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A014825 a(n) = 4*a(n-1) + n with n>1, a(1)=1. 15
1, 6, 27, 112, 453, 1818, 7279, 29124, 116505, 466030, 1864131, 7456536, 29826157, 119304642, 477218583, 1908874348, 7635497409, 30541989654, 122167958635, 488671834560, 1954687338261, 7818749353066 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence.

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

FORMULA

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

G.f.: x/((1-4*x)*(1-x)^2).

a(n) = Sum_{k=0..n} (n-k)*4^k = Sum_{k=0..n} k*4^(n-k). - Paul Barry, Jul 30 2004

a(n) = Sum{k=0..n} binomial(n+2, k+2)*3^k [Offset 0]. - Paul Barry, Jul 30 2004

a(n) = Sum{k=0..n} Sum_{j=0..2k} (-1)^(j+1)*J(j)*J(2k-j), J(n) = A001045(n). - Paul Barry, Oct 23 2009

Convolution square of A006314. - Michael Somos, Jun 20 2012

E.g.f.: (4*exp(4*x) - (4+3*x)*exp(x))/9. - G. C. Greubel, Feb 18 2020

EXAMPLE

G.f. = x + 6*x^2 + 27*x^3 + 112*x^4 + 453*x^5 + 1818*x^6 + 7279*x^7 + ...

MAPLE

a := n -> 1/3*sum(4^j-1, j=1..n): seq(a(n), n=1..22); # Zerinvary Lajos, Jun 27 2007

a := n -> sum(4^(n-j)*j, j=0..n): seq(a(n), n=1..22); # Zerinvary Lajos, Jun 07 2008

MATHEMATICA

RecurrenceTable[{a[1]==1, a[n]==4a[n-1]+n}, a[n], {n, 30}] (* Harvey P. Dale, Oct 12 2011 *)

a[ n_]:= SeriesCoefficient[x/((1-4x)(1-x)^2), {x, 0, n}] (* Michael Somos, Jun 20 2012 *)

PROG

(MAGMA) [(4^(n+1)-3*n-4)/9: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

(PARI) {a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* Michael Somos, Jun 20 2012 */

(Sage) [(4^(n+1) -3*n -4)/9 for n in (1..30)] # G. C. Greubel, Feb 18 2020

CROSSREFS

Cf. A006314, A053142.

Sequence in context: A005284 A198694 A220101 * A141844 A176476 A079742

Adjacent sequences:  A014822 A014823 A014824 * A014826 A014827 A014828

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 8 14:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)