A060924 -id:A060924 - cp's OEIS frontend

A061187 Staircase of coefficients of polynomials used for column g.f.s of triangle A060924.

3, -2, 6, 2, -4, 9, 39, -57, 30, -8, 12, 136, -96, -84, 104, -32, 15, 320, 293, -1260, 1155, -530, 160, -32, 18, 618, 2118, -4242, 890, 2718, -2652, 1088, -192, 21, 1057, 7224, -5037, -19208, 33383, -23793, 9534, -2632, 672

Offset: 0

Wolfdieter Lang, Apr 20 2001

a(n,m) is coefficient of x^m of polynomial pLo(n+1,x) := (((1+x)+(3-2*x)*sqrt(x))^(n+1) - ((1+x)-(3-2*x)*sqrt(x))^(n+1))/(2*sqrt(x)) of degree n+1+floor(n/2)= A001651(n). pLo(n+1,x)= sum(binomial(n+1,2*j+1)*(1+x)^(n-2*j)*(3-2*x)^(2*j+1)*x^j,j=0..floor(n/2)), n >= 0.

pLo(m+1,x) appears as numerator polynomial of the g.f. for column m >= 0 of the triangle A060924 (even part of bisection of Lucas triangle).

			{3, -2}; {6, 2, -4}; {9, 39, -57, 30, -8}; ...; pLo(2, x)= 6+2*x-4*x^2= 2*(1+x)*(3-2*x).

A061186 (companion staircase).

a(n, m)= sum(3*(-9/2)^j*binomial(n+1, 2*j+1)*sum((-3/2)^(k-m)*binomial(n-2*j, k) *binomial(2*j+1, m-k-j), k=max(0, m-3*j-1)..n-2*j), j=0..floor(n/2)), 0<= m <= n+1+floor(n/2); else 0.

A060927 Row sums of triangle A060924 (odd part of bisection of Lucas triangle).

Original entry on oeis.org

3, 13, 65, 337, 1737, 8945, 46073, 237313, 1222345, 6296017, 32429337, 167036065, 860364393, 4431539313, 22825840825, 117570661697, 605579465737, 3119200691985, 16066286106713, 82753748396513

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, 0, 4).

Cf. A060926 (row sums of A060923 companion triangle).

Cf. A060924, A060928.

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (3-2*x)/(1-5*x-4*x^3) )); // G. C. Greubel, Apr 07 2021

CoefficientList[Series[(3-2*x)/(1-5*x-4*x^3), {x, 0, 30}], x] (* G. C. Greubel, Apr 07 2021 *)

def A060927_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (3-2*x)/(1-5*x-4*x^3) ).list()
A060927_list(30) # G. C. Greubel, Apr 07 2021

a(n) = Sum_{j=0..n} A060924(n, j).
G.f.: (3-2*x)/(1-5*x-4*x^3).
a(n) = 3*A060928(n) - 2*A060928(n-1), n >= 1; a(0)=3.

A005248 Bisection of Lucas numbers: a(n) = L(2n) = A000032(2n).

Original entry on oeis.org

2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, 39603, 103682, 271443, 710647, 1860498, 4870847, 12752043, 33385282, 87403803, 228826127, 599074578, 1568397607, 4106118243, 10749957122, 28143753123, 73681302247, 192900153618, 505019158607, 1322157322203

