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 1-6 of 6 results.

A141721 A141631(n) mod 10.

Original entry on oeis.org

2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7, 2, 3, 0, 3, 2, 7, 8, 5, 8, 7
Offset: 1

Views

Author

Paul Curtz, Sep 12 2008

Keywords

Comments

Mentioned in A010716.
The sequence and its first differences (5, 1, -3, 3, -1, -5, 1, -3, 3, -1) are both periodic with period length 10.

Programs

  • Mathematica
    Mod[#,10]&/@Table[3n^2-4n+3,{n,200}] (* Harvey P. Dale, Oct 28 2012 *)

Formula

a(n)=a(n-10), period 10 (2, 7, 8, 5, 8, 7, 2, 3, 0, 3).
Also: decimal expansion of the constant 309541367/1111111111. - R. J. Mathar, Oct 15 2008

Extensions

Edited by R. J. Mathar, Oct 15 2008
More terms from Harvey P. Dale, Oct 28 2012

A016969 a(n) = 6*n + 5.

Original entry on oeis.org

5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(18).
Exponents e such that x^e + x - 1 is reducible.
First differences of A141631. - Paul Curtz, Sep 12 2008
a(n-1), n >= 1, appears as first column in the triangle A239127 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
Odd unlucky numbers in A050505. - Fred Daniel Kline, Feb 25 2017
Intersection of A005408 and A016789. - Bruno Berselli, Apr 26 2018
Numbers that are not divisible by their digital root in base 4. - Amiram Eldar, Nov 24 2022

Links

Crossrefs

Cf. A111863, A007310, A008588, A016921, A016933, A016945, A016957, A049452.
Cf. A050505 (unlucky numbers).
Cf. A000217.

Programs

Formula

a(n) = A003415(A003415(A125200(n+1)))/2. - Reinhard Zumkeller, Nov 24 2006
A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A007310(2*n+1); complement of A016921 with respect to A007310. - Reinhard Zumkeller, Oct 02 2008
From Klaus Brockhaus, Jan 04 2009: (Start)
G.f.: (5+x)/(1-x)^2.
a(0) = 5; for n > 0, a(n) = a(n-1)+6. (End)
a(n) = A016921(n)+4 = A016933(n)+3 = A016945(n)+2 = A016957(n)+1. - Klaus Brockhaus, Jan 04 2009
a(n) = floor((12n-1)/2) with offset 1..a(1)=5. - Gary Detlefs, Mar 07 2010
a(n) = 4*(3*n+1) - a(n-1) (with a(0) = 5). - Vincenzo Librandi, Nov 20 2010
a(n) = floor(1/(1/sin(1/n) - n)). - Clark Kimberling, Feb 19 2010
a(n) = 3*Sum_{k = 0..n} binomial(6*n+5, 6*k+2)*Bernoulli(6*k+2). - Michel Marcus, Jan 11 2016
a(n) = A049452(n+1) / (n+1). - Torlach Rush, Nov 23 2018
a(n) = 2*A000217(n+2) - 1 - 2*A000217(n-1). See Twin Triangular Frames illustration. - Leo Tavares, Aug 25 2021
Sum_{n>=0} (-1)^n/a(n) = Pi/6 - sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021
E.g.f.: exp(x)*(5 + 6*x). - Stefano Spezia, Feb 14 2025

Extensions

More terms from Klaus Brockhaus, Jan 04 2009

A271713 Numbers n such that 3*n - 5 is a square.

Original entry on oeis.org

2, 3, 7, 10, 18, 23, 35, 42, 58, 67, 87, 98, 122, 135, 163, 178, 210, 227, 263, 282, 322, 343, 387, 410, 458, 483, 535, 562, 618, 647, 707, 738, 802, 835, 903, 938, 1010, 1047, 1123, 1162, 1242, 1283, 1367, 1410, 1498, 1543, 1635, 1682, 1778, 1827, 1927, 1978, 2082, 2135, 2243, 2298
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 12 2016

Keywords

Comments

Quasipolynomial of order 2 and degree 2. - Charles R Greathouse IV, Apr 12 2016
From Ray Chandler, Apr 13 2016: (Start)
Square roots of resulting squares gives A001651.
Sequence is the union of A141631 and A271740. (End)

Examples

			a(3) = 7 because 3*7 - 5 = 16 = 4^2.

Links

Crossrefs

Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), this sequence (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6).
Cf. A001651, A141631, A271740.

