cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A270359 Positive integer averages of first n Pell numbers; Sum{k=0..n-1} A000129(k) / n where n is in A270342.

Original entry on oeis.org

1, 2, 4, 17, 36, 369, 1820, 20808, 47280, 246561, 6919153, 16008300, 1086517900, 5924129729, 13855173264, 982740019940, 30127233316440, 167427203210673, 5203545562472737, 12300752138736600, 913640750713307860, 162500024938034177361
Offset: 1

Views

Author

Altug Alkan, Mar 15 2016

Keywords

Examples

			17 is a term because (0 + 1 + 2 + 5 + 12 + 29 + 70) / 7 = 119 / 7 = 17.

Crossrefs

Cf. A000129, A048739, A270342.

Programs

  • PARI
    a048739(n) = local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n;
for(n=1, 1e2, if(a048739(n-1) % (n+1) == 0, print1(a048739(n-1) / (n+1) , ", ")));

A048739 Expansion of 1/((1 - x)*(1 - 2*x - x^2)).

Original entry on oeis.org

1, 3, 8, 20, 49, 119, 288, 696, 1681, 4059, 9800, 23660, 57121, 137903, 332928, 803760, 1940449, 4684659, 11309768, 27304196, 65918161, 159140519, 384199200, 927538920, 2239277041, 5406093003, 13051463048, 31509019100, 76069501249
Offset: 0

Views

Author

Barry E. Williams

Keywords

Comments

Partial sums of Pell numbers A000129.
W(n){1,3;2,-1,1} = Sum_{i=1..n} W(i){1,2;2,-1,0}, where W(n){a,b; p,q,r} implies x(n) = p*x(n-1) - q*x(n-2) + r; x(0)=a, x(1)=b.
Number of 2 X (n+1) binary arrays with path of adjacent 1's from upper left to lower right corner. - R. H. Hardin, Mar 16 2002
Binomial transform of A029744. - Paul Barry, Apr 23 2004
Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba, Jun 16 2004
Equals row sums of triangle A153346. - Gary W. Adamson, Dec 24 2008
Equals the sum of the terms of the antidiagonals of A142978. - J. M. Bergot, Nov 13 2012
a(p-2) == 0 mod p where p is an odd prime, see A270342. - Altug Alkan, Mar 15 2016
Also, the lexicographically earliest sequence of positive integers such that for n > 3, {sqrt(2)*a(n)} is located strictly between {sqrt(2)*a(n-1)} and {sqrt(2)*a(n-2)} where {} denotes the fractional part. - Ivan Neretin, May 02 2017
a(n+1) is the number of weak orderings on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n. - J. Devillet, Oct 06 2017

References

  • Allombert, Bill, Nicolas Brisebarre, and Alain Lasjaunias. "On a two-valued sequence and related continued fractions in power series fields." The Ramanujan Journal 45.3 (2018): 859-871. See Theorem 3, d_{4n+3}.

Links

Crossrefs

First row of table A083087.
With a different offset, a(4n)=A008843(n), a(4n-2)=8*A001110(n), a(2n-1)=A001652(n).
Cf. A001333, A048654, A048655, A083044, A083047, A083050, A153346.

Programs

  • Maple
    a:=n->sum(fibonacci(i,2), i=0..n): seq(a(n), n=1..29); # Zerinvary Lajos, Mar 20 2008
  • Mathematica
    Join[{a=1,b=3},Table[c=2*b+a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/(1-3x+x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,3,8},30] (* Harvey P. Dale, Jun 13 2011 *)
  • PARI
    a(n)=local(w=quadgen(8));-1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n
  • PARI
    vector(100, n, n--; floor((1+sqrt(2))^(n+2)/4)) \\ Altug Alkan, Oct 07 2015
  • PARI
    Vec(1/((1-x)*(1-2*x-x^2)) + O(x^40)) \\ Michel Marcus, May 06 2017

Formula

a(n) = 2*a(n-1) + a(n-2) + 1 with n > 1, a(0)=1, a(1)=3.
a(n) = ((2 + (3*sqrt(2))/2)*(1 + sqrt(2))^n - (2 - (3*sqrt(2))/2)*(1 - sqrt(2))^n )/(2*sqrt(2)) - 1/2.
a(0)=1, a(n+1) = ceiling(x*a(n)) for n > 0, where x = 1+sqrt(2). - Paul D. Hanna, Apr 22 2003
a(n) = 3*a(n-1) - a(n-2) - a(n-3). With two leading zeros, e.g.f. is exp(x)(cosh(sqrt(2)x)-1)/2. a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n+2, 2k+2)2^k. - Paul Barry, Aug 16 2003
-a(-3-n) = A077921(n). - N. J. A. Sloane, Sep 13 2003
E.g.f.: exp(x)(cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2. - N. J. A. Sloane, Sep 13 2003
a(n) = floor((1+sqrt(2))^(n+2)/4). - Bruno Berselli, Feb 06 2013
a(n) = (((1-sqrt(2))^(n+2) + (1+sqrt(2))^(n+2) - 2) / 4). - Altug Alkan, Mar 16 2016
2*a(n) = A001333(n+2)-1. - R. J. Mathar, Oct 11 2017
a(n) = Sum_{k=0..n} binomial(n+1,k+1)*2^floor(k/2). - Tony Foster III, Oct 12 2017

