A152891 -id:A152891 - cp's OEIS frontend

A152895 Partial sums of A152891.

0, 2, 11, 40, 123, 349, 951, 2539, 6710, 17645, 46290, 121302, 317706, 831920, 2178173, 5702734, 14930181, 39087979, 102333945, 267914065, 701408480, 1836311627, 4807526676, 12586268700, 32951279748, 86267570894, 225851433311

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (6,-13,13,-6,1).

Cf. A000217, A001906, A152891.

lst={};a=b=s=0;Do[b+=n+a;a+=n+b;s+=a;AppendTo[lst,s],{n,0,2*4!}];lst
Accumulate[LinearRecurrence[{5,-8,5,-1},{0,2,9,29},30]] (* or *) LinearRecurrence[{6,-13,13,-6,1},{0,2,11,40,123},30] (* Harvey P. Dale, Mar 26 2015 *)

From R. J. Mathar, Jan 08 2009: (Start)
G.f.: x^2*(2-x)/((1-3x+x^2)*(1-x)^3).
a(n) = A001906(n+1)-A000217(n+1). (End)
a(1)=0, a(2)=2, a(3)=11, a(4)=40, a(5)=123, a(n)=6*a(n-1)-13*a(n-2)+ 13*a(n-3)- 6*a(n-4)+a(n-5). - Harvey P. Dale, Mar 26 2015

A035508 a(n) = Fibonacci(2*n+2) - 1.

Original entry on oeis.org

0, 2, 7, 20, 54, 143, 376, 986, 2583, 6764, 17710, 46367, 121392, 317810, 832039, 2178308, 5702886, 14930351, 39088168, 102334154, 267914295, 701408732, 1836311902, 4807526975, 12586269024, 32951280098, 86267571271, 225851433716

Offset: 0

N. J. A. Sloane

Except for 0, numbers whose dual Zeckendorf representation (A104326) has the same number of 0's as 1's. - Amiram Eldar, Mar 22 2021

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, Vol. 117, No. 2 (1993), pp. 313-321.
N. J. A. Sloane, Classic Sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

With different offset: 2nd row of Inverse Stolarsky array A035507.

Cf. A001906, A104326, A112844, A152891 (partial sums).

[Fibonacci(2*n+2)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-1), n=1..26); # Zerinvary Lajos, Mar 22 2009
with(combinat):seq(fibonacci(4*n+2) mod fibonacci(2*n+2),n=0..25);

Fibonacci[2*Range[0, 5!]] - 1 (* Vladimir Joseph Stephan Orlovsky, May 18 2010 *)
LinearRecurrence[{4,-4,1},{0,2,7},40] (* Harvey P. Dale, Jan 15 2025 *)

makelist(fib(2*n+2)-1,n,0,30); /* Martin Ettl, Oct 21 2012 */

numlib::fibonacci(2*n)-1 $ n = 1..38; // Zerinvary Lajos, May 08 2008

[lucas_number1(n, 3, 1)-1 for n in range(1, 27)] # Zerinvary Lajos, Dec 07 2009

a(n) = A001906(n) - 1.
G.f.: x*(2 - x)/((1 - x)*(1 - 3*x + x^2)). a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - R. J. Mathar, Dec 15 2008; adapted to the offset by Bruno Berselli, Apr 19 2011
a(n) = Fibonacci(4*n+2) mod Fibonacci(2*n+2). - Gary Detlefs, Nov 22 2010
a(n+1) = Sum_{k=0..n} Fibonacci(2*k+3). - Gary Detlefs, Dec 24 2010
a(n) = Sum_{i=1..n} A112844(i). - R. J. Mathar, Apr 19 2011
a(n) = floor(Fibonacci(2*n+2) - Fibonacci(n+1)^2/Fibonacci(2*n+2)). - Gary Detlefs, Dec 21 2012
From Peter Bala, Nov 14 2021: (Start)
a(n) = Fibonacci(2*n+4)*(Fibonacci(2*n+1) - 1)/(Fibonacci(2*n+3) - 1).
a(n)= -2 + Sum_{k = 1..2*n+3} (-1)^(k+1)*Fibonacci(k). (End)

A152896 a=b=c=0;c(n)=c+n+a;b(n)=b+n+c;a(n)=a+n+b.

Original entry on oeis.org

0, 3, 15, 55, 183, 588, 1865, 5887, 18550, 58414, 183904, 578936, 1822459, 5736946, 18059358, 56849069, 178955165, 563332829, 1773314909, 5582216334, 17572253459, 55315679765, 174128175040, 548137914348, 1725482812062

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (6, -12, 11, -5, 1).

Cf. A152891, A152895

lst={};a=b=c=0;Do[c+=n+a;b+=n+c;a+=n+b;AppendTo[lst,a],{n,0,2*4!}];lst
LinearRecurrence[{6,-12,11,-5,1},{0,3,15,55,183},30] (* Harvey P. Dale, Dec 11 2014 *)

G.f.: x^2(3-3x+x^2)/((1-x)^2*(1-4x+3x^2-x^3)). a(n)=6*a(n-1)-12*a(n-2)+11*a(n-3)-5*a(n-4)+a(n-5). [From R. J. Mathar, Dec 15 2008]

A152897 Partial sums of A152896.

Original entry on oeis.org

0, 3, 18, 73, 256, 844, 2709, 8596, 27146, 85560, 269464, 848400, 2670859, 8407805, 26467163, 83316232, 262271397, 825604226, 2598919135, 8181135469, 25753388928, 81069068693, 255197243733, 803335158081, 2528817970143

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (7,-18,23,-16,6,-1).

Cf. A152891, A152895, A152896.

lst={};a=b=c=s=0;Do[c+=n+a;b+=n+c;a+=n+b;s+=a;AppendTo[lst,s],{n,0,2*4!}];lst
Rest[CoefficientList[Series[x^2(3-3x+x^2)/((1-x)^3(1-4x+3x^2-x^3)),{x,0,40}],x]]  (* Harvey P. Dale, Mar 11 2011 *)
LinearRecurrence[{7,-18,23,-16,6,-1},{0,3,18,73,256,844},30] (* Harvey P. Dale, Feb 07 2015 *)

a(n)=A124820(n)-A000217(n+1). G.f.: x^2(3-3x+x^2)/((1-x)^3*(1-4x+3x^2-x^3)). [From R. J. Mathar, Dec 15 2008]

a(1)=0, a(2)=3, a(3)=18, a(4)=73, a(5)=256, a(6)=844, a(n)=7*a(n-1)- 18*a(n-2)+ 23*a(n-3)-16*a(n-4)+6*a(n-5)-a (n-6). - Harvey P. Dale, Feb 07 2015

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A152895 Partial sums of A152891.

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A035508 a(n) = Fibonacci(2*n+2) - 1.

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A152896 a=b=c=0;c(n)=c+n+a;b(n)=b+n+c;a(n)=a+n+b.

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A152897 Partial sums of A152896.

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