Programs

Formula

G.f.: x*(2 + x + x^3 + 2*x^4)/((1 - x)^3*(1 + x)^2). - Ilya Gutkovskiy, Apr 12 2016
a(n) = (3/2)*n^2 + O(n). - Charles R Greathouse IV, Apr 12 2016
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5. - Wesley Ivan Hurt, Apr 13 2016

A156140 Accumulation of Stern's diatomic series: a(0)=-1, a(1)=0, and a(n+1) = (2e(n)+1)*a(n) - a(n-1) for n > 1, where e(n) is the highest power of 2 dividing n.

Original entry on oeis.org

-1, 0, 1, 3, 2, 7, 5, 8, 3, 13, 10, 17, 7, 18, 11, 15, 4, 21, 17, 30, 13, 35, 22, 31, 9, 32, 23, 37, 14, 33, 19, 24, 5, 31, 26, 47, 21, 58, 37, 53, 16, 59, 43, 70, 27, 65, 38, 49, 11, 50, 39, 67, 28, 73, 45, 62, 17, 57, 40, 63, 23, 52, 29, 35, 6, 43, 37, 68, 31, 87, 56, 81, 25, 94, 69
Offset: 0

Views

Author

Arie Werksma (Werksma(AT)Tiscali.nl), Feb 04 2009

Keywords

Links

Crossrefs

Cf. A002487, A007814.
From Yosu Yurramendi, Mar 09 2018: (Start)
a(2^m + 0) = A000027(m), m >= 0.
a(2^m + 1) = A002061(m+2), m >= 1.
a(2^m + 2) = A002522(m), m >= 2.
a(2^m + 3) = A033816(m-1), m >= 2.
a(2^m + 4) = A002061(m), m >= 2.
a(2^m + 5) = A141631(m), m >= 3.
a(2^m + 6) = A084849(m-1), m >= 3.
a(2^m + 7) = A056108(m-1), m >= 3.
a(2^m + 8) = A000290(m-1), m >= 3.
a(2^m + 9) = A185950(m-1), m >= 4.
a(2^m + 10) = A144390(m-1), m >= 4.
a(2^m + 12) = A014106(m-2), m >= 4.
a(2^m + 16) = A028387(m-3), m >= 4.
a(2^m + 18) = A250657(m-4), m >= 5.
a(2^m + 20) = A140677(m-3), m >= 5.
a(2^m + 32) = A028872(m-2), m >= 5.
a(2^m - 1) = A005563(m-1), m >= 0.
a(2^m - 2) = A028387(m-2), m >= 2.
a(2^m - 3) = A033537(m-2), m >= 2.
a(2^m - 4) = A008865(m-1), m >= 3.
a(2^m - 7) = A140678(m-3), m >= 3.
a(2^m - 8) = A014209(m-3), m >= 4.
a(2^m - 16) = A028875(m-2), m >= 5.
a(2^m - 32) = A108195(m-5), m >= 6.
(End)

Programs

  • Maple
    A156140 := proc(n)
    option remember ;
    if n <= 1 then
        n-1 ;
    else
        (2*A007814(n-1)+1)*procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A156140(n),n=0..80) ; # R. J. Mathar, Mar 14 2009
  • Mathematica
    Fold[Append[#1, (2 IntegerExponent[#2, 2] + 1) #1[[-1]] - #1[[-2]] ] &, {-1, 0}, Range[73]] (* Michael De Vlieger, Mar 09 2018 *)
  • PARI
    first(n)=my(v=vector(n+1)); v[1]=-1; v[2]=0; for(k=1,n-1,v[k+2]=(2*valuation(k,2)+1)*v[k+1] - v[k]); v \\ Charles R Greathouse IV, Apr 05 2016
  • PARI
    fusc(n)=my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b
a(n)=my(m=1,s,t); if(n==0, return(-1)); while(n%2==0, s+=fusc(n>>=1)); while(n>1, t=logint(n,2); n-=2^t; s+=m*fusc(n)*(t^2+t+1); m*=-t); m*(n-1) + s \\ Charles R Greathouse IV, Dec 13 2016
  • R
    a <- c(0,1)
maxlevel <- 6 # by choice
for(m in 1:maxlevel) {
  a[2^(m+1)] <- m + 1
  for(k in 1:(2^m-1)) {
    r <- m - floor(log2(k)) - 1
    a[2^r*(2*k+1)] <- a[2^r*(2*k)] + a[2^r*(2*k+2)]
}}
a
# Yosu Yurramendi, May 08 2018

