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.

A102083 a(n) = 8*n^2 + 4*n + 1.

Original entry on oeis.org

1, 13, 41, 85, 145, 221, 313, 421, 545, 685, 841, 1013, 1201, 1405, 1625, 1861, 2113, 2381, 2665, 2965, 3281, 3613, 3961, 4325, 4705, 5101, 5513, 5941, 6385, 6845, 7321, 7813, 8321, 8845, 9385, 9941, 10513, 11101, 11705, 12325, 12961, 13613, 14281, 14965, 15665
Offset: 0

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Author

N. J. A. Sloane, Feb 14 2005

Keywords

Comments

If Y and Z are 2-blocks of a 2n-set X then, for n>=2, a(n-2) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Nov 18 2007
Equals binomial transform of [1, 12, 16, 0, 0, 0, ...]. - Gary W. Adamson, Jul 19 2008
Sequence found by reading the line from 1, in the direction 1, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 05 2011
First differences of A100157. - John Molokach, Jul 10 2013

Links

Crossrefs

Cf. A085601, A195241.

Programs

Formula

G.f.: (1+10*x+5*x^2)/(1-x)^3. - Paul Barry, Jun 04 2005
a(n) = 4*(4*n-1)+a(n-1) (with a(0)=1). - Vincenzo Librandi, Nov 16 2010
E.g.f.: (8*x^2 + 12*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017

A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

Original entry on oeis.org

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802
Offset: 0

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Author

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Keywords

Comments

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.
Also A077591 (without first term) and A157914 interleaved.

Links

Crossrefs

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

Programs

  • Magma
    [(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];
  • Mathematica
    CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)
  • PARI
    for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

Formula

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A269342 a(n) = (n + 1)*(2*n + 1)*(4*n + 9)/3.

Original entry on oeis.org

3, 26, 85, 196, 375, 638, 1001, 1480, 2091, 2850, 3773, 4876, 6175, 7686, 9425, 11408, 13651, 16170, 18981, 22100, 25543, 29326, 33465, 37976, 42875, 48178, 53901, 60060, 66671, 73750, 81313, 89376, 97955, 107066, 116725, 126948, 137751, 149150, 161161
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2016

Keywords

Examples

			a(0) = 0*2 + 1*3 = 3;
a(1) = 0*2 + 1*3 + 2*4 + 3*5 = 26;
a(2) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 = 85;
a(3) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 = 196;
a(4) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 + 8*10 + 9*11 = 375, etc.

Links

Crossrefs

Cf. A000466, A005563, A033996, A195241.

Programs

  • Magma
    [(n+1)*(2*n+1)*(4*n+9)/3: n in [0..50]]; // Vincenzo Librandi, Feb 25 2016
  • Mathematica
    Table[(n + 1) (2 n + 1) (4 n + 9)/3, {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {3, 26, 85, 196}, 39]
Table[Sum[8 k^2 + 12 k + 3, {k, 0, n}], {n, 0, 38}]
  • PARI
    Vec((3 + 14*x - x^2)/(1 - x)^4 + O(x^50)) \\ Michel Marcus, Feb 25 2016

Formula

G.f.: (3 + 14*x - x^2)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = Sum_{k=0..n} (8*k^2 + 12*k + 3).
Sum_{n>=0} 1/a(n) = 3*(80*log(2) + 5*Pi - 48)/175 = 0.397024075075621559...
Showing 1-3 of 3 results.

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

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