A002534 -id:A002534 - cp's OEIS frontend

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A111015 Primes in A002535.

Original entry on oeis.org

11, 31, 601, 10711, 45281, 3245551, 4057691201, 87818089575031, 813086055916584907683448771376472778745411281, 16071419731004292876206308878779566599797733387541964081866111137961, 2259503969983505641049567911781316556859822340375755577282633545849516496717511

Offset: 1

Cino Hilliard, Oct 02 2005

Original name: Starting with the fraction 1/1, this sequence gives the prime numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 10 times bottom to get the new top.
Conjecture: Starting with 1/1, there are infinitely many primes in the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 2k times bottom to get the new top, for k>=1.
a(12) has 5304 digits and is not included here. - Bill McEachen, Jan 22 2023
a(12) = A002535(8563) = 1.0733...*10^5303. - Amiram Eldar, Jun 30 2024

			The raw ratios begin 1/1, 11/2, 31/13, 161/44, 601/205, ... = A002535/A002534. Among the numerators, 11, 31, and 601 are primes and are the first three terms here.

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Cf. A002534, A002535.

Select[Numerator/@NestList[(10Denominator[#]+Numerator[#])/ (Denominator[#]+ Numerator[#])&,1/1,200],PrimeQ] (* Harvey P. Dale, Sep 15 2011 *)
Select[LinearRecurrence[{2, 9}, {1, 1}, 150], PrimeQ] (* Amiram Eldar, Jun 30 2024 *)

\\ k=mult,typ=1 num,2 denom. output prime num or denom
primenum(n,k,typ) = {local(a,b,x,tmp,v); a=1;b=1; for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1,v=a,v=b); if(isprime(v), print1(v, ", "); ) ); print(); print(a/b+.)}
primenum(100, 10, 1)

from sympy import isprime
from itertools import islice
from fractions import Fraction
def agen(): # generator of terms
    f = Fraction(1, 1)
    while True:
        n, d = f.numerator + 10*f.denominator, f.numerator + f.denominator
        if isprime(n): yield n
        f = Fraction(n, d)
print(list(islice(agen(), 11))) # Michael S. Branicky, Jan 23 2023

Given t(0)=1, b(0)=1 then for i = 1, 2, ..., t(i)/b(i) = (t(i-1) + 10*b(i-1)) /(t(i-1) + b(i-1)), and sequence consists of the t(i) that are prime.

a(11) from Michel Marcus, Jan 23 2023

Name simplified by Sean A. Irvine, Feb 25 2023

A123007 Expansion of x(1+x)/(1 -2x -9*x^2).

Original entry on oeis.org

1, 3, 15, 57, 249, 1011, 4263, 17625, 73617, 305859, 1274271, 5301273, 22070985, 91853427, 382345719, 1591372281, 6623856033, 27570062595, 114754829487, 477640222329, 1988073910041, 8274909821043, 34442484832455

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,9).

Cf. A002534.

[n le 2 select 3^(n-1) else 2*Self(n-1) +9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 12 2021

M:= {{0, 3}, {3, 2}}; v[1]= {1, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n,30}]
Rest[CoefficientList[Series[(x(x+1))/(1-2x-9x^2),{x,0,30}],x]] (* or *) LinearRecurrence[{2,9},{1,3},30] (* Harvey P. Dale, Aug 07 2015 *)

[(3*i)^(n-2)*(3*i*chebyshev_U(n-1, -i/3) + chebyshev_U(n-2, -i/3)) for n in [1..30]] # G. C. Greubel, Jul 12 2021

From Philippe Deléham, Oct 18 2006: (Start)
a(n) = 2*a(n-1) + 9*a(n-2) for n > 2.
G.f.: x*(1+x)/(1 -2*x -9*x^2). (End)
a(n) = A002534(n) + A002534(n-1). - R. J. Mathar, Sep 17 2013
a(n) = (3*i)^(n-2)*(3*i*chebyshev_U(n-1, -i/3) + chebyshev_U(n-2, -i/3)). - G. C. Greubel, Jul 12 2021

Definition replaced with the Deléham formula by R. J. Mathar, Sep 17 2013

A123008 Expansion of x(1 + 3x)/(1 - 2x - 25x^2).

Original entry on oeis.org

1, 5, 35, 195, 1265, 7405, 46435, 277995, 1716865, 10383605, 63688835, 386967795, 2366156465, 14406507805, 87966927235, 536096549595, 3271366280065, 19945146300005, 121674449601635, 741977556703395, 4525816353447665

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,25).

