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.

A335648 Partial sums of A006010.

Original entry on oeis.org

0, 1, 6, 26, 78, 195, 420, 820, 1476, 2501, 4026, 6222, 9282, 13447, 18984, 26216, 35496, 47241, 61902, 80002, 102102, 128843, 160908, 199068, 244140, 297037, 358722, 430262, 512778, 607503, 715728, 838864, 978384, 1135889, 1313046, 1511658, 1733598, 1980883, 2255604
Offset: 0

Views

Author

Stefano Spezia, Jun 15 2020

Keywords

Links

Crossrefs

Cf. A000290, A001844, A005408, A007204, A053755, A058331, A059722.
Cf. A006010 (1st differences), A186424 (3rd differences), A317614 (2nd differences).

Programs

  • Magma
    I:=[0, 1, 6, 26, 78, 195, 420, 820]; [n le 8 select I[n] else 4*Self(n-1)-4*Self(n-2)-4*Self(n-3)+10*Self(n-4)-4*Self(n-5)-4*Self(n-6)+4*Self(n-7)-Self(n-8): n in [1..39]];
  • Mathematica
    Table[(1+n)(5-5(-1)^n+8n+12n^2+8n^3+2n^4)/80,{n,0,38}]
  • PARI
    a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80;
  • Sage
    (x*(1+2*x+6*x^2+2*x^3+x^4)/((1-x)^6*(1+x)^2)).series(x, 39).coefficients(x, False)

Formula

a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80.
O.g.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^2).
E.g.f.: (cosh(x) - sinh(x))*(-5 + 5*x + (5 + 65*x + 180*x^2 + 130*x^3 + 30*x^4 + 2*x^5)*(cosh(2*x) + sinh(2*x)))/80.
a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n > 7.
a(2*n-1) = n*A053755(n)/5 for n > 0.
a(2*n) = n*A005408(n)*A059722(n-1)/5.
a(2*n+1) - a(2*n-1) = A001844(n)^2 = A007204(n) for n > 0.
a(2*n) - a(2*n-2) = 2*A000290(n)*A058331(n) for n > 0.

A340804 Triangle read by rows: T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k mod 2) with 0 < k <= n.

Original entry on oeis.org

1, 1, 3, 1, 5, 9, 1, 7, 11, 13, 1, 9, 13, 17, 25, 1, 11, 15, 21, 29, 31, 1, 13, 17, 25, 33, 37, 49, 1, 15, 19, 29, 37, 43, 55, 57, 1, 17, 21, 33, 41, 49, 61, 65, 81, 1, 19, 23, 37, 45, 55, 67, 73, 89, 91, 1, 21, 25, 41, 49, 61, 73, 81, 97, 101, 121, 1, 23, 27, 45, 53, 67, 79, 89, 105, 111, 131, 133
Offset: 1

Views

Author

Stefano Spezia, Jan 22 2021

Keywords

Comments

T(n, k) is the k-th diagonal element of an n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern.
It includes exclusively all the odd numbers (A005408). Except the term 1, all the other odd numbers appear a finite number of times.

Examples

			1
1,  3
1,  5,  9,
1,  7, 11, 13
1,  9, 13, 17, 25
1, 11, 15, 21, 29, 31
1, 13, 17, 25, 33, 37, 49
...

Links

Crossrefs

Cf. A005408, A317614 (row sums).
Cf. A000012 (1st column), A006010 (sum of the first n rows), A060747 (2nd column), A074147 (antidiagonals of M matrices), A241016 (row sums of M matrices), A317617 (column sums of M matrices), A322277 (permanent of M matrices), A323723 (subdiagonal sum of M matrices), A323724 (superdiagonal sum of M matrices).

Programs

  • Mathematica
    Table[1+k(n-1)+(2k-n-1)Mod[k,2],{n,12},{k,n}]//Flatten
  • PARI
    T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k % 2); \\ Michel Marcus, Jan 25 2021

Formula

O.g.f.: (1 + y - 3*y^2 + y^3 + x*(-1 - y + 5*y^2 + y^3))/((-1 + x)^2*(-1 + y)^2*(1+y)^2).
E.g.f.: exp(x - y)*(1 + x + 2*y + exp(2*y)*(1 + x*(-1 + 2*y)))/2.

A374668 a(n) is the permanent of the n-th order Hankel matrix M whose generic element is given by M(i,j) = A317614(i+j-1) with i,j = 1, ..., n.

Original entry on oeis.org

1, 1, 31, 10254, 12238276, 41596930860, 309346186680924, 4522151204857137264, 116038936382978521700928, 4918677318766771695942334272, 323424014903141386787887115413440, 31725978444319999354629697685162941056, 4460612938377274751881312432310360618154240
Offset: 0

Views

Author

Stefano Spezia, Jul 15 2024

Keywords

Comments

The Hankel transform of A317614 has the following polynomial as g.f. 1 + x - x^2 - 624*x^3 - 9216*x^4 + 138240*x^5 - 331776*x^6: the matrices are singular for n > 6.

Examples

			a(7) = 4522151204857137264:
  [  1,   4,  15,  32,  65, 108,  175]
  [  4,  15,  32,  65, 108, 175,  256]
  [ 15,  32,  65, 108, 175, 256,  369]
  [ 32,  65, 108, 175, 256, 369,  500]
  [ 65, 108, 175, 256, 369, 500,  671]
  [108, 175, 256, 369, 500, 671,  864]
  [175, 256, 369, 500, 671, 864, 1105]
which is the singular matrix M of minimal order.

Links

Crossrefs

Cf. A317614.
Cf. A000583 (trace of M), A006010 (sum of the 1st row or column of M), A035287 (super- or subdiagonal sum of M), A346174, A374708 (k-th super- or subdiagonal sum of M).

Programs

  • Mathematica
    A317614[n_]:=(1/2)*(n^3 + n*Mod[n,2]); a[n_]:=Permanent[Table[A317614[i+j-1], {i, n}, {j, n}]]; Join[{1}, Array[a, 12]]
Showing 1-3 of 3 results.

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