A000129 -id:A000129 - cp's OEIS frontend

A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.

35, 169, 385, 779, 899, 961, 1121, 1189, 2419, 2555, 2915, 3107, 3827, 6083, 6265, 6441, 6601, 6895, 6965, 7801, 8119, 8339, 9179, 9809, 9881, 10403, 10763, 10835, 10945, 13067, 14027, 14111, 15179, 15841, 18241, 18721, 19097, 20833, 20909, 22499, 23219, 24727, 26795, 27869, 27971

Offset: 1

Jianing Song, Sep 20 2019

nonn

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (a) holds for k, where m = 2.
If k is not required to be coprime to m^2 + 4 (= 8), then there are 1232 such k <= 10^5 and 4973 such k <= 10^6, while there are only 83 terms <= 10^5 and 245 terms <= 10^6 in this sequence.
Also composite numbers k coprime to 8 such that A214028(k) divides k - Kronecker(8,k).

			Pell(36) = 21300003689580 is divisible by 35, so 35 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

             m                       m=1           m=2       m=3
k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  this seq  A327653
k | x(k)-Kronecker(m^2+4,k)        A049062       A099011   A327654
            both                   A212424       A327652   A327655
* k is composite and coprime to m^2 + 4.
Cf. A000129, A214028, A091337 ({Kronecker(8,n)}).

pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327651(n)=!isprime(n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1

A363829 Sum of the divisors of A000129(n) (Pell numbers).

Original entry on oeis.org

1, 3, 6, 28, 30, 144, 183, 1080, 1188, 3780, 5742, 52416, 33462, 131760, 251100, 1290096, 1145124, 5702400, 6804204, 42336000, 50176404, 146352096, 226041700, 2333111040, 1357893000, 4818528000, 9395060400, 47385112320, 44560482150, 251337038400, 264178169640

Offset: 1

Tyler Busby, Oct 19 2023

nonn

			a(9)=1188 because Pell(9)=985 has divisors {1, 5, 197, 985}.

Amiram Eldar, Table of n, a(n) for n = 1..630 (calculated using Jon E. Schoenfield's a-file at A000129)

Cf. A000129, A000203, A272040, A363831, A363833, A364818, A364820.

DivisorSigma[1, LinearRecurrence[{2, 1}, {1, 2}, 32]] (* Amiram Eldar, Oct 19 2023 *)

a(n) = sigma(Pell(n)) = A000203(A000129(n)).

A363831 Number of divisors of A000129(n) (Pell numbers).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 3, 16, 4, 8, 2, 72, 2, 12, 12, 40, 4, 32, 4, 96, 12, 16, 4, 384, 8, 16, 16, 144, 2, 288, 8, 96, 8, 32, 12, 1536, 8, 16, 16, 1024, 2, 288, 4, 384, 96, 32, 4, 3840, 12, 64, 32, 192, 2, 256, 32, 768, 32, 8, 2, 41472, 8, 64, 96, 896, 64, 256, 4

Offset: 1

Tyler Busby, Oct 19 2023

nonn

			a(9)=4 because Pell(9)=985 has divisors {1, 5, 197, 985}.

Amiram Eldar, Table of n, a(n) for n = 1..630 (calculated using Jon E. Schoenfield's a-file at A000129)

Cf. A000005, A000129, A272040, A363829.

DivisorSigma[0,LinearRecurrence[{2,1},{1,2},67]] (* Stefano Spezia, Oct 19 2023 *)

a(n) = sigma0(Pell(n)) = A000005(A000129(n)).

A364818 Number of distinct prime divisors of A000129(n) (Pell numbers).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 2, 5, 2, 6, 3, 4, 2, 7, 3, 4, 4, 6, 1, 7, 3, 5, 3, 5, 3, 9, 3, 4, 4, 9, 1, 7, 2, 8, 6, 5, 2, 10, 3, 6, 5, 7, 1, 8, 5, 8, 5, 3, 1, 13, 3, 6, 6, 8, 6, 8, 2, 9, 4, 8, 3, 13, 2, 7, 8, 9, 5, 10, 4, 12, 7, 5, 2, 14, 7

Offset: 1

Tyler Busby, Oct 21 2023

nonn

			a(8)=3 because Pell(8)=408 has prime factors {2, 2, 2, 3, 17}.

