a(n+3) is diagonal sum of A026729 (as a number triangle), with formula a(n+3) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k} (-1)^(n-k+i)*binomial(n-k, i)*binomial(i+k, i-k). - Paul Barry, Sep 23 2004
a(n) = a(n-1) + a(n-5) = A003520(n-4) + A003520(n-13) = A003520(n-3) - A003520(n-9). - Henry Bottomley, Jan 30 2005
a(n+3) = Sum_{k=0..floor(n/2)} binomial((n-k)/2, k)(1+(-1)^(n-k))/2. - Paul Barry, Sep 09 2005
The sequence 1/(1-x^2-x^3) (a(n+3)) is given by the diagonal sums of the Riordan array (1/(1-x^3), x/(1-x^3)). The row sums are A000930. - Paul Barry, Feb 25 2005
a(n) = A023434(n-7) + 1 for n >= 7. - David Callan, Jul 14 2006
a(n+5) corresponds to the diagonal sums of A030528. The binomial transform of a(n+5) is A052921. a(n+5) = Sum_{k=0..floor(n/2)} Sum_{k=0..n} (-1)^(n-k+i)*binomial(n-k, i)binomial(i+k+1, 2k+1). - Paul Barry, Jun 21 2004
r^(n-1) = (1/r)*a(n) + r*a(n+1) + a(n+2), where r = 1.32471... is the real root of x^3 - x - 1 = 0. Example: r^8 = (1/r)*a(9) + r*a(10) + a(11) = (1/r)*2 + r*3 + 4 = 9.483909... - Gary W. Adamson, Oct 22 2006
a(n) = (r^n)/(2r+3) + (s^n)/(2s+3) + (t^n)/(2t+3) where r, s, t are the three roots of x^3-x-1. - Keith Schneider (schneidk(AT)email.unc.edu), Sep 07 2007
a(n) = -k*a(n-1) + a(n-2) + (k+1)a(n-2) + k*a(n-4), n > 3, for any value of k. - Gary Detlefs, Sep 13 2010
From Francesco Daddi, Aug 04 2011: (Start)
  a(0) + a(2) + a(4) + a(6) + ... + a(2*n) = a(2*n+3).
  a(0) + a(3) + a(6) + a(9) + ... + a(3*n) = a(3*n+2)+1.
  a(0) + a(5) + a(10) + a(15) + ... + a(5*n) = a(5*n+1)+1.
  a(0) + a(7) + a(14) + a(21) + ... + a(7*n) = (a(7*n) + a(7*n+1) + 1)/2.  (End)
a(n+3) = Sum_{k=0..floor((n+1)/2)} binomial((n+k)/3,k), where binomial((n+k)/3,k)=0 for noninteger (n+k)/3. - Nikita Gogin, Dec 07 2012
a(n) = A182097(n-3) for n > 2. - Jonathan Sondow, Mar 14 2014
a(n) = the k-th difference of a(n+5k) - a(n+5k-1), k>=1. For example, a(10)=3 => a(15)-a(14) => 2nd difference of a(20)-a(19) => 3rd difference of a(25)-a(24)... - Bob Selcoe, Mar 18 2014
Construct the power matrix T(n,j) = [A^*j]*[S^*(j-1)] where A=(0,0,1,0,1,0,1,...) and S=(0,1,0,0,...) or A063524. [* is convolution operation] Define S^*0=I with I=(1,0,0,...). Then a(n) = Sum_{j=1...n} T(n,j). - David Neil McGrath, Dec 19 2014
If x=a(n), y=a(n+1), z=a(n+2), then x^3 + 2*y*x^2 - z^2*x - 3*y*z*x + y^2*x + y^3 - y^2*z + z^3 = 1. - Alexander Samokrutov, Jul 20 2015
For the sequence shifted by 6 terms, a(n) = Sum_{k=ceiling(n/3)..ceiling(n/2)} binomial(k+1,3*k-n) [Doslic-Zubac]. - N. J. A. Sloane, Apr 23 2017
From Joseph M. Shunia, Jan 21 2020: (Start)
a(2n) = 2*a(n-1)*a(n) + a(n)^2 + a(n+1)^2, for n > 8.
a(2n-1) = 2*a(n)*a(n+1) + a(n-1)^2, for n > 8.
a(2n+1) = 2*a(n+1)*a(n+2) + a(n)^2, for n > 7. (End)
0*a(0) + 1*a(1) + 2*a(2) + ... + n*a(n) = n*a(n+5) - a(n+9) + 2. - Greg Dresden and Zi Ye, Jul 02 2021
From Greg Dresden and Zi Ye, Jul 06 2021: (Start)
2*a(n) = a(n+2) + a(n-5) for n >= 5.
3*a(n) = a(n+4) - a(n-9) for n >= 9.
4*a(n) = a(n+5) - a(n-9) for n >= 9. (End)

Edited by Charles R Greathouse IV, Mar 17 2010

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).

