A001333 -id:A001333 - cp's OEIS frontend

A002965 Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).

0, 1, 1, 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, 169, 239, 408, 577, 985, 1393, 2378, 3363, 5741, 8119, 13860, 19601, 33461, 47321, 80782, 114243, 195025, 275807, 470832, 665857, 1136689, 1607521, 2744210, 3880899, 6625109, 9369319, 15994428, 22619537

Offset: 0

N. J. A. Sloane

Denominators of Farey fraction approximations to sqrt(2). The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, .... See A082766(n+2) or A119016 for the numerators. "Add" (meaning here to add the numerators and add the denominators, not to add the fractions) 1/0 to 1/1 to make the fraction bigger: 2/1. Now 2/1 is too big, so add 1/1 to make the fraction smaller: 3/2, 4/3. Now 4/3 is too small, so add 3/2 to make the fraction bigger: 7/5, 10/7, ... Because the continued fraction for sqrt(2) is all 2's, it will always take exactly two terms here to switch from a number that's bigger than sqrt(2) to one that's less. A097545/A097546 gives the similar sequence for Pi. A119014/A119015 gives the similar sequence for e. - Joshua Zucker, May 09 2006
The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7; essentially, numerators=A143607, denominators=A002965. - Clark Kimberling, Aug 27 2008
(a(2n)*a(2n+1))^2 is a triangular square. - Hugh Darwen, Feb 23 2012
a(2n) are the interleaved values of m such that 2*m^2+1 and 2*m^2-1 are squares, respectively; a(2n+1) are the interleaved values of their corresponding integer square roots. - Richard R. Forberg, Aug 19 2013
Coefficients of (sqrt(2)+1)^n are a(2n)*sqrt(2)+a(2n+1). - John Molokach, Nov 29 2015
Apart from the first two terms, this is the sequence of denominators of the convergents of the continued fraction expansion sqrt(2) = 1/(1 - 1/(2 + 1/(1 - 1/(2 + 1/(1 - ....))))). - Peter Bala, Feb 02 2017
Limit_{n->infinity} a(2n+1)/a(2n) = sqrt(2); lim_{n->infinity} a(2n)/a(2n-1) = (2+sqrt(2))/2. - Ctibor O. Zizka, Oct 28 2018

			The convergents are rational numbers given by the recurrence relation p/q -> (p + 2*q)/(p + q). Starting with 1/1, the next three convergents are (1 + 2*1)/(1 + 1) = 3/2, (3 + 2*2)/(3 + 2) = 7/5, and (7 + 2*5)/(7 + 5) = 17/12. The sequence puts the denominator first, so a(2) through a(9) are 1, 1, 2, 3, 5, 7, 12, 17. - _Michael B. Porter_, Jul 18 2016

C. Brezinski, History of Continued Fractions and Padé Approximants. Springer-Verlag, Berlin, 1991, p. 24.
Jay Kappraff, Musical Proportions at the Basis of Systems of Architectural Proportion both Ancient and Modern, in Volume I 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. See Eq. 32.7.
Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Guelena Strehler, Chess Fractal, April 2016, p. 24.

T. D. Noe, Table of n, a(n) for n = 0..500
Damanvir Singh Binner, Proofs of Chappelon and Alfonsín Conjectures On Square Frobenius Numbers and its Relationship to Simultaneous Pell's Equations, arXiv:2112.15474 [math.NT], 2021.
Jonathan Chappelon and Jorge Luis Ramírez Alfonsín, The Square Frobenius Number, arXiv:2006.14219 [math.NT], 2020.
H. S. M. Coxeter, The role of intermediate convergents in Tait's explanation for phyllotaxis, J. Algebra 20 (1972), 167-175.
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Pierre Lamothe, En marge du problème des cercles tangents
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
Dave Rusin, Farey fractions on sci.math [Broken link]
Dave Rusin, Farey fractions on sci.math [Cached copy]
K. Williams, The sacred cult revisited: the pavement of the baptistery of San Giovanni, Florence, Math. Intellig., 16 (No. 2, 1994), 18-24.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A000129(n) = a(2n), A001333(n) = a(2n+1).