Cf. A002534, A123004.

[n le 2 select 5^(n-1) else 2*Self(n-1) + 25*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 13 2021

M:= {{0, 5}, {5, 2}}; v[1] = {1, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 30}]

[(5*i)^(n-2)*(3*chebyshev_U(n-2, -i/5) + 5*i*chebyshev_U(n-1, -i/5)) for n in (1..30)] # G. C. Greubel, Jul 13 2021

From Colin Barker, Oct 19 2012: (Start)
a(n) = 2*a(n-1) + 25*a(n-2) for n>2.
G.f.: x*(1+3*x)/(1-2*x-25*x^2). (End)
a(n) = (5*i)^(n-2)*(3*ChebyshevU(n-2, -i/5) + 5*i*ChebyshevU(n-1, -i/5)). - G. C. Greubel, Jul 13 2021

Sequence edited by Joerg Arndt and Colin Barker, Oct 19 2012

A123009 Expansion of x(1 + 5x)/(1 - 2x - 49x^2).

Original entry on oeis.org

1, 7, 63, 469, 4025, 31031, 259287, 2039093, 16783249, 133482055, 1089343311, 8719307317, 70816436873, 568878932279, 4607763271335, 37090594224341, 299961588744097, 2417362294480903, 19532842437422559, 157516437304409365

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 23 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,49).

Cf. A002534, A123005.

[n le 2 select 7^(n-1) else 2*Self(n-1) + 25*Self(n-2): n in [1..31]]; // G. C. Greubel, Jul 13 2021

M:= {{0, 7}, {7, 2}}; v[1]= {1, 1}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 30}]

[(7*i)^(n-2)*(5*chebyshev_U(n-2, -i/7) + 7*i*chebyshev_U(n-1, -i/7)) for n in (1..30)] # G. C. Greubel, Jul 13 2021

From Colin Barker, Oct 19 2012: (Start)
a(n) = 2*a(n-1) + 49*a(n-2) for n>2.
G.f.: x*(1 +5*x)/(1 -2*x -49*x^2). (End)
a(n) = (7*i)^(n-2)*(5*ChebyshevU(n-2, -i/7) + 7*i*ChebyshevU(n-1, -i/7)). - G. C. Greubel, Jul 13 2021

Sequence edited by Joerg Arndt and Colin Barker, Oct 19 2012

A143685 Pascal-(1,9,1) array.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 141, 31, 1, 1, 41, 361, 361, 41, 1, 1, 51, 681, 1991, 681, 51, 1, 1, 61, 1101, 5921, 5921, 1101, 61, 1, 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1, 1, 81, 2241, 24681, 96201, 96201, 24681, 2241, 81, 1, 1, 91, 2961, 41511, 239241, 460251, 239241, 41511, 2961, 91, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 11,   21,    31,     41,      51,       61, ... A017281;
  1, 21,  141,   361,    681,    1101,     1621, ...
  1, 31,  361,  1991,   5921,   13151,    24681, ...
  1, 41,  681,  5921,  29761,   96201,   239241, ...
  1, 51, 1101, 13151,  96201,  460251,  1565301, ...
  1, 61, 1621, 24681, 239241, 1565301,  7272861, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 11,    1;
  1, 21,   21,     1;
  1, 31,  141,    31,     1;
  1, 41,  361,   361,    41,     1;
  1, 51,  681,  1991,   681,    51,    1;
  1, 61, 1101,  5921,  5921,  1101,   61,  1;
  1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A002534, A143680, A143682.

Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8), this sequence (m = 9).

A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143685(n,k,9): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021

Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 10).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021

Square array: T(n, k) = T(n, k-1) + 9*T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(0, k) = 1.
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*10^j.
Riordan array (1/(1-x), x*(1+9*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 10). - Jean-François Alcover, May 24 2013
Sum_{k=0..n} T(n, k) = A002534(n+1). - G. C. Greubel, May 29 2021

A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

Original entry on oeis.org

2, 1, 5, 6, 2, 9, 19, 13, 3, 29, 72, 69, 30, 5, 65, 213, 278, 182, 60, 8, 181, 682, 1084, 928, 451, 118, 13, 441, 1975, 3795, 4065, 2625, 1023, 223, 21, 1165, 5868, 13015, 16590, 13290, 6852, 2221, 414, 34, 2929, 16697, 42404, 63020, 60435, 38799, 16682

Offset: 1

Clark Kimberling, Nov 13 2013

Sum of numbers in row n: A002534(n). Left edge: A006131. Right edge: A000045 (Fibonacci numbers).