Original entry on oeis.org

1, 3, 2, 4, 7, 1, 7, 9, 5, 7, 2, 4, 4, 7, 4, 6, 0, 2, 5, 9, 6, 0, 9, 0, 8, 8, 5, 4, 4, 7, 8, 0, 9, 7, 3, 4, 0, 7, 3, 4, 4, 0, 4, 0, 5, 6, 9, 0, 1, 7, 3, 3, 3, 6, 4, 5, 3, 4, 0, 1, 5, 0, 5, 0, 3, 0, 2, 8, 2, 7, 8, 5, 1, 2, 4, 5, 5, 4, 7, 5, 9, 4, 0, 5, 4, 6, 9, 9, 3, 4, 7, 9, 8, 1, 7, 8, 7, 2, 8, 0, 3, 2, 9, 9, 1

Offset: 1

Fabian Rothelius, Mar 14 2001

Has been also called the silver number, also the plastic number.
This is the smallest Pisot-Vijayaraghavan number.
The name "plastic number" goes back to the Dutch Benedictine monk and architect Dom Hans van der Laan, who gave this name 4 years after the discovery of the number by the French engineer Gérard Cordonnier in 1924, who used the name "radiant number". - Hugo Pfoertner, Oct 07 2018
Sometimes denoted by the symbol rho. - Ed Pegg Jr, Feb 01 2019
Also the solution of 1/x + 1/(1+x+x^2) = 1. - Clark Kimberling, Jan 02 2020
Given any complex p such that real(p)>-1, this constant is the only real solution of the equation z^p+z^(p+1)=z^(p+3), and the only attractor of the complex mapping z->M(z,p), where M(z,p)=(z^p+z^(p+1))^(1/(p+3)), convergent from any complex plane point. - Stanislav Sykora, Oct 14 2021
The Pisot-Vijayaraghavan numbers were named after the French mathematician Charles Pisot (1910-1984) and the Indian mathematician Tirukkannapuram Vijayaraghavan (1902-1955). - Amiram Eldar, Apr 02 2022
The sequence a(n) = v_3^floor(n^2/4) where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1 satisfies the Somos-5 recursion a(n+3)*a(n-2) = a(n+2)*a(n-1) + a(n+1)*a(n) for all n in Z. Also true if floor is removed. - Michael Somos, Mar 24 2023

			1.32471795724474602596090885447809734...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Alex Bellos, The golden ratio has spawned a beautiful new curve: the Harriss spiral, The Guardian, Jan 13 2015.
Gamaliel Cerda-Morales, New Identities for Padovan Numbers, arXiv:1904.05492 [math.CO], 2019.
Brady Haran and Edmund Harriss, The Plastic Ratio, Numberphile video (2019).
Ed Pegg Jr., Pictures based on the plastic constant
Simon Plouffe, Smallest Pisot-Vijayaraghavan number to 50000 digits
Simon Plouffe, The Smallest Pisot-Vijayaraghavan number
F. Rothelius, Formulae.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Dom Hans van der Laan, Le nombre plastique: Quinze leçons sur l’ordonnance architectonique, Brill Academic Pub., Leiden, 1960.
Michel Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Maverick Graph.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Plastic Constant.
Wikipedia, Plastic number.
Index entries for algebraic numbers, degree 3

Cf. A001622. A072117 gives continued fraction.
Cf. A006888, A010527, A051016, A051017, A084252, A075778 (inverse), A126772.
Other Pisot numbers: A086106, A092526, A228777, A293506, A293508, A293509, A293557.
Cf. A002620, A006721, A134816.