Offset: 0

N. J. A. Sloane

Drop initial 2; then iterates of A050411 do not diverge for these starting values. - David W. Wilson
All nonnegative integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = +4 together with b(n)=A001906(n), n>=0. - Wolfdieter Lang, Aug 31 2004
a(n+1) = B^(n)AB(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 3=`10`, 7=`010`, 18=`0010`, 47=`00010`, ..., in Wythoff code. a(0) = 2 = B(1) in Wythoff code.
Output of Tesler's formula (as well as that of Lu and Wu) for the number of perfect matchings of an m X n Möbius band where m and n are both even specializes to this sequence for m=2. - Sarah-Marie Belcastro, Jul 04 2009
Numbers having two 1's in their base-phi representation. - Robert G. Wilson v, Sep 13 2010
Pisano period lengths: 1, 3, 4, 3, 2, 12, 8, 6, 12, 6, 5, 12, 14, 24, 4, 12, 18, 12, 9, 6, ... - R. J. Mathar, Aug 10 2012
From Wolfdieter Lang, Feb 18 2013: (Start)
a(n) is also one half of the total number of round trips, each of length 2*n, on the graph P_4 (o-o-o-o) (the simple path with 4 points (vertices) and 3 lines (or edges)). See the array and triangle A198632 for the general case of the graph P_N (there N is n and the length is l=2*k).
O.g.f. for w(4,l) (with zeros for odd l): y*(d/dy)S(4,y)/S(4,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See also A198632 for a rewritten form. One half of this o.g.f. for x -> sqrt(x) produces the g.f. (2-3x)/(1-3x+x^2) given below. (End)
Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy - 5. - Michel Lagneau, Feb 01 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 7xy + y^2 + 45 = 0. - Colin Barker, Feb 16 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 320 = 0. - Colin Barker, Feb 16 2014
a(n) are the numbers such that a(n)^2-2 are Lucas numbers. - Michel Lagneau, Jul 22 2014
All sequences of this form, b(n+1) = 3*b(n) - b(n-1), regardless of initial values, which includes this sequence, yield this sequence as follows: a(n) = (b(j+n) + b(j-n))/b(j), for any j, except where b(j) = 0. Also note formula below relating this a(n) to all sequences of the form G(n+1) = G(n) + G(n-1).  - Richard R. Forberg, Nov 18 2014
A non-simple continued fraction expansion for F(2n*(k+1))/F(2nk) k>=1 is a(n) + (-1)/(a(n) + (-1)/(a(n) + ... +  (-1)/a(n))) where a(n) appears exactly k times (F(n) denotes the n-th Fibonacci number). E.g., F(16)/F(12) equals 7 + (-1)/(7 + (-1)/7). Furthermore, these a(n) are exactly the positive integers k such that the non-simple infinite continued fraction k + (-1)/(k + (-1)/(k + (-1)/(k + ...))) belongs to Q(sqrt(5)). Compare to Benoit Cloitre and Thomas Baruchel's comments at A002878.  - Greg Dresden, Aug 13 2019
For n >= 1, a(n) is the number of cyclic up-down words of length 2*n over an alphabet of size 3. - Sela Fried, Apr 08 2025