Cf. A001109, A155046.

a:=[0,1];; for n in [3..45] do a[n]:=a[n-1]+a[n-2-((n-1) mod 2)]; od; a; # Muniru A Asiru, Oct 28 2018

import Data.List (transpose)
a002965 n = a002965_list !! n
a002965_list = concat $ transpose [a000129_list, a001333_list]
-- Reinhard Zumkeller, Jan 01 2014

a=new Array(); a[0]=0; a[1]=1;
for (i=2;i<50;i+=2) {a[i]=a[i-1]+a[i-2];a[i+1]=a[i]+a[i-2];}
document.write(a); // Jon Perry, Sep 12 2012

I:=[0,1,1,1]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Nov 30 2015

A002965 := proc(n) option remember; if n <= 0 then 0; elif n <= 3 then 1; else 2*A002965(n-2)+A002965(n-4); fi; end;
A002965:=-(1+2*z+z**2+z**3)/(-1+2*z**2+z**4); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for two leading terms

LinearRecurrence[{0, 2, 0, 1}, {0, 1, 1, 1}, 42] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
With[{c=Convergents[Sqrt[2],20]},Join[{0,1},Riffle[Denominator[c], Numerator[c]]]] (* Harvey P. Dale, Oct 03 2012 *)

PARI
```
a(n)=if(n<4,n>0,2*a(n-2)+a(n-4))
```

x='x+O('x^100); concat(0, Vec((x+x^2-x^3)/(1-2*x^2-x^4))) \\ Altug Alkan, Dec 04 2015

a(n) = 2*a(n-2) + a(n-4) if n>3; a(0)=0, a(1)=a(2)=a(3)=1.
a(2*n) = a(2*n-1) + a(2*n-2) and a(2*n+1) = 2*a(2*n) - a(2*n-1).
G.f.: (x+x^2-x^3)/(1-2*x^2-x^4).
a(0)=0, a(1)=1, a(n) = a(n-1) + a(2*[(n-2)/2]). - Franklin T. Adams-Watters, Jan 31 2006
For n > 0, a(2*n) = a(2*n-1) + a(2*n-2) and a(2*n+1) = a(2*n) + a(2*n-2). - Jon Perry, Sep 12 2012
a(n) = (((sqrt(2) - 2)*(-1)^n + 2 + sqrt(2))*(1 + sqrt(2))^(floor(n/2)) - ((2 + sqrt(2))*(-1)^n -2 + sqrt(2))*(1 - sqrt(2))^(floor(n/2)))/8. - Ilya Gutkovskiy, Jul 18 2016
a(n) = a(n-1) + a(n-2-(n mod 2)); a(0)=0, a(1)=1. - Ctibor O. Zizka, Oct 28 2018

Thanks to Michael Somos for several comments which improved this entry.

A054458 Convolution triangle based on A001333(n), n >= 1.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 17, 23, 9, 1, 41, 76, 48, 12, 1, 99, 233, 204, 82, 15, 1, 239, 682, 765, 428, 125, 18, 1, 577, 1935, 2649, 1907, 775, 177, 21, 1, 1393, 5368, 8680, 7656, 4010, 1272, 238, 24, 1, 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1

Offset: 0

Wolfdieter Lang, Apr 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) (increasing powers of x) is LPell(z)/(1-x*z*LPell(z)) with LPell(z) given in 'Formula'.
Column sequences are A001333(n+1), A054459(n), A054460(n) for m=0..2.
Mirror image of triangle in A209696. - Philippe Deléham, Mar 24 2012
Subtriangle of the triangle given by (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
Riordan array ((1+x)/(1-2*x-x^2), (x+x^2)/(1-2*x-x^2)). - Philippe Deléham, Mar 25 2012

			Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3.
Triangle begins :
  1
  3, 1
  7, 6, 1
  17, 23, 9, 1
  41, 76, 48, 12, 1
  99, 233, 204, 82, 15, 1
  239, 682, 765, 428, 125, 18, 1. - _Philippe Deléham_, Mar 25 2012
(0, 3, -2/3, -1/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 3, 1
  0, 7, 6, 1
  0, 17, 23, 9, 1
  0, 41, 76, 48, 12, 1
  0, 99, 233, 204, 82, 15, 1
  0, 239, 682, 765, 428, 125, 15, 1. - _Philippe Deléham_, Mar 25 2012

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A002203(n+1)/2. Row sums: A055099(n).
Cf. A001333, A054459, A054460.
Cf. A040000, A001333, A055099, A126473, A126501, A126528.

a(n, m) := ((n-m+1)*a(n, m-1) + (2n-m)*a(n-1, m-1) + (n-1)*a(n-2, m-1))/(4*m), n >= m >= 1; a(n, 0)= A001333(n+1); a(n, m) := 0 if n

G.f. for column m: LPell(x)*(x*LPell(x))^m, m >= 0, with LPell(x)= (1+x)/(1-2*x-x^2) = g.f. for A001333(n+1).

G.f.: (1+x)/(1-2*x-y*x-x^2-y*x^2). - Philippe Deléham, Mar 25 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

Sum_{k=0..n} T(n,k)*x^k = A040000(n), A001333(n+1), A055099(n), A126473(n), A126501(n), A126528(n) for x = -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 25 2012

A099088 Indices of prime companion Pell numbers, divided by 2 (A001333).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, 9679, 28753, 79043, 129127, 145969, 165799, 168677, 170413, 172243, 278321, 552283

Offset: 1

T. D. Noe, Sep 24 2004

Note that for A001333(n) to be prime, the index n must be prime or a power of 2. The indices greater than 421 yield probable primes.

Numbers n for which ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 is prime. - Artur Jasinski, Dec 10 2006

F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 62, 1983.

J. B. Cosgrave and K. Dilcher, Pairs of reciprocal quadratic congruences involving primes, Fib. Quart. 51 (2) (2013) 98, after Theorem 3.
Eric Weisstein's World of Mathematics, Pell Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A002203 (companion Pell numbers), A086395 (primes in A001333), A096650 (indices of prime Pell numbers).

Cf. A005850.

lst={}; a=1; b=1; Do[c=a+2b; a=b; b=c; If[PrimeQ[c], AppendTo[lst, n]], {n, 2, 10000}]; lst
(* Second program: *)
Do[If[PrimeQ[Expand[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2]], Print[n]], {n, 0, 1000}] (* Artur Jasinski, Dec 10 2006 *)

isok(n) = isprime(polchebyshev(n, 1, I)/I^n); \\ Michel Marcus, Dec 07 2018

a(24) from Eric W. Weisstein, May 22 2006
a(25) from Eric W. Weisstein, Aug 29 2006
a(26) from Eric W. Weisstein, Nov 11 2006
a(27) from Eric W. Weisstein, Nov 26 2006
a(28) from Eric W. Weisstein, Dec 10 2006
a(29) from Eric W. Weisstein, Jan 25 2007
a(30) from Robert Price, Dec 07 2018
a(31) from Robert Price, Dec 05 2023

A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

Original entry on oeis.org

1, 1, 5, 21, 101, 501, 2561, 13345, 70561, 377281, 2035285, 11059205, 60454005, 332138405, 1832677185, 10150115201, 56398558081, 314273655745, 1755700634981, 9830544087221, 55155558312901, 310027473436821, 1745567243959105, 9843160519978401, 55582528404717601

Offset: 0

Paul D. Hanna, Jan 10 2012

nonn

a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 101*x^4 + 501*x^5 + 2561*x^6 +...
where log(A(x)) = x + 3^2*x^2/2 + 7^2*x^3/3 + 17^2*x^4/4 + 41^2*x^5/5 + 99^2*x^6/6 + 239^2*x^7/7 +...+ A001333(n)^2*x^n/n +...
The last digit of the terms in this sequence seems to be either a '1' or a '5':
by conjecture, a(n) == 0 (mod 5) whenever n has a 2 in its base 5 expansion;
if true, terms a(2*5^k) through a(3*5^k - 1) all end with digit '5' for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..625

Cf. A204062, A026933, A001333, A023729.