SetDefaultRealField(RealField(100)); ((3+Sqrt(23/3))/6)^(1/3) + ((3-Sqrt(23/3))/6)^(1/3); // G. C. Greubel, Mar 15 2019

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%,130) ; # R. J. Mathar, Jan 22 2013

RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Sep 30 2009 *)
s = Sqrt[23/108]; RealDigits[(1/2 + s)^(1/3) + (1/2 - s)^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 12 2017 *)
RealDigits[Root[x^3-x-1,1],10,120][[1]] (* or *) RealDigits[(Surd[9-Sqrt[69],3]+Surd[9+Sqrt[69],3])/(Surd[2,3]Surd[9,3]),10,120][[1]] (* Harvey P. Dale, Sep 04 2018 *)

allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060006.txt", n, " ", d)); \\ Harry J. Smith, Jul 01 2009

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) \\ Altug Alkan, Apr 10 2016

polrootsreal(x^3-x-1)[1] \\ Charles R Greathouse IV, Aug 28 2016

default(realprecision, 110); digits(floor(solve(x=1, 2, x^3 - x - 1)*10^105)) /* Michael Somos, Mar 24 2023 */

numerical_approx(((3+sqrt(23/3))/6)^(1/3) + ((3-sqrt(23/3))/6)^(1/3), digits=100) # G. C. Greubel, Mar 15 2019

Equals (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - Henry Bottomley, May 22 2003
Equals CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - Gerald McGarvey, Nov 26 2004
Equals sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - Gerald McGarvey, Mar 18 2006
Equals (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - Eric Desbiaux, Oct 17 2008
Equals Sum_{k >= 0} 27^(-k)/k!*(Gamma(2*k+1/3)/(9*Gamma(k+4/3)) - Gamma(2*k-1/3)/(3*Gamma(k+2/3))). - Robert Israel, Jan 13 2015
Equals sqrt(Phi) = sqrt(1.754877666246...) (see A109134). - Philippe Deléham, Sep 29 2020
Equals cosh(arccosh(3*c)/3)/c, where c = sqrt(3)/2 (A010527). - Amiram Eldar, May 15 2021
Equals 1/hypergeom([1/5, 2/5, 3/5, 4/5], [2/4, 3/4, 5/4], -5^5/4^4). - Gerry Martens, Mar 16 2025

Edited and extended by Robert G. Wilson v, Aug 03 2002

Removed incorrect comments, Joerg Arndt, Apr 10 2016

A096231 Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {Pi/2, Pi/3, 0}.

Original entry on oeis.org

1, 3, 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, 696081

Offset: 0

Bellovin, Kennedy, Stansifer, Wong (chrkenn(AT)bergen.org), Jul 29 2004

Or, coordination sequence for (2,3,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The generation of a triangle is defined such that exactly one triangle has generation 0 and a triangle has generation n, n > 0, if it is next to a triangle with generation n-1 but not to one with lower generation.
The recursions were found by examining empirical data and have not been proved to be accurate for all n. The generating function was found by assuming that the recursions were accurate; it can be calculated from either recursion. We created a specialized program in Java for finding the sequences of generations for triangles with angles {Pi/p, Pi/q, Pi/r}, p, q, r > 1, that tile the Euclidean or hyperbolic plane; this program was used to calculate the sequence.
The g.f. (1+X)^2 * (1+X+X^2) / (1-X^2-X^3) follows from the Cannon-Wagreich paper, Prop. 3.1, so the g.f. and the recurrence are now a theorem, no longer conjectures, and the additional terms and the Mma program are now justified. - N. J. A. Sloane, Dec 29 2015

			a(1)=3 because exactly three triangles have generation 1, i.e., are adjacent to the triangle with generation 0.

Robert Israel, Table of n, a(n) for n = 0..8110
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257.
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.

Equals A000931(n+10).

I:=[1,3,5,7,9,12,16]; [n le 7 select I[n] else Self(n-1)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

f:= gfun:-rectoproc({a(n) = a(n-2)+a(n-3),
a(0)=1, a(1)=3, a(2)=5, a(3)=7, a(4)=9, a(5)=12}, a(n), remember):
seq(f(n),n=0..50); # Robert Israel, Jan 13 2016

CoefficientList[ Series[(x + 1)^2*(1 + x + x^2)/(1 - x^2 - x^3), {x, 0, 45}], x] (* Robert G. Wilson v, Jul 31 2004 *)
Join[{1, 3, 5}, LinearRecurrence[{0, 1, 1}, {7, 9, 12}, 50]] (* Vincenzo Librandi, Dec 30 2015 *)

a(n)=if(n>2,([0,1,0; 0,0,1; 1,1,0]^n*[1;3;5])[1,1],1) \\ Charles R Greathouse IV, Feb 09 2017

a(n) = a(n-1) + a(n-5) = a(n-2) + a(n-3), for n > 6.

