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 1-4 of 4 results.

A299645 Numbers of the form m*(8*m + 5), where m is an integer.

Original entry on oeis.org

0, 3, 13, 22, 42, 57, 87, 108, 148, 175, 225, 258, 318, 357, 427, 472, 552, 603, 693, 750, 850, 913, 1023, 1092, 1212, 1287, 1417, 1498, 1638, 1725, 1875, 1968, 2128, 2227, 2397, 2502, 2682, 2793, 2983, 3100, 3300, 3423, 3633, 3762, 3982, 4117, 4347, 4488, 4728, 4875
Offset: 1

Views

Author

Bruno Berselli, Feb 26 2018

Keywords

Comments

Equivalently, numbers k such that 32*k + 25 is a square. This means that 4*a(n) + 3 is a triangular number.
Interleaving of A139277 and A139272 (without 0).

Links

Crossrefs

Cf. A139272, A139277.
Subsequence of A011861, A047222.
Cf. numbers of the form m*(8*m + h): A154260 (h=1), A014494 (h=2), A274681 (h=3), A046092 (h=4), this sequence (h=5), 2*A074377 (h=6), A274979 (h=7).

Programs

  • GAP
    List([1..50], n -> (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4);
  • Julia
    [div((8n*(n-1)-(2n-1)*(-1)^n-1), 4) for n in 1:50] # Peter Luschny, Feb 27 2018
  • Magma
    [(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4: n in [1..50]];
  • Maple
    seq((exp(I*Pi*x)*(1-2*x)+8*(x-1)*x-1)/4, x=1..50); # Peter Luschny, Feb 27 2018
  • Mathematica
    Table[(8 n (n - 1) - (2 n - 1) (-1)^n - 1)/4, {n, 1, 50}]
  • Maxima
    makelist((8*n*(n-1)-(2*n-1)*(-1)^n-1)/4, n, 1, 50);
  • PARI
    vector(50, n, nn; (8*n*(n-1)-(2*n-1)*(-1)^n-1)/4)
  • PARI
    concat(0, Vec(x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Feb 27 2018
  • Python
    [(8*n*(n-1)-(2*n-1)*(-1)**n-1)/4 for n in range(1, 60)]
  • Python
    def A299645(n): return (n>>1)*((n<<2)+(1 if n&1 else -5)) # Chai Wah Wu, Mar 11 2025
  • Sage
    [(8*n*(n-1)-(2*n-1)*(-1)^n-1)/4 for n in (1..50)]

Formula

O.g.f.: x^2*(3 + 10*x + 3*x^2)/((1 - x)^3*(1 + x)^2).
E.g.f.: (1 + 2*x - (1 - 8*x^2)*exp(2*x))*exp(-x)/4.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (8*n*(n - 1) - (2*n - 1)*(-1)^n - 1)/4 = (2*n + (-1)^n - 1)*(4*n - 3*(-1)^n - 2)/4. Therefore, 3 and 13 are the only prime numbers in this sequence.
a(n) + a(n+1) = 4*n^2 for even n, otherwise a(n) + a(n+1) = 4*n^2 - 1.
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=2} 1/a(n) = 8/25 + (sqrt(2)-1)*Pi/5.
Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/5 - sqrt(2)*log(2*sqrt(2)+3)/5 - 8/25. (End)
a(n) = (n-1)*(4*n+1)/2 if n is odd and a(n) = n*(4*n-5)/2 if n is even. - Chai Wah Wu, Mar 11 2025

A059403 Quarter-squared applied twice.

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 20, 36, 64, 100, 156, 225, 324, 441, 600, 784, 1024, 1296, 1640, 2025, 2500, 3025, 3660, 4356, 5184, 6084, 7140, 8281, 9604, 11025, 12656, 14400, 16384, 18496, 20880, 23409, 26244, 29241, 32580, 36100, 40000, 44100, 48620, 53361, 58564, 64009
Offset: 0

Views

Author

Henry Bottomley, Mar 21 2001

Keywords

Examples

			a(9)=100 since the ninth quarter-square is 20 and the twentieth quarter-square is 100.