{A001333(n)=polcoeff((1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A001333(k)^2*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(1/(sqrt(1+x+x*O(x^n))*(1-6*x+x^2+x*O(x^n))^(1/4)),n)}

G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).
Self-convolution yields A026933: Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers.
a(n) ~ 2^(1/8) * GAMMA(3/4) * (3+2*sqrt(2))^(n+1/2) / (4 * Pi * n^(3/4)). - Vaclav Kotesovec, Oct 30 2014

A054459 A001333(n), n >= 1, convolved with itself.

Original entry on oeis.org

1, 6, 23, 76, 233, 682, 1935, 5368, 14641, 39406, 104935, 276996, 725849, 1890258, 4896415, 12624752, 32419297, 82951766, 211573047, 538086716, 1364974409, 3454480250, 8724052271, 21989264232, 55326056977, 138975010110

Offset: 0

Wolfdieter Lang, Apr 26 2000

a(n) = A054458(n+1,1) (second column of convolution triangle).

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (The sequence t_n, also the sequence lev_{n-1}.)

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A054458, A001333, A002203.

LinearRecurrence[{4, -2, -4, -1}, {1, 6, 23, 76}, 30] (* Paolo Xausa, Feb 06 2024 *)

a(n) = ((4*n+3)*LP(n)+(2*n+1)*LP(n-1))/4, n >= 1, with LP(n) = A001333(n+1), a(0) = 1.
G.f.: ((1+x)/(1-2*x-x^2))^2.
a(n) = 2*A006645(n+2) - A000129(n+1). - R. J. Mathar, Feb 05 2024

A107769 a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.

Original entry on oeis.org

0, 1, 2, 8, 19, 54, 130, 334, 806, 1995, 4816, 11746, 28357, 68748, 165972, 401388, 969036, 2341141, 5652014, 13649228, 32952151, 79563330, 192082870, 463752730, 1119598130, 2703006111, 6525634012, 15754412038, 38034515209, 91823775384, 221682203880, 535188986904, 1292060510616, 3119311948585

Offset: 0

Emeric Deutsch, Jun 12 2005

a(n) is the number of free polyominoes of width 2 and height n+1 which have no symmetry, i.e., rotations by 180 degrees, flips along the short or long axis generate a different free polyomino. The three elements t, g+ and g- of sequences by Tasi et al. represent a domino in the short cross-section where either both, only the "upper" or only the "lower" square of the domino is occupied. E.g., a(3) = 8 represents 3 5-ominoes of shape the 2x4, 3 6-ominoes of shape 2x4, and 2 7-ominoes of shape 2x4. - R. J. Mathar, Jun 17 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63, eq 25).
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,3,-1,1,1).

Cf. A000129, A001333, A002203, A005409, A126877, A135153.

Table[(LucasL[n+2, 2] -4*Fibonacci[Floor[n/2]+2, 2] +2)/8, {n,0,40}] (* G. C. Greubel, May 24 2021 *)

[(lucas_number2(n+2,2,-1) -4*lucas_number1(2+(n//2),2,-1) +2)/8 for n in (0..40)] # G. C. Greubel, May 24 2021

4*a(n) = Pell(n+3) - Pell(n+2) - 2*Pell(floor((n+4)/2)) + 1, with Pell(n) = A000129(n). - Ralf Stephan, Jun 02 2007
G.f.: x*(1-x+x^2)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)). - Colin Barker, Apr 08 2013
4*a(n) = A001333(n+2) -2*A135153(n+4) +1. - R. J. Mathar, Jun 17 2020
From G. C. Greubel, May 24 2021: (Start)
a(n) = (1/4)*(A001333(n+2) - 2*A000129(floor(n/2)+2) + 1).
a(n) = (1/8)*(A002203(n+2) - 4*A000129(floor(n/2)+2) + 2). (End)

Entry revised by N. J. A. Sloane, Jul 29 2011

A152113 A001333 with terms repeated.

