A130624 -id:A130624 - cp's OEIS frontend

A130625 First differences of A130624.

1, 4, 7, 11, 20, 41, 85, 172, 343, 683, 1364, 2729, 5461, 10924, 21847, 43691, 87380, 174761, 349525, 699052, 1398103, 2796203, 5592404, 11184809, 22369621, 44739244, 89478487, 178956971, 357913940, 715827881, 1431655765, 2863311532

Offset: 0

Paul Curtz, Jun 18 2007

nonn

a(n) = A130624(n+1) - A130624(n).

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A130624, A130626 (second differences).

m:=33; S:=[ [0, 1, 3][ (n-1) mod 3 +1 ]: n in [1..m] ]; T:=[ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; [ T[n+1]-T[n]: n in[1..m-1] ]; /* Klaus Brockhaus, Jun 21 2007 */

LinearRecurrence[{3,-3,2},{1,4,7},40] (* Harvey P. Dale, Apr 27 2015 *)

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

a(n) = 3a(n-1) - 3a(n-2) + 2a(n-3). Sequence is identical to its third differences. Binomial transform of 1, 3, 0. - Paul Curtz, Nov 23 2007

Edited and extended by Klaus Brockhaus, Jun 21 2007

A130626 Second differences of A130624.

Original entry on oeis.org

3, 3, 4, 9, 21, 44, 87, 171, 340, 681, 1365, 2732, 5463, 10923, 21844, 43689, 87381, 174764, 349527, 699051, 1398100, 2796201, 5592405, 11184812, 22369623, 44739243, 89478484, 178956969, 357913941, 715827884, 1431655767, 2863311531

Offset: 0

Paul Curtz, Jun 18 2007

nonn

First differences of A130625: a(n) = A130625(n+1) - A130625(n).

Index entries for linear recurrences with constant coefficients, signature (3, -3, 2).

Cf. A130624, A130625.

m:=34; S:=[ [0, 1, 3][ (n-1) mod 3 +1 ]: n in [1..m] ]; T:=[ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; U:=[ T[n+1]-T[n]: n in[1..m-1] ]; [ U[n+1]-U[n]: n in[1..m-2] ]; /* Klaus Brockhaus, Jun 21 2007 */

Differences[LinearRecurrence[{3,-3,2},{0,1,5},40],2] (* or *) LinearRecurrence[{3,-3,2},{3,3,4},40] (* Harvey P. Dale, Aug 05 2024 *)

G.f.: (3-6*x+4*x^2)/((1-2*x)*(1-x+x^2)).

a(n) = 3a(n-1) - 3a(n-2) + 2a(n-3). - Paul Curtz, Apr 24 2008

Edited and extended by Klaus Brockhaus, Jun 21 2007

cp's OEIS Frontend

A130625 First differences of A130624.

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