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.

A174114 Even central polygonal numbers (A193868) divided by 2.

Original entry on oeis.org

1, 2, 8, 11, 23, 28, 46, 53, 77, 86, 116, 127, 163, 176, 218, 233, 281, 298, 352, 371, 431, 452, 518, 541, 613, 638, 716, 743, 827, 856, 946, 977, 1073, 1106, 1208, 1243, 1351, 1388, 1502, 1541, 1661, 1702, 1828, 1871, 2003, 2048, 2186, 2233, 2377, 2426, 2576
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 08 2010

Keywords

Comments

Central terms of A170950, seen as a triangle of rows with an odd number of terms.
Equivalently, numbers of the form m*(4*m+3)+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Jan 05 2016
Conjecure: the sequence terms are the exponents in the expansion of Sum_{n >= 1} q^n * (Product_{k >= 2*n} 1 - q^k) = q + q^2 + q^8 + q^11 + q^23 + q^28 + .... Cf. A266883. - Peter Bala, May 10 2025

Links

Crossrefs

Cf. A002061, A002522, A010696, A170950, A193868, A266883.
Cf. A033951: numbers of the form m*(4*m+3)+1 for nonnegative m.

Programs

  • Mathematica
    Select[Table[(n (n + 1)/2 + 1)/2, {n, 600}], IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)
(Select[PolygonalNumber@ Range@ 100, OddQ] + 1 )/2 (* Version 10.4, or *)
Rest@ CoefficientList[Series[-x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)
  • PARI
    a(n)=(2*n-1)*(2*n-1-(-1)^n)\4+1 \\ Charles R Greathouse IV, Jun 11 2015

Formula

a(n+3) - a(n+2) - a(n+1) + a(n) = A010696(n+1).
a(n) = A170950(A002061(n)).
a(n) = A193868(n)/2. - Omar E. Pol, Aug 16 2011
G.f.: -x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 18 2011
E.g.f.: ((2 + x + 2*x^2)*cosh(x) + (1 - x + 2*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Nov 16 2024
Sum_{n>=1} 1/a(n) = 4*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 12 2025

Extensions

New name from Omar E. Pol, Aug 16 2011

A014493 Odd triangular numbers.

Original entry on oeis.org

1, 3, 15, 21, 45, 55, 91, 105, 153, 171, 231, 253, 325, 351, 435, 465, 561, 595, 703, 741, 861, 903, 1035, 1081, 1225, 1275, 1431, 1485, 1653, 1711, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 3003, 3081, 3321, 3403, 3655, 3741, 4005, 4095, 4371, 4465, 4753, 4851
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Comments

Odd numbers of the form n*(n+1)/2.
For n such that n(n+1)/2 is odd see A042963 (congruent to 1 or 2 mod 4).
Even central polygonal numbers minus 1. - Omar E. Pol, Aug 17 2011
Odd generalized hexagonal numbers. - Omar E. Pol, Sep 24 2015

References

  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 68.

Links

Crossrefs

Cf. A000217, A000796, A014494, A019669, A042963, A067589, A128880.

Programs

  • GAP
    List([1..50], n -> (2*n-1)*(2*n-1-(-1)^n)/2); # G. C. Greubel, Feb 09 2019
  • Magma
    [(2*n-1)*(2*n-1-(-1)^n)/2: n in [1..50]]; // Vincenzo Librandi, Aug 18 2011
  • Maple
    [(2*n-1)*(2*n-1-(-1)^n)/2$n=1..50]; # Muniru A Asiru, Mar 10 2019
  • Mathematica
    Select[ Table[n(n + 1)/2, {n, 93}], OddQ[ # ] &] (* Robert G. Wilson v, Nov 05 2004 *)
LinearRecurrence[{1,2,-2,-1,1},{1,3,15,21,45},50] (* Harvey P. Dale, Jun 19 2011 *)
  • PARI
    a(n)=(2*n-1)*(2*n-1-(-1)^n)/2 \\ Charles R Greathouse IV, Sep 24 2015
  • Python
    def A014493(n): return ((n<<1)-1)*(n-(n&1^1)) # Chai Wah Wu, Feb 12 2023
  • Sage
    [(2*n-1)*(2*n-1-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Feb 09 2019

Formula

From Ant King, Nov 17 2010: (Start)
a(n) = (2*n-1)*(2*n - 1 - (-1)^n)/2.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
G.f.: x*(1 + 2*x + 10*x^2 + 2*x^3 + x^4)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = A000217(A042963(n)). - Reinhard Zumkeller, Feb 14 2012, Oct 04 2004
a(n) = A193868(n) - 1. - Omar E. Pol, Aug 17 2011
Let S = Sum_{n>=0} x^n/a(n), then S = Q(0) where Q(k) = 1 + x*(4*k+1)/(4*k + 3 - x*(2*k+1)*(4*k+3)^2/(x*(2*k+1)*(4*k+3) + (4*k+5)*(2*k+3)/Q(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 27 2013
E.g.f.: (2*x^2+x+1)*cosh(x)+x*(2*x-1)*sinh(x)-1. - Ilya Gutkovskiy, Apr 24 2016
Sum_{n>=1} 1/a(n) = Pi/2 (A019669). - Robert Bilinski, Jan 20 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2). - Amiram Eldar, Mar 06 2022

Extensions

More terms from Erich Friedman

A193867 Odd central polygonal numbers.

Original entry on oeis.org

1, 7, 11, 29, 37, 67, 79, 121, 137, 191, 211, 277, 301, 379, 407, 497, 529, 631, 667, 781, 821, 947, 991, 1129, 1177, 1327, 1379, 1541, 1597, 1771, 1831, 2017, 2081, 2279, 2347, 2557, 2629, 2851, 2927, 3161, 3241, 3487, 3571, 3829, 3917, 4187, 4279
Offset: 1

Views

Author

Omar E. Pol, Aug 15 2011

Keywords

Comments

Even triangular numbers plus 1.
Union of A188135 and A185438 without repetitions (A188135 is a bisection of this sequence. Another bisection is A185438 but without its initial term).

Links

Crossrefs

Cf. A000124, A014494, A014601, A185438, A188135, A193868.

Programs

  • Mathematica
    Select[Accumulate[Range[0,100]],EvenQ]+1 (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,7,11,29,37},50] (* Harvey P. Dale, Nov 29 2014 *)
  • PARI
    Vec(-x*(x^2+1)*(x^2+6*x+1) / ((1+x)^2*(x-1)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016

Formula

a(n) = A000124(A014601(n-1)).
a(n) = 1 + A014494(n-1).
G.f.: -x*(x^2+1)*(x^2+6*x+1) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011
From Colin Barker, Jan 27 2016: (Start)
a(n) = (4*n^2+2*(-1)^n*n-4*n-(-1)^n+3)/2.
a(n) = 2*n^2-n+1 for n even.
a(n) = 2*n^2-3*n+2 for n odd. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(2*cosh(sqrt(7)*Pi/4) - sqrt(2))). - Amiram Eldar, May 11 2025
Showing 1-3 of 3 results.

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