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 17, 17, 41, 41, 99, 99, 239, 239, 577, 577, 1393, 1393, 3363, 3363, 8119, 8119, 19601, 19601, 47321, 47321, 114243, 114243, 275807, 275807, 665857, 665857, 1607521, 1607521, 3880899, 3880899, 9369319, 9369319, 22619537, 22619537, 54608393

Offset: 1

N. J. A. Sloane, Sep 21 2009

Suggested by an email message from Hugo van der Sanden, Mar 23 2009, who says: Consider the partitions of a 2 X n rectangle into connected pieces consisting of unit squares cut along lattice lines. Then a(n) is the number of distinct pieces with rotational symmetry that extend to opposite corners.

a(n+2) is the number of palindromic words of length n on a 3-letter alphabet {a,b,c} which do not contain the "ab" subword. See A001906 for the words of length n on a 3-letter alphabet without "ab" subword but not necessarily palindromic. Example length 1: "a" or "b" or "c". Example length 2: "aa", "bb", "cc". Example length 3: There are 9 palindromic words but "aba" and "bab" are not admitted and only 7 remain. - R. J. Mathar, Jul 10 2019

			The pieces illustrating a(3) = 3 are:
 AAA BB. .CC
 AAA .BB CC.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A001333, A078469, A152124.

From Colin Barker, Jul 14 2013: (Start)
a(n) = 2*a(n-2) + a(n-4).
G.f.: -x*(x+1)*(x^2+1) / (x^4+2*x^2-1). (End)
a(n+1) = A135153(n) + A135153(n+2). - R. J. Mathar, Jul 10 2019

A048777 First partial sums of A005409; second partial sums of A001333.

Original entry on oeis.org

1, 5, 16, 44, 113, 281, 688, 1672, 4049, 9789, 23648, 57108, 137889, 332913, 803744, 1940432, 4684641, 11309749, 27304176, 65918140, 159140497, 384199177, 927538896, 2239277016, 5406092977, 13051463021, 31509019072, 76069501220, 183648021569, 443365544417

Offset: 0

Barry E. Williams

Form an array having the first column all 1's and the first row the squares 1, 4, 9, ..., so m(n,1) = 1 and m(1,n) = n^2 for n = 1, 2, 3, ..., and let the interior terms be m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j). Then the sums of the terms in the antidiagonals are the terms of this sequence. - J. M. Bergot, Nov 16 2012

Define a triangle with T(n,n)=n+1 and T(n,0)=n*(n+1)+1 for n >= 0. Define the interior terms via T(r,c) = T(r-2,c-1) + T(r-1,c-1) + T(r-1,c). Then the row sums are a(n) = Sum_{k=0..n} T(n,k). - J. M. Bergot, Feb 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Cf. A000129, A001333, A002203, A005409, A048776.

I:=[1,5,16,44]; [n le 4 select I[n] else 4*Self(n-1) -4*Self(n-2) +Self(n-4): n in [1..36]]; // G. C. Greubel, Apr 23 2021

LinearRecurrence[{4,-4,0,1},{1,5,16,44},40] (* Harvey P. Dale, Nov 12 2017 *)
Table[(LucasL[n+3, 2] -2*(2n+5))/4, {n,0,35}] (* G. C. Greubel, Apr 23 2021 *)