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

More terms from Robert G. Wilson v, Jul 31 2004

A182097 Expansion of 1/(1-x^2-x^3).

Original entry on oeis.org

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, 396655, 525456, 696081, 922111, 1221537, 1618192, 2143648, 2839729, 3761840, 4983377, 6601569, 8745217

Offset: 0

N. J. A. Sloane, Apr 11 2012

Number of compositions (ordered partitions) into parts 2 and 3. - Joerg Arndt, Aug 21 2013
a(n) is the top left entry of the n-th power of any of the 3X3 matrices [0, 1, 1; 0, 0, 1; 1, 0, 0], [0, 1, 0; 1, 0, 1; 1, 0, 0], [0, 1, 1; 1, 0, 0; 0, 1, 0] or [0, 0, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
Conjectured values of d(n), the dimension of a Z-module in MZV(conv). See the Waldschmidt link. - Michael Somos, Mar 14 2014
Shannon et al. (2006) call these the Van der Laan numbers. - N. J. A. Sloane, Jan 11 2022

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...

A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See R_n.
Michel Waldschmidt, "Multiple Zeta values and Euler-Zagier numbers", in Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Hoffman, The algebra of multiharmonic series, Journ. of Alg., Vol. 192, Issue 2 (Aug 1997), 477-495.
I. E. Leonard and A. C. F. Liu, A familiar recurrence occurs again, Amer. Math. Monthly, 119 (2012), 333-336.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (32).
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.
Michel Waldschmidt, Multiple Zeta values and Euler-Zagier numbers, Slides, Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: 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. A000931, A001608.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3))); // G. C. Greubel, Aug 11 2018

a[ n_] := If[n < 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, n}]]; (* Michael Somos, Dec 13 2013 *)
CoefficientList[Series[1/(1-x^2-x^3),{x,0,60}],x] (* or *) LinearRecurrence[ {0,1,1},{1,0,1},70] (* Harvey P. Dale, Dec 04 2014 *)

{a(n) = if( n<0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n))}; /* Michael Somos, Dec 13 2013 */

Vec(1/(1-x^2-x^3) + O(x^99)) \\ Altug Alkan, Sep 02 2016

G.f.: 1 / (1 - x^2 - x^3).
a(n) = A000931(n+3).
From Michael Somos, Dec 13 2013: (Start)
a(n) = A176971(-n).
a(n) = a(n-2) + a(n-3) for all n in Z.
a(n-7) = A133034(n).
a(n-5) = A078027(n).
a(n-3) = A000931(n).
a(n+2) = A134816(n).
a(n+4) = A164001(n) if n > 1. - (End)
a(n) = (A001608(n) - A000931(n))/2. - Elmo R. Oliveira, Dec 31 2022

A171861 Expansion of x(1+x+x^2) / ( (x-1)(x^3+x^2-1) ).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 18, 25, 34, 46, 62, 83, 111, 148, 197, 262, 348, 462, 613, 813, 1078, 1429, 1894, 2510, 3326, 4407, 5839, 7736, 10249, 13578, 17988, 23830, 31569, 41821, 55402, 73393, 97226, 128798, 170622, 226027, 299423, 396652, 525453, 696078, 922108

Offset: 1

Ed Pegg Jr, Oct 16 2010

Number of wins in Penney's game if the two players start HHT and TTT and HHT beats TTT.