			G.f. = 2 + 3*x + 7*x^2 + 18*x^3 + 47*x^4 + 123*x^5 + 322*x^6 + 843*x^7 + ... - _Michael Somos_, Aug 11 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Richard P. Stanley, Enumerative combinatorics, Vol. 2. Volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard André-Jeannin, Summation of Certain Reciprocal Series Related to Fibonacci and Lucas Numbers, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 200-204.
Peter Bala, Some simple continued fraction expansions for an infinite product, Part 1.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Pooja Bhadouria, Deepika Jhala and Bijendra Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Vol. 8, No. 1 (2014), pp. 81-92; sequences B_1, T_1 and R_1.
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 51, 56.
Noureddine Chair, Exact two-point resistance, and the simple random walk on the complete graph minus N edges, Ann. Phys., Vol. 327, No. 12 (2012), pp. 3116-3129, Eq. (18).
Tony Crilly, Double sequences of positive integers, Math. Gaz., Vol. 69 (1985), pp. 263-271.
Pedro P. B. de Oliveira, Enrico Formenti, Kévin Perrot, Sara Riva and Eurico L. P. Ruivo, Non-maximal sensitivity to synchronism in periodic elementary cellular automata: exact asymptotic measures, arXiv:2004.07128 [cs.FL], 2020.
Robert G. Donnelly, Molly W. Dunkum, Murray L. Huber and Lee Knupp, Sign-alternating Gibonacci polynomials, arXiv:2012.14993 [math.CO], 2020.
Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, Vol. 5, No. 15 (2014), pp. 2226-2234.
Margherita Maria Ferrari and Norma Zagaglia Salvi, Aperiodic Compositions and Classical Integer Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.
Sela Fried, A formula for the number of up-down words, arXiv:2503.02005 [math.CO], 2025.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
Dale Gerdemann, Collision of Digits "Also interesting are the two bisections of the Lucas numbers A005248 (digit minimizer) and A002878 (digit maximizer). I particularly like the multiples of A005248 because I have this image of the two digits piling up on top of each other and then spreading out like waves".
André Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding Part 1 Part 2, Fib. Quart., Vol. 9, No. 3 (1971), pp. 277-295, 298.
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1986.
Tanya Khovanova, Recursive Sequences.
Emrah Kılıç, Yücel Türker Ulutaş and Neşe Ömür, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq., Vol. 14 (2011), Article 11.5.6, Table 2.
Wentao T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, Vol. 293, No. 5-6 (2002), pp. 235-246. [From Sarah-Marie Belcastro, Jul 04 2009]
Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee and Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.
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.
Sara Riva, Factorisation of Discrete Dynamical Systems, Ph.D. Thesis, Univ. Côte d'Azur (France 2023).
Jeffrey Shallit, An interesting continued fraction, Math. Mag., Vol. 48, No. 4 (1975), pp. 207-211.
Jeffrey Shallit, An interesting continued fraction, Math. Mag., Vol. 48, No. 4 (1975), pp. 207-211. [Annotated scanned copy]
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.
Jeffrey Shallit and N. J. A. Sloane, Correspondence, 1975.
Paweł J. Szabłowski, On moments of Cantor and related distributions, arXiv preprint arXiv:1403.0386 [math.PR], 2014.
Glenn Tesler, Matchings in Graphs on Non-Orientable Surfaces, Journal of Combinatorial Theory, Series B, Vol. 78, No. 2 (2000), pp. 198-231.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Eric Weisstein's World of Mathematics, Phi Number System.
A. V. Zarelua, On Matrix Analogs of Fermat’s Little Theorem, Mathematical Notes, Vol. 79, No. 6 (2006), pp. 783-796. Translated from Matematicheskie Zametki, Vol. 79, No. 6 (2006), pp. 840-855.
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2).
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A002878 (odd-indexed Lucas numbers), A001906 (Chebyshev S(n-1, 3)), a(n) = sqrt(4+5*A001906(n)^2), A228842.
a(n) = A005592(n)+1 = A004146(n)+2 = A065034(n)-1.
First differences of A002878. Pairwise sums of A001519. First row of array A103997.
Cf. A153415, A201157. Also Lucas(k*n): A000032 (k = 1), A014448 (k = 3), A056854 (k = 4), A001946 (k = 5), A087215 (k = 6), A087281 (k = 7), A087265 (k = 8), A087287 (k = 9), A065705 (k = 10), A089772 (k = 11), A089775 (k = 12).

a005248 n = a005248_list !! n
a005248_list = zipWith (+) (tail a001519_list) a001519_list
-- Reinhard Zumkeller, Jan 11 2012

[Lucas(2*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

a:= n-> (<<2|3>>. <<3|1>, <-1|0>>^n)[1$2]: seq(a(n), n=0..30); # Alois P. Heinz, Jul 31 2008
with(combinat): seq(5*fibonacci(n)^2+2*(-1)^n, n= 0..26);

a[0] = 2; a[1] = 3; a[n_] := 3a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 27}] (* Robert G. Wilson v, Jan 30 2004 *)
Fibonacci[1 + 2n] + 1/2 (-Fibonacci[2n] + LucasL[2n]) (* Tesler. Sarah-Marie Belcastro, Jul 04 2009 *)
LinearRecurrence[{3, -1}, {2, 3}, 50] (* Sture Sjöstedt, Nov 27 2011 *)
LucasL[Range[0,60,2]] (* Harvey P. Dale, Sep 30 2014 *)

{a(n) = fibonacci(2*n + 1) + fibonacci(2*n - 1)}; /* Michael Somos, Jun 23 2002 */

