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.

A174341 a(n) = Numerator of Bernoulli(n, 1) + 1/(n+1).

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, -37, 1, 37, 1, -211, 1, 2311, 1, -407389, 1, 37153, 1, -1181819909, 1, 76977929, 1, -818946931, 1, 277930363757, 1, -84802531453217, 1, 90219075042851, 1, -711223555487930419, 1, 12696640293313423, 1, -6367871182840222481, 1, 35351107998094669831, 1, -83499808737903072705023, 1, 12690449182849194963361, 1
Offset: 0

Views

Author

Paul Curtz, Mar 16 2010

Keywords

Comments

a(n) is numerator of (A164555(n)/A027642(n) + 1/(n+1)).
1/(n+1) and Bernoulli(n,1) are autosequences in the sense that they remain the same (up to sign) under inverse binomial transform. This feature is kept for their sum, a(n)/A174342(n) = 2, 1, 1/2, 1/4, 1/6, 1/6, 1/6, 1/8, 7/90, 1/10, ...
Similar autosequences are also A000045, A001045, A113405, A000975 preceded by two zeros, and A140096.
Conjecture: the numerator of (A164555(n)/(n+1) + A027642(n)/(n+1)^2) is a(n) and the denominator of this fraction is equal to 1 if and only if n+1 is prime or 1. Cf. A309132. - Thomas Ordowski, Jul 09 2019
The "if" part of the conjecture is true: see the theorems in A309132 and A326690. The values of the numerator when n+1 is prime are A327033. - Jonathan Sondow, Aug 15 2019

Links

Crossrefs

Cf. A164555, A027642, A174342 (denominators), A025529, A003506, A309132, A326690, A327033.

Programs

  • Magma
    [2,1] cat [Numerator(Bernoulli(n)+1/(n+1)): n in [2..40]]; // Vincenzo Librandi, Jul 18 2019
  • Maple
    A174341 := proc(n) bernoulli(n,1)+1/(n+1); numer(%) end proc: # R. J. Mathar, Nov 19 2010
  • Mathematica
    a[n_] := Numerator[BernoulliB[n, 1] + 1/(n + 1)];
Table[a[n], {n, 0, 47}] (* Peter Luschny, Jul 13 2019 *)
  • PARI
    B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
a(n)=numerator(B(n) + 1/(n + 1));
for(n=0, 50, print1(a(n),", ")) \\ Indranil Ghosh, Jun 19 2017
  • PARI
    a(n)=numerator(bernpol(n, 1) + 1/(n + 1)); \\ Michel Marcus, Jun 26 2025
  • Python
    from sympy import bernoulli, Integer
def a(n): return (bernoulli(n) + 1/Integer(n + 1)).numerator # Indranil Ghosh, Jun 19 2017

Extensions

Reformulation of the name by Peter Luschny, Jul 13 2019

A185874 Second accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 11, 10, 10, 21, 26, 20, 15, 34, 48, 50, 35, 21, 50, 76, 90, 85, 56, 28, 69, 110, 140, 150, 133, 84, 36, 91, 150, 200, 230, 231, 196, 120, 45, 116, 196, 270, 325, 350, 336, 276, 165, 55, 144, 248, 350, 435, 490, 504, 468, 375, 220, 66, 175, 306, 440, 560, 651, 700, 696, 630, 495, 286, 78, 209, 370, 540, 700, 833, 924, 960, 930, 825, 638, 364, 91, 246, 440, 650, 855, 1036, 1176, 1260, 1275, 1210, 1056, 806, 455, 105, 286, 516, 770, 1025, 1260, 1456, 1596, 1665, 1650, 1540, 1326, 1001, 560
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2011

Keywords

Comments

A member of the accumulation chain: A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

Examples

			Northwest corner:
.   1,   3,   6,   10,   15,   21,   28,   36,   45,   55, ...
.   4,  11,  21,   34,   50,   69,   91,  116,  144,  175, ...
.  10,  26,  48,   76,  110,  150,  196,  248,  306,  370, ...
.  20,  50,  90,  140,  200,  270,  350,  440,  540,  650, ...
.  35,  85, 150,  230,  325,  435,  560,  700,  855, 1025, ...
.  56, 133, 231,  350,  490,  651,  833, 1036, 1260, 1505, ...
.  84, 196, 336,  504,  700,  924, 1176, 1456, 1764, 2100, ...
. 120, 276, 468,  696,  960, 1260, 1596, 1968, 2376, 2820, ...
. 165, 375, 630,  930, 1275, 1665, 2100, 2580, 3105, 3675, ...
. 220, 495, 825, 1210, 1650, 2145, 2695, 3300, 3960, 4675, ...
...

Links

Crossrefs

Cf. A144112, A051340, A141419, A185875, A185876.
Row 1 to 5: A000217, A115056, 2*A140096, 10*A000096, 5*A059845.
Column 1 to 3: A000292, A051925, A267370 and 3*A005581.
Main diagonal: A117066.

Programs

  • Mathematica
    f[n_, k_] := (1/12)*k*n*(1 + n)*(1 + 3*k + 2*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten

Formula

T(n,k) = k*n*(n+1)*(2*n+3*k+1)/12 for k>=1, n>=1.

Extensions

Edited by Bruno Berselli, Jan 14 2016

A141325 a(n) = A000045(n) + A131531(n+3).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 9, 13, 21, 33, 55, 89, 145, 233, 377, 609, 987, 1597, 2585, 4181, 6765, 10945, 17711, 28657, 46369, 75025, 121393, 196417, 317811, 514229, 832041, 1346269, 2178309, 3524577, 5702887, 9227465, 14930353, 24157817, 39088169
Offset: 0

Views

Author

Paul Curtz, Aug 03 2008

Keywords

Links

Crossrefs

Cf. A014437, A140096, A140413.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2+x^4)/((1+x^3)*(1-x-x^2)) )); // G. C. Greubel, Jun 11 2019
  • Mathematica
    LinearRecurrence[{1,1,-1,1,1}, {1,1,1,1,3}, 40] (* Jean-François Alcover, Aug 16 2017 *)
Table[Fibonacci@ n + Boole[Mod[n, 3] == 0] - 2 Boole[Mod[n, 6] == 3], {n, 0, 40}] (* Michael De Vlieger, Aug 16 2017 *)
  • PARI
    my(x='x+O('x^40)); Vec((1-x^2+x^4)/((1+x^3)*(1-x-x^2))) \\ G. C. Greubel, Jun 11 2019
  • Sage
    ((1-x^2+x^4)/((1+x^3)*(1-x-x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 11 2019

Formula

G.f.: (1-x^2+x^4)/((1+x)*(1-x+x^2)*(1-x-x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

Extensions

Definition corrected by R. J. Mathar, Sep 16 2009
More terms from R. J. Mathar, Sep 27 2009
Showing 1-3 of 3 results.

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