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 1-4 of 4 results.

A080961 Fourth binomial transform of A010686 (period 2: repeat 1,5).

Original entry on oeis.org

1, 9, 57, 321, 1713, 8889, 45417, 230001, 1158753, 5820009, 29178777, 146130081, 731358993, 3658920729, 18300980937, 91524036561, 457677578433, 2288560079049, 11443316955897, 57218134461441, 286095321353073, 1430490553902969, 7152494610927657, 35762598578876721
Offset: 0

Views

Author

Paul Barry, Mar 03 2003

Keywords

Examples

			G.f. = 1 + 9*x + 57*x^2 + 321*x^3 + 1713*x^4 + 8889*x^5 + 45417*x^6 + 230001*x^7 + ...

Links

Crossrefs

Cf. A003948, A005059, A010686, A080960, A080962.

Programs

  • Magma
    binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >;
binomtf(binomtf(binomtf(binomtf(&cat[ [1, 5]: n in [1..11] ])))); // Klaus Brockhaus, Nov 26 2009
  • Magma
    [3*5^n - 2*3^n: n in [0..30]]; // Vincenzo Librandi, Dec 07 2012
  • Maple
    A080961:=n->3*5^n-2*3^n: seq(A080961(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2016
  • Mathematica
    CoefficientList[Series[(1 + x)/((1 - 3*x) * (1 - 5*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 07 2012 *)

Formula

a(n) = 5*a(n-1) + 4*3^(n-1).
a(n) = 3*5^n - 2*3^n.
G.f.: (1+x)/((1-3*x)*(1-5*x)). - Klaus Brockhaus, Nov 26 2009
From Mario C. Enriquez, Dec 08 2016: (Start)
a(n) = A005059(n+1) + A005059(n) = (5^(n+1)+5^n-3^(n+1)-3^n)/2.
a(n) = Sum_{k=0..n} A003948(n-k)*3^k = Sum_{k=0..n} (3^k * ceiling(Sum_{v=0..n-k} (5^v - 5^(v-2)))). (End)
a(n) = 8*a(n-1) - 15*a(n-2) for n > 1. - Wesley Ivan Hurt, Dec 08 2016
E.g.f.: exp(3*x)*(3*exp(2*x) - 2). - Stefano Spezia, Jul 23 2024

Extensions

Definition corrected, edited by Klaus Brockhaus, Nov 26 2009

A080962 5th binomial transform of the periodic sequence (1,6,1,1,6,1...).

Original entry on oeis.org

1, 11, 86, 596, 3896, 24656, 153056, 938816, 5714816, 34616576, 209010176, 1259303936, 7576795136, 45544656896, 273603485696, 1642963091456, 9863147257856, 59200358383616, 355288049647616, 2132071895269376, 12793805761150976, 76768332125044736, 460631982982823936
Offset: 0

Views

Author

Paul Barry, Mar 03 2003

Keywords

Links

Crossrefs

Cf. A080960, A080961, A081033.

Programs

  • Magma
    [(7/2)*6^n-(5/2)*4^n: n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
  • Mathematica
    CoefficientList[Series[(1 + x) / ((1 - 4 x) (1 - 6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{10,-24},{1,11},20] (* Harvey P. Dale, Sep 06 2016 *)

Formula

a(n) = 6*a(n-1) + 5*4^(n-1).
a(n) = (7/2)*6^n - (5/2)*4^n.
G.f.: (1+x)/((1-4*x)*(1-6*x)). - Vincenzo Librandi, Aug 06 2013
E.g.f.: exp(4*x)*(7*exp(2*x) - 5). - Stefano Spezia, Jul 23 2024

A171478 a(n) = 6*a(n-1) - 8*a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 8.

Original entry on oeis.org

1, 8, 42, 190, 806, 3318, 13462, 54230, 217686, 872278, 3492182, 13974870, 55911766, 223671638, 894735702, 3579041110, 14316361046, 57265837398, 229064136022, 916258116950, 3665035613526, 14660148745558, 58640607565142
Offset: 0

Views

Author

Klaus Brockhaus, Dec 09 2009

Keywords

Comments

Second binomial transform of A168648.
Partial sums of A080960.

Links

Crossrefs

Cf. A168648 ((10*2^n+2*(-1)^n)/3, a(0)=1), A080960 (third binomial transform of A010685), A171472, A171473.

Programs

  • GAP
    a:=[1,8];; for n in [3..25] do a[n]:=6*a[n-1]-8*a[n-2]+2; od; a; # Muniru A Asiru, Mar 22 2018
  • Magma
    [(10*4^n-9*2^n+2)/3: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011
  • Maple
    a:= proc(n) option remember: if n = 0 then 1 elif n = 1 then 8 elif  n >= 2 then 6*procname(n-1) - 8*procname(n-2) + 2 fi; end:
seq(a(n), n = 0..25); # Muniru A Asiru, Mar 22 2018
  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==8,a[n]==6a[n-1]-8a[n-2]+2},a,{n,30}] (* or *) LinearRecurrence[{7,-14,8},{1,8,42},30] (* Harvey P. Dale, May 04 2012 *)
  • PARI
    {m=23; v=concat([1, 8], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]+2); v}

Formula

a(n) = (10*4^n - 9*2^n + 2)/3.
G.f.: (1+x)/((1-x)*(1-2*x)*(1-4*x)).
a(0)=1, a(1)=8, a(2)=42, a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Harvey P. Dale, May 04 2012
a(n) = A203241(n+1) + 2^n*(2^(n+1)-1), n>0. - J. M. Bergot, Mar 21 2018

A171475 a(n) = 6*a(n-1) - 8*a(n-2), for n > 2, with a(0) = 1, a(1) = 6, a(2) = 27.

Original entry on oeis.org

1, 6, 27, 114, 468, 1896, 7632, 30624, 122688, 491136, 1965312, 7862784, 31454208, 125822976, 503304192, 2013241344, 8053014528, 32212156416, 128848822272, 515395682304, 2061583515648, 8246335635456, 32985345687552
Offset: 0

Views

Author

Klaus Brockhaus, Dec 09 2009

Keywords

Comments

Binomial transform of A037480; second binomial transform of A133600.
First differences of A080960.

Links

Crossrefs

Cf. A037480 ((5*3^n +(-1)^n -6)/8), A133600 (row sums of triangle A133599), A080960 (third binomial transform of A010685).

Programs

  • Magma
    I:=[6,27]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 02 2021
  • Mathematica
    Table[If[n==0, 1, 3*(5*4^n - 2*2^n)/8],{n,0,30}] (* G. C. Greubel, Dec 02 2021 *)
LinearRecurrence[{6,-8},{1,6,27},30] (* Harvey P. Dale, Oct 25 2023 *)
  • PARI
    {m=21; v=concat([1, 6, 27], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]); v}
  • Sage
    [1]+[3*(5*4^n - 2*2^n)/8 for n in (1..30)] # G. C. Greubel, Dec 02 2021

Formula

a(n) = 3*(5*4^n - 2*2^n)/8 for n > 0.
G.f.: (1-x)*(1+x)/((1-2*x)*(1-4*x)).
E.g.f.: (1/8)*(-1 - 6*exp(2*x) + 15*exp(4*x)). - G. C. Greubel, Dec 02 2021
Showing 1-4 of 4 results.

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