HHT beats TTT 70% of the time. - Geoffrey Critzer, Mar 01 2014

			a(n) enumerates length n+2 sequences on {H,T} that end in HHT but do not contain the contiguous subsequence TTT.
a(3)=4 because we have: TTHHT, THHHT, HTHHT, HHHHT.
a(4)=6 because we have: TTHHHT, THTHHT, THHHHT, HTTHHT, HTHHHT, HHHHHT. - _Geoffrey Critzer_, Mar 01 2014

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

Related sequences are A000045 (HHH beats HHT, HTT beats TTH), A006498 (HHH beats HTH), A023434 (HHH beats HTT), A000930 (HHH beats THT, HTH beats HHT), A000931 (HHH beats TTH), A077868 (HHT beats HTH), A002620 (HHT beats HTT), A000012 (HHT beats THH), A004277 (HHT beats THT), A070550 (HTH beats HHH), A000027 (HTH beats HTT), A097333 (HTH beats THH), A040000 (HTH beats TTH), A068921 (HTH beats TTT), A054405 (HTT beats HHH), A008619 (HTT beats HHT), A038718 (HTT beats THT), A128588 (HTT beats TTT).

Cf. A164315 (essentially the same sequence).

A171861 := proc(n) option remember; if n <=4 then op(n,[1,2,4,6]); else procname(n-1)+procname(n-2)-procname(n-4) ; end if; end proc:

nn=44;CoefficientList[Series[x(1+x+x^2)/(1-x-x^2+x^4),{x,0,nn}],x] (* Geoffrey Critzer, Mar 01 2014 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,1,1]^(n-1)*[1;2;4;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = a(n-1) +a(n-2) -a(n-4) = A000931(n+10)-3 = A134816(n+6)-3 = A078027(n+12)-3.

a(n) = A164315(n-1). - Alois P. Heinz, Oct 12 2017

A152198 Triangle read by rows, A007318 rows repeated.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1

Offset: 0

Gary W. Adamson, Nov 28 2008

Eigensequence of the triangle = A051163: (1, 2, 5, 12, 30, 76,...)
Another version of A152815. - Philippe Deléham, Dec 13 2008
Row sums : A016116(n); Diagonal sums: A000931(n+5). - Philippe Deléham, Dec 13 2008
Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012
Sums along rising diagonals are A134816. - John Molokach, Jul 09 2013

			The triangle starts
1;
1;
1, 1;
1, 1;
1, 2, 1;
1, 2, 1;
1, 3, 3, 1;
1, 3, 3, 1;
1, 4, 6, 4, 1;
1, 4, 6, 4, 1;
1, 5, 10, 10, 5, 1;
1, 5, 10, 10, 5, 1;
...
Triangle (1,0,-1,0,0,...) DELTA (0,1,-1,0,0,...) begins:
1
1, 0
1, 1, 0
1, 1, 0, 0
1, 2, 1, 0, 0
1, 2, 1, 0, 0, 0
1, 3, 3, 1, 0, 0, 0
1, 3, 3, 1, 0, 0, 0, 0
1, 4, 6, 4, 1, 0, 0, 0, 0
1, 4, 6, 4, 1, 0, 0, 0, 0, 0
1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0...

Cf. A007318, A133156, A051163.

t[n_, k_] := Binomial[ Floor[n/2], k]; Table[t[n, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Sep 13 2012 *)

Triangle read by rows, Pascal's triangle rows repeated.
Equals inverse binomial transform of A133156 unsigned.
G.f. : (1+x)/(1-(1+y)*x^2). - Philippe Deléham, Jan 16 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057077(n), A019590(n+1), A000012(n), A016116(n), A108411(n), A074872(n+1) for x = -2, -1, 0, 1, 2, 4 respectively. - Philippe Deléham, Jan 16 2012
T(n,k) = A065941(n-k, n-2*k) = abs(A108299(n-k, n-2*k)). - Johannes W. Meijer, Sep 05 2013

More terms from Philippe Deléham, Dec 14 2008

A164001 Spiral of triangles around a hexagon.

Original entry on oeis.org

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).

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).

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

cp's OEIS Frontend

A152545 Padovan-Fibonacci triangle, read by rows, where the first column equals the Padovan spiral numbers (A134816), while the row sums equal the Fibonacci numbers (A000045).

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A285551 Volume of each square prism building the next 3-dimensional box in A100538 where side lengths form the Padovan spiral number sequence (A134816), starting with 1 X 1 X 1, 1 X 1 X 2, 2 X 2 X 2, 2 X 2 X 3, 4 X 4 X 5, ...

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A000931 Padovan sequence (or Padovan numbers): a(n) = a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 0.

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A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).

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A096231 Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {Pi/2, Pi/3, 0}.

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A182097 Expansion of 1/(1-x^2-x^3).

A171861 Expansion of x(1+x+x^2) / ( (x-1)(x^3+x^2-1) ).