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.

A131056 A007318 * A131055.

Original entry on oeis.org

1, 3, 7, 17, 41, 97, 225, 513, 1153, 2561, 5633, 12289, 26625, 57345, 122881, 262145, 557057, 1179649, 2490369, 5242881, 11010049, 23068673, 48234497, 100663297, 209715201, 436207617, 905969665
Offset: 1

Views

Author

Gary W. Adamson, Jun 12 2007

Keywords

Examples

			a(4) = 17 = (1, 3, 3, 1) dot (1, 2, 2, 4) = (1 + 6 + 6 + 4).

Links

Crossrefs

Cf. A131055, A131054, A141222.

Programs

Formula

Binomial transform of A131055: (1, 2, 2, 4, 4, 6, 6, ...). A131056 = A131054 as an infinite lower triangular matrix * [1,2,3,...] as a vector.
G.f.: x*(1-2*x+2*x^3)/((1-x)*(1-2*x)^2); a(n)=-0^n/2+2^(n-1)*(n+1)+1. - Paul Barry, Jun 14 2008
a(n) = 2+A099035(n-1), n>1. - Juri-Stepan Gerasimov, Oct 02 2011

Extensions

More terms from Paul Barry, Jun 14 2008

A080827 Rounded up staircase on natural numbers.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405, 1459
Offset: 1

Views

Author

Paul Barry, Feb 28 2003

Keywords

Comments

Represents the 'rounded up' staircase diagonal on A000027, arranged as a square array. A000982 is the 'rounded down' staircase.
Partial sums of A131055. - Paul Barry, Jun 14 2008
The same sequence arises in the triangular array of integers >= 1 according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array and the second row of that subarray (with apex a(n-1)) contains just two numbers, one odd one even. The one with the same (odd) parity as a(n-1) is a(n). - David James Sycamore, Jul 29 2018

Links

Crossrefs

Apart from leading term identical to A099392.
Cf. A000027, A000982, A007590, A131055.

Programs

Formula

a(n) = ceiling((n^2+1)/2).
a(1) = 1, a(2n) = a(2n-1) + 2n, a(2n+1) = a(2n) + 2n. - Amarnath Murthy, May 07 2003
From Paul Barry, Apr 12 2008: (Start)
G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3).
a(n) = n*(n+1)/2-floor((n-1)/2). [corrected by R. J. Mathar, Jul 14 2013] (End)
From Wesley Ivan Hurt, Sep 08 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4.
a(n) = (n^2 + 2 - (1 - (-1)^n)/2)/2.
a(n) = floor(n^2/2) + 1 = A007590(n-1) + 1. (End)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) - 1/2. - Amiram Eldar, Sep 15 2022
E.g.f.: ((2 + x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 27 2024

A141222 Expansion of -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).

Original entry on oeis.org

1, 5, 22, 95, 406, 1722, 7260, 30459, 127270, 529958, 2200276, 9111830, 37650172, 155266100, 639191160, 2627302995, 10784089350, 44208873390, 181025067300, 740483276610, 3026059513620, 12355464845100
Offset: 0

Views

Author

Paul Barry, Jun 14 2008

Keywords

Comments

Apply Riordan array (1/sqrt(1-4x), xc(x)) to A131056, c(x) the g.f. of A000108.
Apply Riordan array (c(x)/sqrt(1-4*x), x*c(x)^2) to A131055.
Hankel transform appears to be (-1)^n*A085046(n).
Coefficients T(2*n+1,n) of triangle A103450. [Emanuele Munarini, Jun 01 2012, corrected by Werner Schulte, Nov 27 2021]

Links

Programs

  • Mathematica
    Table[((1+3*n+n^2)*Binomial[2*n, n])/(n+1),{n,0,20}] (* Vaclav Kotesovec, Feb 13 2014 *)
CoefficientList[Series[-1/(2*x)+(2*x-1)^2/(2*x*(1-4x)^(3/2)),{x,0,20}],x] (* Vaclav Kotesovec, Feb 13 2014 *)
a[n_] := (1 + 3 n + n^2) CatalanNumber[n];
Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 28 2021 *)
  • Maxima
    a(n):=sum(binomial(2*n,k)*binomial(n+1,2*n-k),k,0,n); makelist(a(n),n,0,40); /* Emanuele Munarini, Jun 01 2012 */

Formula

a(n) = Sum_{k=0..n} (1 + (k+1)*2^(k-1) - 0^k/2)*C(2n-k,n-k); a(n) = Sum_{k=0..n} C(2n,k)*C(n+1,2n-k).
Equals the Narayana transform (A001263) of integer squares. - Gary W. Adamson, Jul 29 2011
Conjecture: (n+1)*a(n) + 2*(-3*n-1)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
From Vaclav Kotesovec, Feb 13 2014: (Start)
G.f.: -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
a(n) = (1 + 3*n + n^2) * C(2*n,n) / (n+1).
Recurrence: (n+1)*(n^2 + n - 1)*a(n) = 2*(2*n-1)*(n^2 + 3*n + 1)*a(n-1).
(End)

Extensions

Name of the sequence corrected by Vaclav Kotesovec, Feb 13 2014

A053064 Alternately append n to end or beginning of previous term.

Original entry on oeis.org

1, 12, 312, 3124, 53124, 531246, 7531246, 75312468, 975312468, 97531246810, 1197531246810, 119753124681012, 13119753124681012, 1311975312468101214, 151311975312468101214, 15131197531246810121416, 1715131197531246810121416, 171513119753124681012141618
Offset: 1

Views

Author

Felice Russo, Feb 25 2000

Keywords

Comments

A055642(a(n)) = A058183(n); A007953(a(n)) = A037123(n); A010879(a(n)) = A010879(2*floor(n/2)) = A010879(A131055(n)); A000030(a(n)) = A000030(ceiling(n/2)). - Reinhard Zumkeller, Oct 10 2008

References

  • Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Links

Crossrefs

Cf. A007908, A053063.

Extensions

More terms from Sean A. Irvine, Dec 05 2021
Showing 1-4 of 4 results.

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