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 1-6 of 6 results.

A062394 a(n) = 6^n + 1.

Original entry on oeis.org

2, 7, 37, 217, 1297, 7777, 46657, 279937, 1679617, 10077697, 60466177, 362797057, 2176782337, 13060694017, 78364164097, 470184984577, 2821109907457, 16926659444737, 101559956668417, 609359740010497, 3656158440062977
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2001

Keywords

Links

Crossrefs

Sequences of the form m^n + 1: A000012 (m=0), A007395 (m=1), A000051 (m=2), A034472 (m=3), A052539 (m=4), A034474 (m=5), this sequence (m=6), A034491 (m=7), A062395 (m=8), A062396 (m=9), A062397 (m=10), A034524 (m=11), A178248 (m=12), A141012 (m=13), A228081 (m=64).
Cf. A000400.

Programs

  • Magma
    [6^n + 1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011
  • Mathematica
    6^Range[0,30] +1
LinearRecurrence[{7,-6},{2,7},30] (* Harvey P. Dale, Aug 11 2015 *)
  • PARI
    vector(20, n, n--; 6^n + 1) \\ Michel Marcus, Jun 11 2015
  • SageMath
    [6^n+1 for n in range(31)] # G. C. Greubel, Mar 11 2023

Formula

a(n) = 6*a(n-1) - 5.
a(n) = A000400(n) + 1.
a(n) = 7*a(n-1) - 6*a(n-2).
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-6*x).
E.g.f.: exp(x) + exp(6*x). (End)

A034524 a(n) = 11^n + 1.

Original entry on oeis.org

2, 12, 122, 1332, 14642, 161052, 1771562, 19487172, 214358882, 2357947692, 25937424602, 285311670612, 3138428376722, 34522712143932, 379749833583242, 4177248169415652, 45949729863572162, 505447028499293772
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form m^n + 1: A000012 (m=0), A007395 (m=1), A000051 (m=2), A034472 (m=3), A052539 (m=4), A034474 (m=5), A062394 (m=6), A034491 (m=7), A062395 (m=8), A062396 (m=9), A062397 (m=10), this sequence (m=11), A178248 (m=12), A141012 (m=13), A228081 (m=64).
Cf. A001020.

Programs

Formula

From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-11*x).
E.g.f.: exp(x) + exp(11*x). (End)
From G. C. Greubel, Mar 11 2023: (Start)
a(n) = 11*a(n-1) - 10.
a(n) = A001020(n) + 1. (End)

A141013 E.g.f. Sum_{d|M} (exp(d*x)-1)/d, M=14.

Original entry on oeis.org

0, 4, 24, 250, 3096, 40834, 554664, 7647250, 106237176, 1481554114, 20701400904, 289537131250, 4051542498456, 56707753666594, 793811662272744, 11112685048647250, 155572843119354936
Offset: 0

Views

Author

R. J. Mathar, Jul 11 2008

Keywords

Links

Crossrefs

Cf. A141012 (M=13), A141014 (M=15).

Programs

  • Magma
    [0] cat [1+2^(n-1)+7^(n-1)+14^(n-1): n in [1..20]]; // Vincenzo Librandi, Dec 12 2012
  • Maple
    A141013 := proc(n) local d; add(d^(n-1),d=numtheory[divisors](14)) ; end proc: seq(A141013(n),n=1..20) ; # R. J. Mathar, Mar 05 2010
  • Mathematica
    CoefficientList[Series[- 2 x (-2 + 36 x - 163 x^2 + 168 x^3)/((x-1) (14*x-1) (2*x-1) (7*x-1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 12 2012 *)

Formula

From R. J. Mathar, Mar 05 2010: (Start)
a(n) = sum_{d|14} d^(n-1) = 1+2^(n-1)+7^(n-1)+14^(n-1).
a(n)= 24*a(n-1) -163*a(n-2) +336*a(n-3) -196*a(n-4), n>4.
G.f: -2*x*(-2+36*x-163*x^2+168*x^3)/((x-1)*(14*x-1)*(2*x-1)*(7*x-1)).
(End)
a(n) = A000051(n-1)*A034491(n-1). - R. J. Mathar, May 26 2016

A224384 a(n) = 1 + 17^n.

Original entry on oeis.org

2, 18, 290, 4914, 83522, 1419858, 24137570, 410338674, 6975757442, 118587876498, 2015993900450, 34271896307634, 582622237229762, 9904578032905938, 168377826559400930, 2862423051509815794, 48661191875666868482, 827240261886336764178, 14063084452067724991010
Offset: 0

Views

Author

Philippe Deléham, Apr 05 2013

Keywords

Comments

Sum of n-th powers of divisors of 17.

Links

Crossrefs

Cf. A001026.
Similar sequences: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660, A141012, A141013, A141014, A020514, A034661, A034662, A034663, A034664, A034665, A034666, A034667, A034668, A034669, A020516, A034671, A034672, A034673, A034674, A034675.

Programs

Formula

a(n) = A001026(n) + 1.
G.f.: 1/(1-x) + 1/(1-17*x).
E.g.f.: exp(x) + exp(17*x).
a(n) = 18*a(n-1) - 17*a(n-2) with a(0) = 2, a(1) = 18.

A153079 a(n) = 13^n + 2.

Original entry on oeis.org

3, 15, 171, 2199, 28563, 371295, 4826811, 62748519, 815730723, 10604499375, 137858491851, 1792160394039, 23298085122483, 302875106592255, 3937376385699291, 51185893014090759, 665416609183179843, 8650415919381337935, 112455406951957393131, 1461920290375446110679
Offset: 0

Views

Author

Vincenzo Librandi, Feb 10 2009

Keywords

Links

Crossrefs

Cf. A141012.

Programs

  • Magma
    I:=[3, 15]; [n le 2 select I[n] else 14*Self(n-1)-13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 25 2012
  • Mathematica
    LinearRecurrence[{14,-13},{3,15},30]
13^Range[0,20]+2 (* Harvey P. Dale, Aug 13 2019 *)

Formula

From R. J. Mathar, Jan 05 2011: (Start)
a(n) = 1 + A141012(n+1).
G.f.: (3-27*x)/((13*x-1)*(x-1)). (End)
From Elmo R. Oliveira, May 07 2025: (Start)
E.g.f.: (2 + exp(12*x))*exp(x).
a(n) = 14*a(n-1) - 13*a(n-2). (End)

Extensions

a(9) corrected and a(17), a(18) added by R. J. Mathar, Jan 05 2011

A193578 a(n) = (13^n + 1)/2.

Original entry on oeis.org

1, 7, 85, 1099, 14281, 185647, 2413405, 31374259, 407865361, 5302249687, 68929245925, 896080197019, 11649042561241, 151437553296127, 1968688192849645, 25592946507045379, 332708304591589921, 4325207959690668967, 56227703475978696565, 730960145187723055339, 9502481887440399719401
Offset: 0

Views

Author

Vincenzo Librandi, Sep 17 2011

Keywords

Links

Crossrefs

Cf. A141012.

Programs

Formula

a(n) = 13*a(n-1) - 6, a(0)=1.
From Elmo R. Oliveira, Mar 25 2025: (Start)
G.f.: (1-7*x)/((1-x)*(1-13*x)).
E.g.f.: exp(7*x)*cosh(6*x).
a(n) = 14*a(n-1) - 13*a(n-2).
a(n) = A141012(n+1)/2. (End)
Showing 1-6 of 6 results.

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

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