Links

Crossrefs

Cf. A008233 for an alternative approach.

Programs

  • Mathematica
    Floor[Floor[Range[0,50]^2/4]^2/4] (* or *) LinearRecurrence[{2,1,-4,2,0,-2,4,-1,-2,1},{0,0,0,1,4,9,20,36,64,100},50] (* Harvey P. Dale, Dec 13 2014 *)
  • PARI
    a(n) = { (n^2\4)^2\4 } \\ Harry J. Smith, Jun 26 2009

Formula

a(n) = floor(floor(n^2/4)^2/4) = A002620(A002620(n)).
a(4*n) = 4n^4; a(4*n+1) = n^2*(2*n+1)^2;
a(4*n+2) = 2*n*(n+1)*(2*n*(n+1)+1); a(4*n+3) = (n+1)^2*(2*n+1)^2.
a(2n) = A060494(2n); a(2n-1) = A060494(2n-1)-A011861(n).
G.f.: x^3*(1 + 2*x + 2*x^3 + x^4)/((1 - x)^5*(1 + x)^3*(1 + x^2)). - R. J. Mathar, Sep 09 2008

A060494 a(n) = floor(n^4/64).

Original entry on oeis.org

0, 0, 0, 1, 4, 9, 20, 37, 64, 102, 156, 228, 324, 446, 600, 791, 1024, 1305, 1640, 2036, 2500, 3038, 3660, 4372, 5184, 6103, 7140, 8303, 9604, 11051, 12656, 14430, 16384, 18530, 20880, 23447, 26244, 29283, 32580, 36147, 40000, 44152, 48620, 53418, 58564, 64072
Offset: 0

Views

Author

Henry Bottomley, Mar 21 2001

Keywords

Examples

			a(9) = floor(9^4/64) = floor(6561/64) = floor(102.51562...) = 102.

Links

Programs

  • Mathematica
    Floor[Range[0,50]^4/64] (* or *) LinearRecurrence[ {4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1},{0,0,0,1,4,9,20,37,64,102,156,228,324,446,600,791,1024,1305,1640,2036},50] (* Harvey P. Dale, May 30 2014 *)
  • PARI
    a(n) = { n^4\64 } \\ Harry J. Smith, Jul 06 2009

Formula

a(n) = floor(A000583(n)/64) = floor(A011863(n-1)/4). a(2n) = A059403(2n); a(2n-1) = A059403(2n-1) + A011861(n).
From R. J. Mathar, Mar 24 2011: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-16) - 4*a(n-17) + 6*a(n-18) - 4*a(n-19) + a(n-20).
G.f.: -x^3 *(1 - x^2 + 4*x^3 - 4*x^4 - 3*x^6 + 4*x^7 - 3*x^8 + 4*x^9 - 4*x^10 + 4*x^11 - x^12 + x^14 + 4*x^5) / ( (1+x) *(x^2+1) *(x^4+1) *(x^8+1) *(x-1)^5 ). (End)

A011869 a(n) = floor(n*(n-1)/16).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 26, 28, 31, 34, 37, 40, 43, 47, 50, 54, 58, 62, 66, 70, 74, 78, 83, 87, 92, 97, 102, 107, 112, 118, 123, 129, 135, 141, 147, 153, 159, 165, 172, 178, 185, 192, 199, 206, 213, 221, 228, 236, 244, 252, 260
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011861.

Programs

  • Magma
    [(n*(n-1) div 16) : n in [0..70]]; // Vincenzo Librandi, May 26 2019
  • Mathematica
    Table[Floor[(n(n-1))/16],{n,0,70}] (* or *) LinearRecurrence[{3,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3,1},{0,0,0,0,0,1,1,2,3,4,5,6,8,9,11,13,15},70] (* Harvey P. Dale, May 23 2019 *)

Formula

G.f.: -((x^5 (1-x+x^2) (1-x+x^2-x^3+x^4-x^5+x^6))/((-1+x)^3 (1+x^2) (1+x^4) (1+x^8)))
Showing 1-4 of 4 results.

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

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