Amiram Eldar, Table of n, a(n) for n = 1..630

Cf. A000129, A001221, A272040, A363829, A363831, A363833.

PrimeNu[LinearRecurrence[{2, 1}, {1, 2}, 85]] (* Amiram Eldar, Oct 21 2023 *)

a(n) = omega(Pell(n)) = A001221(A000129(n)).

A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2Pell(n+1)+2Pell(n)-2^n, with Pell = A000129.

Original entry on oeis.org

1, 4, 10, 26, 66, 166, 414, 1026, 2530, 6214, 15214, 37154, 90546, 220294, 535230, 1298946, 3149506, 7630726, 18476494, 44714786, 108168210, 261575494, 632367774, 1528408194, 3693378466, 8923553734, 21557263150, 52071634466

Offset: 0

Johannes W. Meijer, Aug 06 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
The sequence above corresponds to 24 A[5] vectors with decimal values 23, 29, 53, 83, 86, 89, 92, 113, 116, 149, 209, 212, 275, 278, 281, 284, 305, 308, 338, 344, 368, 401, 404 and 464. These vectors lead for the side squares to A000079 and for the corner squares to 2*A094723 (a(n)=2*Pell(n+1)-2^n).
From Clark Kimberling, Aug 23 2017 (Start)
p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3.
Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453). See A291000 for a guide to related sequences. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2).

Cf. A175654, A175655 (central square).

Cf. A000129 (Pell(n)), A078057 (Pell(n)+Pell(n+1)), A094723 (Pell(n+2)-2^n).

I:=[1,4,10]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013

nmax:=27; m:=5; A[5]:= [0,0,0,0,1,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{4,-3,-2},{1,4,10},30] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[Series[(1 - 3 x^2) / (1 - 4 x + 3 x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)

Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017

G.f.: ( 1-3*x^2 ) / ( (2*x-1)*(x^2+2*x-1) ).
a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=10.
Limit_{n->oo} a(n+1)/a(n) = 1+sqrt(2).
a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - Colin Barker, Aug 29 2017

A261331 Expansion of Product_{k>=1} (1+x^k)^(A000129(k)).

Original entry on oeis.org

1, 1, 2, 7, 18, 52, 143, 396, 1083, 2971, 8087, 21981, 59533, 160857, 433467, 1165542, 3126951, 8372451, 22374172, 59684669, 158941356, 422582925, 1121814072, 2973703449, 7871754065, 20809918535, 54943916547, 144891525408, 381647503607, 1004149670985

Offset: 0

Vaclav Kotesovec, Aug 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
Eric Weisstein's World of Mathematics, Pell Number
Wikipedia, Pell number

Cf. A000129, A261329, A261330, A261332, A261050, A261051.

nmax=40; Pell[0]=0; Pell[1]=1; Pell[n_]:=Pell[n] = 2*Pell[n-1] + Pell[n-2]; CoefficientList[Series[Product[(1+x^k)^Pell[k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (1+sqrt(2))^n * exp(-1/8 + 2^(1/4)*sqrt(n) + s) / (2^(11/8) * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} (-1)^(k+1)/(((sqrt(2)+1)^k - (sqrt(2)-1)^k - 2)*k) = -0.1149083344289588668149210160138124159112948627968378825745674888...

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - 2*x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

A354005 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a positive Pell number (A000129).

Original entry on oeis.org

3, 4, 6, 7, 10, 11, 17, 24, 41, 58, 99, 116, 140, 239, 280, 338, 577, 816, 1393, 1970, 3363, 3940, 4756, 8119, 9512, 11482, 19601, 27720, 47321, 66922, 114243, 133844, 161564, 275807, 323128, 390050, 665857, 941664, 1607521, 2273378, 3880899, 4546756, 5488420

Offset: 1

Rémy Sigrist, May 13 2022

nonn

The sequence is well defined:
- a(1) = 3,
- for n > 0, let k be such that A000129(k) + 1 + a(1) + ... + a(n) < A000129(k+1),
- then a(n+1) <= A000129(k) + 1.

Rémy Sigrist, C++ program

See A353889 for similar sequences.

Cf. A000129, A354006, A354007.