Formula

Let b(n) = A002487(n), Stern's diatomic series.
a(n+1)*b(n) - a(n)*b(n+1) = 1 for n >= 0.
a(2n+1) = a(n) + a(n+1) + b(n) + b(n+1) for n >= 0.
a(2n) = a(n) + b(n) for n >= 0.
a(2^n + k) = -n*a(k) + (n^2 + n + 1)*b(k) for 0 <= k <= 2^n.
b(2^n + k) = -a(k) + (n + 1)*b(k) for 0 <= k <= 2^n.
a(2^m + k) = b(2^m+k)*m + b(k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 09 2018
a(2^(m+1)+2^m+1) = 2*m+1, m >= 0. - Yosu Yurramendi, Mar 09 2018
From Yosu Yurramendi, May 08 2018: (Start)
a(2^m) = m, m >= 0.
a(2^r*(2*k+1)) = a(2^r*(2*k)) + a(2^r*(2*k+2)), r = m - floor(log_2(k)) - 1, m > 0, 1 <= k < 2^m.
(End)

A133146 Antidiagonal sums of the triangle A133128.

Original entry on oeis.org

2, 5, 7, 14, 18, 29, 35, 50, 58, 77, 87, 110, 122, 149, 163, 194, 210, 245, 263, 302, 322, 365, 387, 434, 458, 509, 535, 590, 618, 677, 707, 770, 802, 869, 903, 974, 1010, 1085, 1123, 1202, 1242, 1325, 1367, 1454, 1498, 1589, 1635, 1730, 1778, 1877, 1927, 2030
Offset: 0

Views

Author

Paul Curtz, Aug 27 2008

Keywords

Examples

			a(2) = A133128(2,0) + A133128(1,1) = 10 - 3 = 7.
a(3) = A133128(3,0) + A133128(2,1) = 17 - 3 = 14.

Links

Programs

  • Magma
    [19/8 +5*n/4 +3*n^2/4 -(-1)^n*(n/4+3/8): n in [0..60]]; // Vincenzo Librandi, Aug 10 2011
  • Mathematica
    CoefficientList[Series[(1+2x)(2-x+x^3)/((1-x)^3(1+x)^2),{x,0,60}],x] (* or *) LinearRecurrence[{1,2,-2,-1,1},{2,5,7,14,18},60] (* Harvey P. Dale, Aug 26 2013 *)

Formula

First differences: a(n+1) - a(n) = A059029(n+1).
Bisections: a(2n+1) = A005918(n+1). a(2n) = A141631(n+1).
G.f.: (1+2*x)(2 - x + x^3)/((1-x)^3*(1+x)^2). - R. J. Mathar, Oct 15 2008
a(n) = 19/8 + 5*n/4 + 3*n^2/4 - (-1)^n*(n/4 + 3/8). - R. J. Mathar, Oct 15 2008
From Harvey P. Dale, Aug 26 2013: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5); a(0)=2, a(1)=5, a(2)=7, a(3)=14, a(4)=18. (End)

Extensions

Edited and extended by R. J. Mathar, Oct 15 2008

A271740 a(n) = 3*n^2 - 2*n + 2.

Original entry on oeis.org

2, 3, 10, 23, 42, 67, 98, 135, 178, 227, 282, 343, 410, 483, 562, 647, 738, 835, 938, 1047, 1162, 1283, 1410, 1543, 1682, 1827, 1978, 2135, 2298, 2467, 2642, 2823, 3010, 3203, 3402, 3607, 3818, 4035, 4258, 4487, 4722, 4963, 5210, 5463, 5722, 5987, 6258, 6535, 6818, 7107, 7402, 7703
Offset: 0

Views

Author

Ray Chandler, Apr 13 2016

Keywords

Comments

3*a(n) - 5 is a square. - Vincenzo Librandi, Apr 13 2016

Links

Crossrefs

Cf. A141631, A271713.

Programs

Formula

A271713 = A141631 union A271740.
From Bruno Berselli, Apr 13 2016: (Start)
O.g.f.: (3 + x + 2*x^2)/(1 - x)^3.
E.g.f.: (3 + 7*x + 3*x^2)*exp(x). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Apr 13 2016

Extensions

Definition changed so sequence starts one term earlier. Some formulas may need adjusting. - N. J. A. Sloane, Jun 22 2021
Showing 1-6 of 6 results.

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

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