{a(n) = 2 * subst( poltchebi(n), x, 3/2)}; /* Michael Somos, Jun 28 2003 */

[lucas_number2(n,3,1) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

a(n) = Fibonacci(2*n-1) + Fibonacci(2*n+1).
G.f.: (2-3*x)/(1-3*x+x^2). - Simon Plouffe in his 1992 dissertation.
a(n) = S(n, 3) - S(n-2, 3) = 2*T(n, 3/2) with S(n-1, 3) = A001906(n) and S(-2, x) = -1. U(n, x)=S(n, 2*x) and T(n, x) are Chebyshev's U- and T-polynomials.
a(n) = a(k)*a(n - k) - a(n - 2k) for all k, i.e., a(n) = 2*a(n) - a(n) = 3*a(n - 1) - a(n - 2) = 7*a(n - 2) - a(n - 4) = 18*a(n - 3) - a(n - 6) = 47*a(n - 4) - a(n - 8) etc., a(2n) = a(n)^2 - 2. - Henry Bottomley, May 08 2001
a(n) = A060924(n-1, 0) = 3*A001906(n) - 2*A001906(n-1), n >= 1. - Wolfdieter Lang, Apr 26 2001
a(n) ~ phi^(2*n) where phi=(1+sqrt(5))/2. - Joe Keane (jgk(AT)jgk.org), May 15 2002
a(0)=2, a(1)=3, a(n) = 3*a(n-1) - a(n-2) = a(-n). - Michael Somos, Jun 28 2003
a(n) = phi^(2*n) + phi^(-2*n) where phi=(sqrt(5)+1)/2, the golden ratio. E.g., a(4)=47 because phi^(8) + phi^(-8) = 47. - Dennis P. Walsh, Jul 24 2003
With interpolated zeros, trace(A^n)/4, where A is the adjacency matrix of path graph P_4. Binomial transform is then A049680. - Paul Barry, Apr 24 2004
a(n) = (floor((3+sqrt(5))^n) + 1)/2^n. - Lekraj Beedassy, Oct 22 2004
a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2^n (Note: substituting the number 1 for 3 in the last equation gives A000204, substituting 5 for 3 gives A020876). - Creighton Dement, Apr 19 2005
a(n) = (1/(n+1/2))*Sum_{k=0..n} B(2k)*L(2n+1-2k)*binomial(2n+1, 2k) where B(2k) is the (2k)-th Bernoulli number. - Benoit Cloitre, Nov 02 2005
a(n) = term (1,1) in the 1 X 2 matrix [2,3] . [3,1; -1,0]^n. - Alois P. Heinz, Jul 31 2008
a(n) = 2*cosh(2*n*psi), where psi=log((1+sqrt(5))/2). - Al Hakanson, Mar 21 2009
From Sarah-Marie Belcastro, Jul 04 2009: (Start)
a(n) - (a(n) - F(2n))/2 - F(2n+1) = 0. (Tesler)
Product_{r=1..n} (1 + 4*(sin((4r-1)*Pi/(4n)))^2). (Lu/Wu) (End)
a(n) = Fibonacci(2n+6) mod Fibonacci(2n+2), n > 1. - Gary Detlefs, Nov 22 2010
a(n) = 5*Fibonacci(n)^2 + 2*(-1)^n. - Gary Detlefs, Nov 22 2010
a(n) = A033888(n)/A001906(n), n > 0. - Gary Detlefs, Dec 26 2010
a(n) = 2^(2*n) * Sum_{k=1..2} (cos(k*Pi/5))^(2*n). - L. Edson Jeffery, Jan 21 2012
From Peter Bala, Jan 04 2013: (Start)
Let F(x) = Product_{n>=0} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(3 - sqrt(5)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.31829 56058 81914 31334 ... = 2 + 1/(3 + 1/(7 + 1/(18 + ...))).
Also F(-alpha) = 0.64985 97768 07374 32950 has the continued fraction representation 1 - 1/(3 - 1/(7 - 1/(18 - ...))) and the simple continued fraction expansion 1/(1 + 1/((3-2) + 1/(1 + 1/((7-2) + 1/(1 + 1/((18-2) + 1/(1 + ...))))))).
F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((3^2-4) + 1/(1 + 1/((7^2-4) + 1/(1 + 1/((18^2-4) + 1/(1 + ...))))))).
Added Oct 13 2019: 1/2 + 1/2*F(alpha)/F(-alpha) = 1.5142923542... has the simple continued fraction expansion 1 + 1/((3 - 2) + 1/(1 + 1/((18 - 2) + 1/(1 + 1/(123 - 2) + 1/(1 + ...))))). (End)
G.f.: (W(0)+6)/(5*x), where W(k) = 5*x*k + x - 6 + 6*x*(5*k-9)/W(k+1) (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
Sum_{n >= 1} 1/( a(n) - 5/a(n) ) = 1. Compare with A001906, A002878 and A023039. - Peter Bala, Nov 29 2013
0 = a(n) * a(n+2) - a(n+1)^2 - 5 for all n in Z. - Michael Somos, Aug 24 2014
a(n) = (G(j+2n) + G(j-2n))/G(j), for n >= 0 and any j, positive or negative, except where G(j) = 0, and for any sequence of the form G(n+1) = G(n) + G(n-1) with any initial values for G(0), G(1), including non-integer values. G(n) includes Lucas, Fibonacci. Compare with A081067 for odd number offsets from j. - Richard R. Forberg, Nov 16 2014
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 5*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
From J. M. Bergot, Oct 28 2015: (Start)
For n>0, a(n) = F(n-1) * L(n) + F(2*n+1) - (-1)^n with F(k) = A000045(k).
For n>1, a(n) = F(n+1) * L(n) + F(2*n-1) - (-1)^n.
For n>2, a(n) = 5*F(2*n-3) + 2*L(n-3) * L(n) + 8*(-1)^n. (End)
For n>1, a(n) = L(n-2)*L(n+2) -7*(-1)^n. - J. M. Bergot, Feb 10 2016
a(n) = 6*F(n-1)*L(n-1) - F(2*n-6) with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, Apr 21 2017
a(n) = F(2*n) + 2*F(n-1)*L(n) with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, May 01 2017
E.g.f.: exp(4*x/(1+sqrt(5))^2) + exp((1/4)*(1+sqrt(5))^2*x). - Stefano Spezia, Aug 13 2019
From Peter Bala, Oct 14 2019: (Start)
a(n) = F(2*n+2) - F(2*n-2) = A001906(n+1) - A001906(n-1).
a(n) = trace(M^n), where M is the 2 X 2 matrix [0, 1; 1, 1]^2 = [1, 1; 1, 2].
Consequently the Gauss congruences hold: a(n*p^k) = a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. See Zarelua and also Stanley (Ch. 5, Ex. 5.2(a) and its solution).
Sum_{n >= 1} (-1)^(n+1)/( a(n) + 1/a(n) ) = 1/5.
Sum_{n >= 1} (-1)^(n+1)/( a(n) + 3/(a(n) + 2/(a(n))) ) = 1/6.
Sum_{n >= 1} (-1)^(n+1)/( a(n) + 9/(a(n) + 4/(a(n) + 1/(a(n)))) ) = 1/9.
x*exp(Sum_{n >= 1} a(n)*x^/n) = x + 3*x^2 + 8*x^3 + 21*x^4 + ... is the o.g.f. for A001906. (End)
a(n) = n + 2 + Sum_{k=1..n-1} k*a(n-k). - Yu Xiao, May 30 2020
Sum_{n>=1} 1/a(n) = A153415. - Amiram Eldar, Nov 11 2020
Sum_{n>=0} 1/(a(n) + 3) = (2*sqrt(5) + 1)/10 (André-Jeannin, 1991). - Amiram Eldar, Jan 23 2022
a(n) = 2*cosh(2*n*arccsch(2)) = 2*cosh(2*n*asinh(1/2)). - Peter Luschny, May 25 2022
a(n) = (5/2)*(Sum_{k=-n..n} binomial(2*n, n+5*k)) - (1/2)*4^n. - Greg Dresden, Jan 05 2023
a(n) = Sum_{k>=0} Lucas(2*n*k)/(Lucas(2*n)^(k+1)). - Diego Rattaggi, Jan 12 2025

