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.

A006324 a(n) = n*(n + 1)*(2*n^2 + 2*n - 1)/6.

Original entry on oeis.org

1, 11, 46, 130, 295, 581, 1036, 1716, 2685, 4015, 5786, 8086, 11011, 14665, 19160, 24616, 31161, 38931, 48070, 58730, 71071, 85261, 101476, 119900, 140725, 164151, 190386, 219646, 252155, 288145, 327856, 371536, 419441, 471835, 528990, 591186
Offset: 1

Views

Author

Albert Rich (Albert_Rich(AT)msn.com), Jun 14 1998

Keywords

Comments

4-dimensional analog of centered polygonal numbers.
Partial sums of A000447. - Zak Seidov, May 19 2006
From Johannes W. Meijer, Jun 27 2009: (Start)
Equals the absolute values of the coefficients that precede the a(n-1) factors of the recurrence relations RR(n) of A162011.
This sequence enabled the analysis of A162012 and A162013. (End)
Equals the number of integer quadruples (x,y,z,w) such that min(x,y) < min(z,w), max(x,y) < max(z,w), and 0 <= x,y,z,w <= n. - Andrew Woods, Apr 21 2014
For n>3 a(n)=twice the area of an irregular quadrilateral with vertices at the points (C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), (C(n+2,4),C(n+3,4)), and (C(n+3,4),C(n+4,4)). - J. M. Bergot, Jun 14 2014

Links

Crossrefs

Cf. A000447, A000539, A000217, A002415, A053134.
Cf. A162011, A162012, a(n-2), and A162013, a(n-3). - Johannes W. Meijer, Jun 27 2009

Programs

Formula

a(n) = 8*C(n + 2, 4) + C(n + 1, 2).
a(n) = (Sum_{k=1..n} k^5) / (Sum_{k=1..n} k) = A000539(n) / A000217(n). - Alexander Adamchuk, Apr 12 2006
From Johannes W. Meijer, Jun 27 2009: (Start)
Recurrence relation 0 = Sum_{k=0..5} (-1)^k*binomial(5,k)*a(n-k).
G.f.: (1+6*z+z^2)/(1-z)^5. (End)
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, May 02 2021
Sum_{n>=1} 1/a(n) = 6 + 2*sqrt(3)*Pi*tan(sqrt(3)*Pi/2). - Amiram Eldar, Aug 23 2022
a(n) = A053134(n-1) - 4*A002415(n). - Yasser Arath Chavez Reyes, Feb 12 2024

Extensions

Simpler definition from Alexander Adamchuk, Apr 12 2006
More terms from Zak Seidov

A162011 A sequence related to the recurrence relations of the right hand columns of the EG1 triangle A162005.

Original entry on oeis.org

1, -1, 1, -11, 19, -9, 1, -46, 663, -3748, 7711, -6606, 2025, 1, -130, 6501, -163160, 2236466, -17123340, 71497186, -154127320, 174334221, -98986050, 22325625, 1, -295, 36729, -2549775, 109746165, -3080128275, 57713313405, -727045264875
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Comments

The recurrence relation RR(n) = 0 of the n-th right hand column can be found with RR(n) = expand( product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) = 0 and replacing z^p by a(n-p).
The polynomials in the numerators of the generating functions GF(z) of the coefficients that precede the a(n), a(n-1), a(n-2) and a(n-3) sequences, see A000012, A006324, A162012 and A162013, are symmetrical. This phenomenon leads to the sequence [1, 1, 6, 1, 19, 492, 1218, 492, 19 , 9, 3631, 115138, 718465, 1282314, 718465, 115138, 3631, 9].

Examples

			The recurrence relations for the first few right hand columns:
n = 1: a(n) = 1*a(n-1)
n = 2: a(n) = 11*a(n-1)-19*a(n-2)+9*a(n-3)
n = 3: a(n) = 46*a(n-1)-663*a(n-2)+3748*a(n-3)-7711*a(n-4)+6606*a(n-5)-2025*a(n-6)
n = 4: a(n) = 130*a(n-1)-6501*a(n-2)+163160*a(n-3)-2236466*a(n-4)+17123340*a(n-5)-71497186*a(n-6)+154127320*a(n-7)-174334221*a(n-8)+98986050*a(n-9)-22325625*a(n-10)

Crossrefs

A000012, A004004 (2x), A162008, A162009 and A162010 are the first five right hand columns of EG1 triangle A162005.
A000124 (the Lazy Caterer's sequence) gives the number of terms of the RR(n).
A006324, A162012 and A162013 equal the absolute values of the coefficients that precede the a(n-1), a(n-2) and a(n-3) factors of the RR(n).

Programs

  • Maple
    nmax:=5; for n from 1 to nmax do RR(n) := expand(product((1-(2*k-1)^2*z)^(n-k+1), k=1..n), z) od: T:=1: for n from 1 to nmax do for m from 0 to(n)*(n+1)/2 do a(T):= coeff(RR(n), z, m): T:=T+1 od: od: seq(a(k), k=1..T-1);

Formula

RR(n) = expand( product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) with n = 1, 2, 3, .. . The coefficients of these polynomials lead to the sequence given above.

A162012 The sequence of the absolute values of the a(n-2) coefficients of A162011.

Original entry on oeis.org

0, 19, 663, 6501, 36729, 149842, 491274, 1375206, 3413982, 7710813, 16133689, 31690659, 59028879, 105082068, 179893252, 297641916, 477906924, 747198807, 1140797259, 1704931921, 2499346773, 3600290694, 5103978990, 7130572930
Offset: 1

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Crossrefs

Equals the absolute values of the coefficients that precede the a(n-2) factors of the recurrence relations RR(n) of A162011.
Cf. A006324 [a(n-1)] and A162013 [a(n-3)].

Programs

  • Maple
    nmax:=26; for n from 1 to nmax do RR(n) := expand(product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) od: T:=1: for n from 1 to nmax do a(T):=coeff(RR(n),z,2): T:=T+1 od: seq(a(k),k=1..T-1);

Formula

a(n) = (20*n^8+80*n^7+4*n^6-268*n^5-155*n^4+230*n^3+131*n^2-42*n)/360
Recurrence relation sum((-1)^k*binomial(9,k)*a(n-k), k= 0 .. 9) = 0
GF(z) = z*(19+492*z+1218*z^2+492*z^3+19*z^4)/(1-z)^9
Showing 1-3 of 3 results.

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