A363833 Number of prime factors of A000129(n) (Pell numbers) (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 2, 5, 2, 3, 1, 7, 1, 4, 4, 7, 2, 5, 2, 7, 4, 4, 2, 10, 3, 4, 4, 8, 1, 9, 3, 9, 3, 5, 4, 12, 3, 4, 4, 11, 1, 9, 2, 9, 7, 5, 2, 14, 4, 6, 5, 8, 1, 8, 5, 11, 5, 3, 1, 17, 3, 6, 7, 13, 6, 8, 2, 10, 4, 9, 3, 17, 2, 7, 10, 10, 6, 10, 4, 15, 7, 5, 2

Offset: 1

Tyler Busby, Oct 19 2023

nonn

			a(8)=5 because Pell(8)=408 has prime factors {2, 2, 2, 3, 17}.

Amiram Eldar, Table of n, a(n) for n = 1..630 (calculated using Jon E. Schoenfield's a-file at A000129)

Cf. A000129, A001222, A272040, A363829, A363831, A364818.

PrimeOmega[LinearRecurrence[{2,1},{1,2},83]] (* Stefano Spezia, Oct 19 2023 *)

a(n) = bigomega(Pell(n)) = A001222(A000129(n)).

A048624 Essentially a duplicate of A000129.

Original entry on oeis.org

2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689

Offset: 0

dead

A104565 Reversion of Pell numbers A000129(n+1).

Original entry on oeis.org

1, -2, 3, -2, -6, 28, -61, 54, 158, -860, 2062, -2004, -5804, 33720, -84509, 86054, 247862, -1492908, 3838298, -4019452, -11537556, 71101832, -185868978, 198310460, 567902572, -3555617432, 9404104764, -10168382696, -29069700056, 184127171952, -491229517661

Offset: 0

Paul Barry, Mar 15 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000

a:= proc(n) a(n):= `if`(n<2, 1-3*n,
      ((8-8*n)*a(n-2)-(4*n+2)*a(n-1))/(n+2))
    end:
seq (a(n), n=0..40);  # Alois P. Heinz, Nov 09 2012

CoefficientList[Series[(Sqrt[1+4*x+8*x^2]-1-2*x)/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
Table[(-2)^n Hypergeometric2F1[1/2-n/2, -n/2, 2, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)

def A104565_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a - 2*b - c
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u)
    return R
A104565_list(30)  # Peter Luschny, Nov 01 2012

G.f.: (sqrt(1+4*x+8*x^2)-1-2*x)/(2*x^2).
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*C(k)*(-1)^(n-k)2^(n-2k)}, where C(n) is A000108. - Paul Barry, May 16 2005
G.f. 1/G(0) where G(k)=  1 + 2*x + x^2/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 10 2012
G.f.: (2/W(0)-1)/x  where W(k)=  1 + 1/(1 + 2*x/(1 + 2*x/W(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 21 2012
D-finite with recurrence (n+2)*a(n) +2*(2*n+1)*a(n-1) +8*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 09 2012
G.f.: G(0)/x^2 - 1/x - 1/x^2 where G(k)=  1 + 2*x/(1 + 1/(1 + 2*x/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
G.f.: 1/(x^2*Q(0)) - 1/(x^2) - 1/x, where Q(k)= 1 - (4*k+1)*x*(1+2*x)/(k+1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3) - (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(2). - Vaclav Kotesovec, Feb 08 2014
a(n) = (-2)^n*hypergeom([1/2-n/2,-n/2], [2], -1). - Vladimir Reshetnikov, Nov 07 2015

cp's OEIS Frontend

A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.

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A363829 Sum of the divisors of A000129(n) (Pell numbers).

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A363831 Number of divisors of A000129(n) (Pell numbers).

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A364818 Number of distinct prime divisors of A000129(n) (Pell numbers).

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A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2*Pell(n+1)+2*Pell(n)-2^n, with Pell = A000129.

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Magma

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A261331 Expansion of Product_{k>=1} (1+x^k)^(A000129(k)).

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A354005 Lexicographically earliest sequence of distinct positive integers with no finite subset summing to a positive Pell number (A000129).

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A363833 Number of prime factors of A000129(n) (Pell numbers) (counted with multiplicity).

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A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2Pell(n+1)+2Pell(n)-2^n, with Pell = A000129.