Additional comments from Michael Somos, Jun 23 2001

A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 3, 1, 4, 6, 1, 7, 17, 9, 1, 11, 38, 39, 12, 1, 18, 80, 120, 70, 15, 1, 29, 158, 315, 280, 110, 18, 1, 47, 303, 753, 905, 545, 159, 21, 1, 76, 566, 1687, 2568, 2120, 942, 217, 24, 1, 123, 1039, 3612, 6666, 7043, 4311

Offset: 0

Wolfdieter Lang, Apr 20 2001

In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) is (1+2*z)/(1-(1+x)*z-(1+2*x)*z^2).
Row sums give A060925. Column sequences (without leading zeros) are, for m=0..6: A000032(n+1)= A000204(n+1) (Lucas), A004799(n+1), A060929-33.
Bisection of this triangle gives triangles A060923 (even part) and A060924 (odd part).
For the m-th column sequence (without leading zeros) one has: a(n+m,m)= (pL1(m,n)*L(n+2)+pL2(m,n)*L(n+1))/(m!*5^m), m >= 0, with the Lucas numbers L(n)=A000032(n), n >= 0 and the row polynomials pL1(n,x) := sum(A061188(n,m)*x^n,m=0..n) and pL2(n,x) := sum(A061189(n,m)*x^m,m=0..n).
Riordan array ((1+2*x)/(1-x-x^2), x*(1+2*x)/(1-x-x^2)). - Philippe Deléham, Jan 21 2014
T is the convolution triangle of A000204 (see A357368). - Peter Luschny, Oct 19 2022

			p(2,x) = 4+6*x+x^2.
Triangle begins:
1 ;
3, 1;
4, 6, 1;
7, 17, 9, 1;
11, 38, 39, 12, 1;
18, 80, 120, 70, 15, 1;
29, 158, 315, 280, 110, 18, 1;
47, 303, 753, 905, 545, 159, 21, 1;

