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.

A111711 Leading column of triangle mentioned in A111710.

Original entry on oeis.org

1, 2, 5, 8, 14, 19, 28, 35, 47, 56, 71, 82, 100, 113, 134, 149, 173, 190, 217, 236, 266, 287, 320, 343, 379, 404, 443, 470, 512, 541, 586, 617, 665, 698, 749, 784, 838, 875, 932, 971, 1031, 1072, 1135, 1178, 1244, 1289, 1358, 1405, 1477, 1526, 1601, 1652
Offset: 1

Views

Author

Amarnath Murthy, Aug 24 2005

Keywords

Comments

Also partial sums of A257143. - Reinhard Zumkeller, Apr 17 2015

Links

Crossrefs

Cf. A111710, A111712, A080512, A257143.

Programs

  • Haskell
    a111711 n = a111711_list !! (n-1)
a111711_list = 1 : zipWith (+) a111711_list a080512_list
-- Reinhard Zumkeller, Apr 17 2015
  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{1,2,5,8,14},60] (* Harvey P. Dale, Jun 21 2023 *)

Formula

a(1)=1, a(2n) = a(2n-1)+2n-1, a(2n+1)=a(2n)+3n; a(n) = A111710(n-1)+1. - Franklin T. Adams-Watters, May 01 2006
From Chai Wah Wu, Mar 05 2021: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
G.f.: x*(-x^4 - x^3 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2). (End)
a(n) = (10*n*(n-1) + (-1)^n*(1-2*n)+15)/16. - Eric Simon Jacob, Jun 11 2022

Extensions

More terms from Franklin T. Adams-Watters, May 01 2006

A111712 Arithmetic mean of the n-th row of triangle mentioned in A111710.

Original entry on oeis.org

1, 3, 6, 10, 16, 22, 31, 39, 51, 61, 76, 88, 106, 120, 141, 157, 181, 199, 226, 246, 276, 298, 331, 355, 391, 417, 456, 484, 526, 556, 601, 633, 681, 715, 766, 802, 856, 894, 951, 991, 1051, 1093, 1156, 1200, 1266, 1312, 1381, 1429, 1501, 1551, 1626, 1678
Offset: 1

Views

Author

Amarnath Murthy, Aug 24 2005

Keywords

Examples

			a(4) = (8+9+10+13)/4 =10

Links

Crossrefs

Cf. A111710, A111711.
Partial sums of A195013 prepended with 1.

Programs

  • Haskell
    a111712 n = a111712_list !! (n-1)
a111712_list = scanl (+) 1 a195013_list
-- Reinhard Zumkeller, Apr 06 2015
  • Mathematica
    With[{r = Range[50]}, Accumulate[Join[{1}, Riffle[2*r, 3*r]]]] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 3, 6, 10, 16}, 100] (* Paolo Xausa, Feb 09 2024 *)

Formula

a(1)=1, a(2n) = a(2n-1)+2n, a(2n+1)=a(2n)+3n. a(n) = A111711(n)+floor(n/2). - Franklin T. Adams-Watters, May 01 2006

Extensions

More terms from Franklin T. Adams-Watters, May 01 2006

A147875 Second heptagonal numbers: a(n) = n*(5*n+3)/2.

Original entry on oeis.org

0, 4, 13, 27, 46, 70, 99, 133, 172, 216, 265, 319, 378, 442, 511, 585, 664, 748, 837, 931, 1030, 1134, 1243, 1357, 1476, 1600, 1729, 1863, 2002, 2146, 2295, 2449, 2608, 2772, 2941, 3115, 3294, 3478, 3667, 3861, 4060, 4264, 4473, 4687, 4906, 5130, 5359, 5593
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Keywords

Comments

Zero followed by partial sums of A016897.
Apparently = every 2nd term of A111710 and A085787.
Bisection of A085787. Sequence found by reading the line from 0, in the direction 0, 13, ... and the line from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
Numbers of the form m^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - Bruno Berselli, Feb 11 2015

Examples

			G.f. = 4*x + 13*x^2 + 27*x^3 + 46*x^4 + 70*x^5 + 99*x^6 + 133*x^7 + ... - _Michael Somos_, Jan 25 2019

Links

Crossrefs

Cf. A016897, A111710, A000217, A085787, A224419 (positions of squares).
Second n-gonal numbers: A005449, A014105, A045944, A179986, A033954, A062728, A135705.
Cf. A000566.
Cf. A003215, A000096, A000290.

Programs

  • GAP
    List([0..50], n-> n*(5*n+3)/2); # G. C. Greubel, Jul 04 2019
  • Magma
    [n*(5*n+3)/2: n in [0..50]]; // G. C. Greubel, Jul 04 2019
  • Mathematica
    Table[(n(5n+3))/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 4, 13}, 50] (* Harvey P. Dale, May 15 2013 *)
  • PARI
    a(n)=n*(5*n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015
  • Sage
    [n*(5*n+3)/2 for n in (0..50)] # G. C. Greubel, Jul 04 2019

Formula

G.f.: x*(4+x)/(1-x)^3.
a(n) = Sum_{k=0..n-1} A016897(k).
a(n) - a(n-1) = 5*n -1. - Vincenzo Librandi, Nov 26 2010
G.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3) + (2*k+2)*(2*k+3)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 14 2012
E.g.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - 2*x*(k+2)^2*(k+3)/(2*x*(k+2)*(k+3) + (2*k+2)^2*(2*k+3)/U(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 14 2012
a(n) = A130520(5n+3). - Philippe Deléham, Mar 26 2013
a(n) = A131242(10n+7)/2. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=13. - Harvey P. Dale, May 15 2013
Sum_{n>=1} 1/a(n) = 10/9 + sqrt(1 - 2/sqrt(5))*Pi/3 - 5*log(5)/6 + sqrt(5)*log((1 + sqrt(5))/2)/3 = 0.4688420784500060750083432... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n) + A000217(2*n). - Bruno Berselli, Jul 01 2016
From Ilya Gutkovskiy, Jul 01 2016: (Start)
E.g.f.: x*(8 + 5*x)*exp(x)/2.
Dirichlet g.f.: (5*zeta(s-2) + 3*zeta(s-1))/2. (End)
a(n) = A000566(-n) for all n in Z. - Michael Somos, Jan 25 2019
From Leo Tavares, Feb 14 2022: (Start)
a(n) = A003215(n) - A000217(n+1). See Sliced Hexagons illustration in links.
a(n) = A000096(n) + 2*A000290(n). (End)

Extensions

Edited by Klaus Brockhaus and R. J. Mathar, Nov 20 2008
New name from Bruno Berselli, Jan 13 2011
Showing 1-3 of 3 results.

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