			First 3 rows:
2 ... 1
5 ... 6 .... 2
9 ... 19 ... 13 ... 3
First 3 polynomials:  2 + x, 5 + 6*x + 2*x^2, 9 + 19*x + 13*x^2 + 3*x^3.

Cf. A230000, A231775, A000045.

t[n_] := t[n] = Table[(x + 1)/(x + 2), {k, 0, n}];
b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
p[x_, n_] := p[x, n] = Last[Expand[Numerator[b]]][[n]];
u = Table[p[x, n], {n, 1, 10}]
v = CoefficientList[u, x]; Flatten[v]

A231775 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

Original entry on oeis.org

2, 1, 2, 3, 1, 10, 17, 10, 2, 18, 47, 45, 19, 3, 58, 173, 210, 129, 40, 5, 130, 491, 769, 642, 302, 76, 8, 362, 1545, 2850, 2940, 1830, 687, 144, 13, 882, 4391, 9565, 11925, 9315, 4671, 1469, 265, 21, 2330, 12901, 31898, 46195, 43170, 26994, 11294, 3049, 482

Offset: 1

Clark Kimberling, Nov 13 2013

Sum of numbers in row n: 3*A002534(n). Left edge: 2*A006131. Right edge: A000045 (Fibonacci numbers).

			First 3 rows:
2 .... 1
2 .... 3 .... 1
10 ... 17 ... 10 ... 2
First 3 polynomials:  2 + x, 2 + 3*x + x^2, 10 + 17*x + 10*x^2 + 2*x^3.

Cf. A230000, A231774, A000045.

t[n_] := t[n] = Table[(x + 1)/(x + 2), {k, 0, n}];
b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];
p[x_, n_] := p[x, n] = Last[Expand[Denominator[b]]][[n]];
u = Table[p[x, n], {n, 1, 10}]
v = CoefficientList[u, x]; Flatten[v]

A094726 Let M = the 2 X 2 matrix [ 0 3 / 3 2]. Take (M^n * [1 1])/3 = [p q]; then a(n) = p.

Original entry on oeis.org

1, 5, 19, 83, 337, 1421, 5875, 24539, 101953, 424757, 1767091, 7356995, 30617809, 127448573, 530457427, 2207952011, 9190020865, 38251609829, 159213407443, 662691303347, 2758303273681, 11480828277485, 47786386018099

Offset: 1

Gary W. Adamson, May 23 2004

			a(6) = 1421 since (M^n * [1 1])/3 = [1421 q].

Index entries for linear recurrences with constant coefficients, signature (2,9).

Cf. A002534.

a[n_] := (MatrixPower[{{0, 3}, {3, 2}}, n].{{1}, {1}})[[1, 1]]/3; Table[ a[n], {n, 22}] (* Robert G. Wilson v, Jun 05 2004 *)

From Colin Barker, Nov 08 2012: (Start)
a(n) = 2*a(n-1)+9*a(n-2).
G.f.: -x*(3*x+1)/(9*x^2+2*x-1). (End)

Edited, corrected and extended by Robert G. Wilson v, Jun 05 2004

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

Original entry on oeis.org

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549

Offset: 1

XU Pingya, Sep 24 2017

nonn

			For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]

g(n,k) = ([0,1;2*n-1,2]^k*[0;1])[1,1]
a(n) = for(k=3,oo,if(ispseudoprime(g(n,k)),return(g(n,k)))) \\ Jason Yuen, Apr 12 2025

When 2*n + 3 = p is prime, a(n) = p.

cp's OEIS Frontend

A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).

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A111015 Primes in A002535.

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A123007 Expansion of x*(1+x)/(1 -2*x -9*x^2).

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A123008 Expansion of x*(1 + 3*x)/(1 - 2*x - 25*x^2).

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A123009 Expansion of x*(1 + 5*x)/(1 - 2*x - 49*x^2).

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Magma

Mathematica

Sage

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A143685 Pascal-(1,9,1) array.

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Magma

Mathematica

Sage

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A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

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A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

A123007 Expansion of x(1+x)/(1 -2x -9*x^2).

A123008 Expansion of x(1 + 3x)/(1 - 2x - 25x^2).

A123009 Expansion of x(1 + 5x)/(1 - 2x - 49x^2).

A231774 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

A231775 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)x + ... + c(n)x^(n) which is the denominator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x + 2).

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).