Cf. A000032.

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000204); # Peter Luschny, Oct 19 2022

a(n, m)=((n-m+1)*a(n, m-1)+2*(2*n-m)*a(n-1, m-1)+4*(n-1)*a(n-2, m-1))/(5*m), n >= m >= 1, a(n, 0)= A000204(n+1)= A000032(n+1).
G.f. for m-th column: ((1+2*x)/(1-x-x^2))* ((x*(1+2*x))/(1-x-x^2))^m.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 21 2014

Example improved by Philippe Deléham, Jan 21 2014

A060923 Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).

Original entry on oeis.org

1, 4, 1, 11, 17, 1, 29, 80, 39, 1, 76, 303, 315, 70, 1, 199, 1039, 1687, 905, 110, 1, 521, 3364, 7470, 6666, 2120, 159, 1, 1364, 10493, 29634, 37580, 20965, 4311, 217, 1, 3571, 31885, 109421, 181074, 148545

Offset: 0

Wolfdieter Lang, Apr 20 2001

			Triangle begins:
  {1};
  {4,1};
  {11,17,1};
  {29,80,39,1};
  ...
pLe(2,x) = 1+11*x-11*x^2+4*x^3.

Row sums give A060926.
Column sequences (without leading zeros) are, for m=0..3: A002878, A060934-A060936.
Companion triangle A060924 (odd part).
Cf. A060922.

a(n, m) = A060922(2*n-m, m).
a(n, m) = ((2*(n-m)+1)*A060924(n-1, m-1) + 2*(4*n-3*m)*a(n-1, m-1) + 4*(2*n-m-1)*A060924(n-2, m-1))/(5*m), m >= n >= 1; a(n, 0)= A002878(n); else 0.
G.f. for column m >= 0: x^m*pLe(m+1, x)/(1-3*x+x^2)^(m+1), where pLe(n, x) := Sum_{m=0..n+floor(n/2)} A061186(n, m)*x^m are the row polynomials of the (signed) staircase A061186.
T(n,k) = 3*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2) + 4*T(n-3,k-2), T(0,0) = 1, T(1,0) = 4, T(1,1) = 1, T(2,0) = 11, T(2,1) = 17, T(2,2) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 21 2014

A061171 One half of second column of Lucas bisection triangle (odd part).

Original entry on oeis.org

3, 19, 79, 283, 940, 2982, 9171, 27581, 81557, 237995, 687158, 1966764, 5588259, 15780103, 44323195, 123920827, 345062176, 957403026, 2647935987, 7302634865, 20087869313, 55128445259, 150971982314

Offset: 0

Wolfdieter Lang, Apr 20 2001

Numerator of g.f. is on half of row polynomial Sum_{m=0..2} A061187(1,m) * x^m.

G. C. Greubel, Table of n, a(n) for n = 0..1000
É. Czabarka, R. Flórez, L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000032, A000045, A001906, A002878, A005248.

I:=[3,19,79,283]; [n le 4 select I[n] else 6*Self(n-1) - 11*Self(n-2) + 6*Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[(1+x)(3-2x)/(1-3x+x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{6,-11,6,-1},{3,19,79,283},30] (* Harvey P. Dale, Oct 11 2012 *)

my(x='x+O('x^30)); Vec((1+x)*(3-2*x)/(1-3*x+x^2)^2) \\ G. C. Greubel, Dec 21 2017

2*a(n) = A060924(n+1, 1).
G.f.: (1+x)*(3-2*x)/(1-3*x+x^2)^2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4), with a(0)=3, a(1)=19, a(2)=79, a(3)=283. - Harvey P. Dale, Oct 11 2012
a(n) = Fibonacci(2*n+4) + n*Lucas(2*n+3). - Lechoslaw Ratajczak, May 06 2020

A061172 Third column of Lucas bisection triangle (odd part).

