A178626 Convolution of Pell(n) and 10^n.

1, 12, 125, 1262, 12649, 126560, 1265769, 12658098, 126581965, 1265822028, 12658226021, 126582274070, 1265822774161, 12658227822392, 126582278418945, 1265822784660282, 12658227847739509, 126582278480139300

Offset: 1

Author

Mark Dols, May 31 2010

Keywords

• nonn
• easy

Comments

Row sums of:

1

10....2

100...20...5

1000..200..50..12

10000.2000.500.120.29

or:

1

10....2

100...21...4

1000..210..44..8

10000.2100.441.92.16

or without the first column:

1

10

100......1

1000.....12

10000....124....1

100000...1248...14

Links

• G. C. Greubel, Table of n, a(n) for n = 1..995
• Index entries for linear recurrences with constant coefficients, signature (12,-19,-10).

Crossrefs

Cf. A021083, A000129, A038207, A009965.

Programs

GAP

a:=[1,12,125];; for n in [4..30] do a[n]:=12*a[n-1]-19*a[n-2] - 10*a[n-3]; od; a; # G. C. Greubel, Jan 28 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x/((10*x-1)*(x^2+2*x-1)) )); // G. C. Greubel, Jan 28 2019

Mathematica

LinearRecurrence[{12, -19, -10}, {1, 12, 125}, 30] (* G. C. Greubel, Jan 28 2019 *)

PARI

my(x='x+O('x^30)); Vec( x/((10*x-1)*(x^2+2*x-1)) ) \\ G. C. Greubel, Jan 28 2019

Sage

a=(x/((10*x-1)*(x^2+2*x-1))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 28 2019

Formula

a(n) = 10*a(n-1) + Pell(n) with a(1)= Pell(1) = 1.

From R. J. Mathar, May 31 2010: (Start)

a(n) = +12*a(n-1) -19*a(n-2) -10*a(n-3).

G.f.: x/ ( (10*x-1)*(x^2+2*x-1) ). (End)

a(n) = (2^(3+n)*5^(1+n) + (1-sqrt(2))^n*(-20+11*sqrt(2)) - (1+sqrt(2))^n*(20+11*sqrt(2))) / 316. - Colin Barker, Jan 29 2019

Extensions

a(10), a(11) corrected and sequence extended by R. J. Mathar, May 31 2010