A129361 -id:A129361 - cp's OEIS frontend

A227356 Partial sums of A129361.

1, 2, 5, 10, 20, 36, 65, 112, 193, 324, 544, 900, 1489, 2442, 4005, 6534, 10660, 17336, 28193, 45760, 74273, 120408, 195200, 316216, 512257, 829458, 1343077, 2174130, 3519412, 5696124, 9219105, 14919408, 24144289

Offset: 1

Kival Ngaokrajang, Jul 08 2013

nonn

Sum of labeled numbers of boxes arranged as Pyramid type-II with base Fibonacci(n).
Let us call a Pyramid "type-I" when each row starts with the same number as the leftmost base number, and "type-II" when each column has the same number as the base.
The Pyramid arrangements are related to other sequences as follows:
   Base Number     Type-I     Type-II
   -----------     ------     -------
   Natural         A002623    A034828
   Odd             A000292    A128624
   Fibonacci       A129696    a(n)
   1               A002620    A002620
   1,0             A008805
See illustration in links.

Kival Ngaokrajang, Illustration for some small n.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,1,-1,0,1).

Cf. A002623, A034828, A002620, A000292, A128624, A129696, A008805.

LinearRecurrence[{2,1,-3,1,-1,0,1},{1,2,5,10,20,36,65},40] (* Harvey P. Dale, Jun 30 2025 *)

For n >=2, a(n) = a(n-1) + A129361(n-1).
G.f. -x*(1+x)*(x^2-x+1) / ( (x-1)*(x^2+x-1)*(x^4+x^2-1) ). - Joerg Arndt, Jul 10 2013
a(n) = 2 + A000045(n+4) - A096748(n+6). - R. J. Mathar, Jul 20 2013

A129362 a(n) = Sum_{k=floor((n+1)/2)..n} J(k+1), J(k) = A001045(k).

Original entry on oeis.org

1, 1, 4, 8, 19, 37, 80, 160, 331, 661, 1344, 2688, 5419, 10837, 21760, 43520, 87211, 174421, 349184, 698368, 1397419, 2794837, 5591040, 11182080, 22366891, 44733781, 89473024, 178946048, 357903019, 715806037

Offset: 0

Paul Barry, Apr 11 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).

Cf. A001045, A129361.

A001045:= func< n | (2^n - (-1)^n)/3 >;
[(&+[A001045(n-j+1): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jan 31 2024

LinearRecurrence[{1,3,-1,0,-2,-4},{1,1,4,8,19,37},30] (* Harvey P. Dale, Oct 22 2011 *)

def A001045(n): return (2^n - (-1)^n)/3
def A129362(n): return sum(A001045(n-j+1) for j in range(1+(n//2)))
[A129362(n) for n in range(31)] # G. C. Greubel, Jan 31 2024

G.f.: (1+2*x^3)/((1-x-2*x^2)*(1-x^2-2*x^4)).
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6).
a(n) = Sum_{k=0..n} ( J(k+1) - J((k+1)/2)*(1-(-1)^k)/2 ).
a(n) = Sum_{j=0..floor(n/2)} A001045(n-j+1). - G. C. Greubel, Jan 31 2024

cp's OEIS Frontend

A227356 Partial sums of A129361.

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