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 1-3 of 3 results.

A174655 Partial sums of A049486.

Original entry on oeis.org

1, 5, 15, 36, 70, 123, 197, 298, 428, 593, 795, 1040, 1330, 1671, 2065, 2518, 3032, 3613, 4263, 4988, 5790, 6675, 7645, 8706, 9860, 11113, 12467, 13928, 15498, 17183, 18985, 20910, 22960, 25141, 27455, 29908, 32502, 35243, 38133, 41178, 44380
Offset: 1

Views

Author

Jonathan Vos Post, Mar 25 2010

Keywords

Comments

Partial sums of maximum length of non-crossing path on n X n square lattice. The subsequence of primes in this partial sum begins: 5, 197, 593, 3613, 11113, 17183.

Examples

			a(7) = 1 + 4 + 10 + 21 + 34 + 53 + 74 = 197 is prime.

Links

Crossrefs

Cf. A049486, A049487.

Programs

  • Mathematica
    Accumulate[Join[{1,4},LinearRecurrence[{2,0,-2,1},{10,21,34,53},40]]] (* or *) Join[{1,5},LinearRecurrence[{3,-2,-2,3,-1},{15,36,70,123,197},40]] (* Harvey P. Dale, Aug 21 2013 *)

Formula

a(n) = SUM[i=1..n] A049486(i).
Conjecture: a(n) = (3*(-9+(-1)^n)+34*n-12*n^2+8*n^3)/12 for n>1. G.f.: x*(x^5-x^4+3*x^3+2*x^2+2*x+1) / ((x-1)^4*(x+1)). - Colin Barker, May 02 2013

A266883 Numbers of the form m*(4*m+1)+1, where m = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

1, 4, 6, 15, 19, 34, 40, 61, 69, 96, 106, 139, 151, 190, 204, 249, 265, 316, 334, 391, 411, 474, 496, 565, 589, 664, 690, 771, 799, 886, 916, 1009, 1041, 1140, 1174, 1279, 1315, 1426, 1464, 1581, 1621, 1744, 1786, 1915, 1959, 2094, 2140, 2281, 2329, 2476, 2526
Offset: 0

Views

Author

Bruno Berselli, Jan 05 2016

Keywords

Comments

Also, numbers m such that 16*m-15 is a square. Therefore, the terms 1 and 4 are the only squares in this sequence.
Conjecture: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n-1} 1 - q^k) = q + q^4 + q^6 + q^15 + q^19 + q^34 + .... Cf. A174114. - Peter Bala, May 10 2025

Links

Crossrefs

Cf. A001057, A049486, A127365, A130472.
Cf. A002061: m*(4*m+2)+1 for m = 0,0,-1,1,-2,2,-3,3, ...
Cf. A174114: m*(4*m+3)+1 for m = 0,-1,1,-2,2,-3,3,-4,4, ...
Cf. A054556: m*(4*m+1)+1 for nonpositive m.
Cf. A054567: m*(4*m+1)+1 for nonnegative m.
Cf. A074378: numbers m such that 16*m+1 is a square.

Programs

  • Magma
    [n*(n+1)+1-((2*n+1)*(-1)^n-1)/4: n in [0..50]];
  • Magma
    I:=[1,4,6,15,19]; [n le 5 select I[n] else Self(n-1) + 2*Self(n-2) -2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 06 2016
  • Mathematica
    Table[n (n + 1) + 1 - ((2 n + 1) (-1)^n - 1)/4, {n, 0, 50}]
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 6, 15, 19}, 60] (* Vincenzo Librandi, Jan 06 2016 *)
  • PARI
    vector(50, n, n--; n*(n+1)+1-((2*n+1)*(-1)^n-1)/4)
  • PARI
    Vec((1+3*x+3*x^3+x^4)/((1+x)^2*(1-x)^3) + O(x^100)) \\ Altug Alkan, Jan 06 2016
  • Python
    [n*(n+1)+1-((2*n+1)*(-1)**n-1)/4 for n in range(60)]
  • Sage
    [n*(n+1)+1-((2*n+1)*(-1)^n-1)/4 for n in range(50)]

Formula

O.g.f.: (1 + 3*x + 3*x^3 + x^4)/((1 + x)^2*(1 - x)^3).
E.g.f.: (5 + 8*x + 4*x^2)*exp(x)/4 -(1 - 2*x)*exp(-x)/4.
a(n) = a(-n-1) = n*(n + 1) + 1 - ((2*n + 1)*(-1)^n - 1)/4 = (2*n + 1)*floor((n + 1)/2) + 1.
a(n) = A002061(n+1) + A001057(n) = A074378(n)+1.
a(n+1) + a(n+2) = A049486(n+3).

A049487 Number of maximal-length non-crossing paths on n X n square lattice.

Original entry on oeis.org

0, 2, 38, 13, 1674, 790526
Offset: 1

Views

Author

Arlin Anderson (starship1(AT)gmail.com)

Keywords

Crossrefs

Cf. A049486.
Showing 1-3 of 3 results.

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