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 1-9 of 9 results.

A083076 Third row of number array A083075.

Original entry on oeis.org

1, 5, 33, 229, 1601, 11205, 78433, 549029, 3843201, 26902405, 188316833, 1318217829, 9227524801, 64592673605, 452148715233, 3165041006629, 22155287046401, 155087009324805, 1085609065273633, 7599263456915429, 53194844198408001
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Comments

Binomial transform of A067411. Inverse binomial transform of A082412.
Trinomial transform of Jacobsthal numbers A001045. - Paul Barry, Sep 10 2007

Links

Crossrefs

Cf. A034478, A066443, A082761, A083077.

Programs

Formula

a(n) = (2*7^n + 1)/3.
G.f.: (1-3*x)/((1-x)*(1-7*x)).
E.g.f.: (2*exp(7*x) + exp(x))/3.
a(n) = Sum_{k=0..2*n} trinomial(n,k)*Fibonacci(k+1), where trinomial(n,k) are the trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n) = 7*a(n-1) - 2, a(n) = 8*a(n-1) - 7*a(n-2). - Vincenzo Librandi, Nov 06 2011

A083082 First subdiagonal of number array A083075.

Original entry on oeis.org

1, 5, 61, 1065, 23801, 650893, 21120373, 794552657, 34040573425, 1637411510421, 87420145670573, 5131440522974329, 328542298690792681, 22789709877615373085, 1702810951672313083621, 136361486383932674632353
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083080, A083083.

Programs

  • Magma
    [((n+1)*(2*n+5)^n+1)/(n+2): n in [0..20]]; // Vincenzo Librandi, Nov 12 2011
  • Mathematica
    Table[((n+1)(2n+5)^n+1)/(n+2),{n,0,20}] (* Harvey P. Dale, Oct 06 2019 *)

Formula

a(n) = ((n+1)*(2n+5)^n + 1)/(n+2).

A083077 Fifth row of number array A083075.

Original entry on oeis.org

1, 9, 97, 1065, 11713, 128841, 1417249, 15589737, 171487105, 1886358153, 20749939681, 228249336489, 2510742701377, 27618169715145, 303799866866593, 3341798535532521, 36759783890857729, 404357622799435017
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083076, A066443, A083078.

Programs

Formula

a(n) = (4*11^n + 1)/5.
G.f.: (1-3*x)/((1-x)*(1-11*x)).
E.g.f.: (4*exp(11*x) + exp(x))/5.
a(n) = 11*a(n-1) - 2, a(n) = 12*a(n-1) - 11*a(n-2). - Vincenzo Librandi, Nov 10 2011

A083080 Main diagonal of number array A083075.

Original entry on oeis.org

1, 3, 33, 547, 11713, 309411, 9763393, 359046339, 15096500481, 714852041923, 37660464739681, 2185503641764323, 138548894120306881, 9527726662032987747, 706481006206076565633, 56192761405186331759491
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083081, A083082.

Programs

Formula

a(n) = (n*(2n+3)^n + 1)/(n+1).

A083078 6th row of number array A083075.

Original entry on oeis.org

1, 11, 141, 1831, 23801, 309411, 4022341, 52290431, 679775601, 8837082811, 114882076541, 1493466995031, 19415070935401, 252395922160211, 3281146988082741, 42654910845075631, 554513840985983201
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083075, A083077.

Programs

  • Mathematica
    LinearRecurrence[{14,-13},{1,11},30] (* Harvey P. Dale, Dec 27 2013 *)

Formula

a(n) = (5*13^n + 1)/6.
G.f.: (1-3*x)/((1-x)*(1-13*x)).
E.g.f.: (5*exp(13*x) + exp(x))/6.
a(n) = 13*a(n-1) - 2; a(n) = 14*a(n-1) - 13*a(n-2). - Vincenzo Librandi, Nov 11 2011

A083081 First superdiagonal of number array A083075.

Original entry on oeis.org

1, 13, 229, 4921, 128841, 4022341, 146450893, 6103787761, 286833509137, 15011892880381, 866190689012661, 54637591044108073, 3740820141248285785, 276304073198956644661, 21900911192388373534621
Offset: 1

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083080, A083082.

Programs

Formula

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

A083083 A diagonal of number array A083075.

Original entry on oeis.org

1, 7, 97, 1831, 43393, 1242375, 41818561, 1620979687, 71191804801, 3496805826823, 190053352702753, 11329044782441127, 735151931535979777, 51600331868857972231, 3896042468112362132353, 314921475287825567805799
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083082, A083084.

Programs

  • Magma
    [((n+2)*(2*n+7)^n+1)/(n+3): n in [0..20]]; // Vincenzo Librandi, Nov 12 2011
  • Mathematica
    Table[((n+2)(2n+7)^n+1)/(n+3),{n,0,20}] (* Harvey P. Dale, Nov 18 2021 *)

Formula

a(n) = ((n+2)*(2n+7)^n + 1)/(n+3).

A083084 A diagonal of number array A083075.

Original entry on oeis.org

1, 9, 141, 2893, 73081, 2200977, 77189509, 3095295861, 139872233073, 7039013063065, 390656716635901, 23714578436644509, 1563646420268423401, 111315499088924632353, 8511391223995285616373, 695805743909499121660357
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A083083, A083082.

Programs

Formula

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

A083079 4th column of number array A083075.

Original entry on oeis.org

1, 63, 229, 547, 1065, 1831, 2893, 4299, 6097, 8335, 11061, 14323, 18169, 22647, 27805, 33691, 40353, 47839, 56197, 65475, 75721, 86983, 99309, 112747, 127345, 143151, 160213, 178579, 198297, 219415, 241981, 266043, 291649, 318847, 347685
Offset: 0

Views

Author

Paul Barry, Apr 23 2003

Keywords

Links

Crossrefs

Cf. A082109.

Programs

  • Magma
    [8*n^3+28*n^2+26*n+1: n in [0..30]]; // Vincenzo Librandi, Nov 12 2011
  • Mathematica
    Table[8 n^3 + 28 n^2 + 26 n + 1, {n, 0, 40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{1,63,229,547},40] (* Harvey P. Dale, May 14 2011 *)

Formula

a(n) = 8*n^3 + 28*n^2 + 26*n + 1 = (n*(2n+3)^3 + 1)/(n+1).
G.f.: (1 + 59*x - 17*x^2 + 5*x^3)/(1-x)^4. - Harvey P. Dale, May 14 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=63, a(2)=229, a(3)=547. - Harvey P. Dale, May 14 2011
Showing 1-9 of 9 results.

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

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