Original entry on oeis.org

9, 120, 753, 3612, 15040, 57366, 206115, 709152, 2360943, 7659870, 24340184, 76031100, 234116493, 712166952, 2143779645, 6394719216, 18923041360, 55601888562, 162350117703, 471371537040, 1361642740059

Offset: 0

Wolfdieter Lang, Apr 20 2001

Numerator of g.f. is row polynomial Sum_{m=0..4} A061187(2,m)*x^m.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-30,45,-30,9,-1).

Cf. A002548, 2*A061171.

I:=[9, 120, 753, 3612, 15040, 57366]; [n le 6 select I[n] else 9*Self(n-1)-30*Self(n-2)+45*Self(n-3)-30*Self(n-4)+9*Self(n-5) - Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[(3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3, {x,0,50}], x] (* or *) LinearRecurrence[{9,-30,45,-30,9,-1}, {9, 120, 753, 3612, 15040, 57366}, 30] (* G. C. Greubel, Dec 21 2017 *)

x='x+O('x^30); Vec((3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3) \\ G. C. Greubel, Dec 21 2017

a(n) = A060924(n+2, 2).

G.f.: (3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3.

A060926 Row sums of triangle A060923 (even part of bisection of Lucas triangle).

Original entry on oeis.org

1, 5, 29, 149, 765, 3941, 20301, 104565, 538589, 2774149, 14289005, 73599381, 379093501, 1952623525, 10057515149, 51803949749, 266830242845, 1374381274821, 7079122173101, 36462931836885

Offset: 0

Wolfdieter Lang, Apr 20 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,0,4).

Cf. A060927 (Row sums of A060924).

I:=[1,5,29]; [n le 3 select I[n] else 5*Self(n-1) + 4*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 09 2021

LinearRecurrence[{5,0,4}, {1,5,29}, 31] (* G. C. Greubel, Apr 09 2021 *)

def A060926_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+4*x^2)/(1-5*x-4*x^3) ).list()
A060926_list(30) # G. C. Greubel, Apr 09 2021

a(n) = Sum_{j=0..n} A060923(n, j).
a(n) = A060928(n) + 4*A060928(n-2), n >= 2, otherwise A060928(n).
G.f.: (1+4*x^2)/(1-5*x-4*x^3).

A061173 One-fourth of fourth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

3, 70, 642, 4055, 20945, 95372, 398290, 1561683, 5836190, 21001410, 73300478, 249476600, 831342517, 2720979140, 8768966810, 27881856697, 87610794135, 272424413508, 839229287580, 2563768831145, 7773145679478

Offset: 0

Wolfdieter Lang, Apr 20 2001

Numerator of g.f. is row polynomial sum(A061187(3,m)*x^m,m=0..5)/4.

Cf. A005248, 2*A061171, A061172.

a(n)=A060924(n+3, 3)/4.

G.f.: (x+1)*(3-2*x)*(1+11*x-11*x^2+4*x^3)/(1-3*x+x^2)^4.

A061174 Fifth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

15, 545, 7043, 57560, 365045, 1970905, 9520315, 42385132, 177293730, 705980760, 2701362950, 10001654350, 36020160943, 126701700755, 436709397085, 1478813477920, 4930328078835, 16212542696607

Offset: 0

Wolfdieter Lang, Apr 20 2001

Numerator of g.f. is row polynomial sum(A061187(4,m)*x^m,m=0..7).

Cf. A002548, 2*A061171, A061172, 4*A061173.

a(n)=A060924(n+4, 4).

G.f.: (3-2*x)*(16*x^6-56*x^5+181*x^4-306*x^3+171*x^2+110*x+5)/(1-3*x+x^2)^5.

cp's OEIS Frontend

A061187 Staircase of coefficients of polynomials used for column g.f.s of triangle A060924.

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A060927 Row sums of triangle A060924 (odd part of bisection of Lucas triangle).

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A005248 Bisection of Lucas numbers: a(n) = L(2*n) = A000032(2*n).

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A060922 Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

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A060923 Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).

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A061171 One half of second column of Lucas bisection triangle (odd part).

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A061172 Third column of Lucas bisection triangle (odd part).

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A005248 Bisection of Lucas numbers: a(n) = L(2n) = A000032(2n).