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 11 2002

A335668 Even composites m such that A002203(m) == 2 (mod m).

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 120, 128, 144, 168, 192, 216, 240, 256, 264, 272, 288, 336, 360, 384, 432, 480, 504, 512, 528, 544, 576, 600, 648, 672, 720, 768, 792, 816, 840, 864, 960, 1008, 1024, 1056, 1080, 1088, 1152, 1176, 1200, 1296, 1320, 1344, 1440, 1512
Offset: 1

Views

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

Comments

If p is a prime, then A002203(p)==2 (mod p).
Even composites for which the congruence holds.
Even composites m for which the sum of the Pell numbers A000129(0) + ... + A000129(m-1) is divisible by m.

Examples

			4 is the first composite number m for which A002203(m)==2 (mod m) since A002203(4)=34==2 (mod 4), so a(1)=4.
The next even composite for which the congruence holds is m = 8 since A002203(8)=1154==2 (mod 8), so a(2)=8.

References

  • D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Links

Crossrefs

Cf. A270342 (all positive integers), A270345 (all composites), A330276 (odd composites),

Programs

  • Mathematica
    Select[Range[4, 2000, 2], Divisible[LucasL[#, 2] - 2, #] &] (* Amiram Eldar, Jun 18 2020 *)

A270345 Composite integers n such that the sum of the Pell numbers A000129(0) + ... + A000129(n-1) is divisible by n.

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 120, 128, 144, 168, 169, 192, 216, 240, 256, 264, 272, 288, 336, 360, 384, 385, 432, 480, 504, 512, 528, 544, 576, 600, 648, 672, 720, 768, 792, 816, 840, 864, 960, 961, 1008, 1024, 1056, 1080, 1088, 1105, 1121, 1152, 1176, 1200, 1296, 1320, 1344
Offset: 1

Views

Author

Altug Alkan, Mar 15 2016

Keywords

Comments

Nonprime terms of A270342.
Terms that are not divisible by 4 are 169, 385, 961, 1105, 1121, 3827, 4901, 6265, 6441, 6601, 7107, 7801, 8119, ...

Examples

			4 is a term because 0 + 1 + 2 + 5 = 8 is divisible by 4.
8 is a term because 0 + 1 + 2 + 5 + 12 + 29 + 70 + 169 = 288 is divisible by 8.

Crossrefs

Cf. A000129, A048739, A270342.

Programs

  • PARI
    a048739(n) = local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n;
for(n=1, 1e3, if(a048739(n-1) % (n+1) == 0 && !isprime(n+1), print1(n+1, ", ")));

A270449 Odd integers n such that the sum of the Pell numbers A000129(0) + ... + A000129(n-1) is divisible by n*(n+1)/2.

Original entry on oeis.org

13, 61, 157, 181, 193, 337, 385, 397, 541, 673, 733, 769, 877, 1153, 1201, 1213, 1453, 1873, 1933, 2017, 2029, 2557, 2593, 2797, 3217, 3313, 3517, 4177, 4273, 4561, 4621, 4657, 5101, 5233, 5437, 5581, 5641, 6337, 6637, 6781, 7057, 7213, 7393, 7481, 7537, 7561, 7933, 8221, 8317
Offset: 1

Views

Author

Altug Alkan, Mar 17 2016

Keywords

Comments

Sequence contains the prime numbers most of the time. Nonprime terms of this sequence are 385, 111361, 111841, 155041, 186961 ...

Examples

			13 is a term because (0 + 1 + 2 + 5 + 12 + 29 + 70 + 169 + 408 + 985 + 2378 + 5741 + 13860) / (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13) = 260.

Crossrefs

Cf. A000129, A048739, A270342.

Programs

  • PARI
    a048739(n) = local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n;
for(n=1, 1e4, if(a048739(n-1) % ((n+1)*(n+2)/2) == 0 && (n+1) % 2 == 1, print1((n+1), ", ")));
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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