[(lucas_number2(n+3,2,-1) -2*(2*n+5))/4 for n in (0..35)] # G. C. Greubel, Apr 23 2021

a(n) = 2*a(n-1) + a(n-2) + 2*n+1 with a(0)=1, a(1)=5.
a(n) = ( {(5+(7/2)*sqrt(2))*(1+sqrt(2))^n - (5-(7/2)*sqrt(2))*(1-sqrt(2))^n}/2*sqrt(2) ) - (2*n+5)/2.
a(n) = (1/2)*( Pell(n+3) + Pell(n+2) -2*n -5 ), with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007
From Colin Barker, Sep 20 2012: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
G.f.: (1+x)/((1-x)^2*(1-2*x-x^2)). (End)
a(n) = A048776(n-1) + A048776(n). - R. J. Mathar, Feb 28 2013
a(n) = (A002203(n+3) - 2*(2*n+5))/4. - G. C. Greubel, Apr 23 2021
E.g.f.: exp(x)*(7*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x) - 2*x - 5)/2. - Stefano Spezia, May 13 2023

More terms from Harvey P. Dale, Nov 12 2017

A054460 A001333(n), n >= 1, convolved twice with itself.

Original entry on oeis.org

1, 9, 48, 204, 765, 2649, 8680, 27312, 83313, 247985, 723624, 2077164, 5880797, 16454865, 45577200, 125130432, 340882113, 922265721, 2479938368, 6631802220, 17646603933, 46744464745, 123314065944, 324085913136, 848801213425

Offset: 0

Wolfdieter Lang, Apr 26 2000

a(n) = A054458(n+2,2) (third column of convolution triangle).

Index entries for linear recurrences with constant coefficients, signature (6,-9,-4,9,6,1)

Cf. A054458, A054459, A001333, A002203.

LinearRecurrence[{6, -9, -4, 9, 6, 1}, {1, 9, 48, 204, 765, 2649}, 30] (* Paolo Xausa, Feb 06 2024 *)

a(n) = (n+1)*(LP2(n+1)+2*LP2(n)+LP2(n-1))/8, n >= 1, LP2(n) = A054459(n), a(0) = 1.
a(n) = (n+1)*((10*n+11)*LP(n)+(4*n+5)*LP(n-1))/16, n >= 1, LP(n) = A001333(n+1).
G.f.: ((1+x)/(1-2*x-x^2))^3.
a(n) = +6*a(n-1) -9*a(n-2) -4*a(n-3) +9*a(n-4) +6*a(n-5) +a(n-6). - R. J. Mathar, Feb 05 2024

A062134 Triangle of coefficients of polynomials (rising powers) useful for convolutions of A001333(n+1), n >= 0 (associated Pell numbers).

Original entry on oeis.org

1, 2, 0, 8, 24, 16, 336, 832, 576, 128, 12480, 28480, 23680, 8960, 1280, 481920, 1208832, 1167360, 552960, 130560, 12288, 22786560, 61834752, 65709056, 35911680, 10895360, 1763328, 118784, 1280885760, 3645444096

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials pPL1(n,x) := Sum_{m=0..n} A062133(n,m)*x^m and pPL2(n,x) := Sum_{m=0..n} a(n,m)*x^m appear in the k-fold convolution of the associated Pell numbers PL(n) := A001333(n+1), n >= 0, as follows: PL(k; n) := A054458(n+k,k) = (2*pPL1(k,n)*PL(n+1)+pPL2(k,n)*PL(n))/(k!*8^k), k >= 0.

			Triangle begins:
  {1};
  {2,0};
  {8,24,16};
  {336,832,576,128};
  ...
pPL1(1,n) = 1+2*n.
pPL2(1,n) = 2.
PL(1; n) = A054459(n) = ((1+2*n)*PL(n+1)+PL(n))/4.

Cf. A062133(n, m) (companion triangle), A054458(n, m) (convolution triangle).

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cp's OEIS Frontend

A002965 Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).

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A054458 Convolution triangle based on A001333(n), n >= 1.

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A099088 Indices of prime companion Pell numbers, divided by 2 (A001333).

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A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

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A054459 A001333(n), n >= 1, convolved with itself.

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A107769 a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.

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A152113 A001333 with terms repeated.

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