A173862 -id:A173862 - cp's OEIS frontend

A200672 Partial sums of A173862.

1, 2, 3, 5, 7, 9, 13, 17, 21, 29, 37, 45, 61, 77, 93, 125, 157, 189, 253, 317, 381, 509, 637, 765, 1021, 1277, 1533, 2045, 2557, 3069, 4093, 5117, 6141, 8189, 10237, 12285, 16381, 20477, 24573, 32765, 40957, 49149, 65533, 81917, 98301, 131069, 163837, 196605

Offset: 1

Jeremy Gardiner, Nov 20 2011

nonn

Partial sums of powers of 2 repeated 3 times.

			a(4) = 1+1+1+2 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Shatalov A. A, The Cupola Algorithm Data And The Modulation-37 The Natural Sciences Aspect And The Using For Analysis Of Ancient Layouts, European Journal Of Natural History, 2007 No. 1, p. 35.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2).

Cf. A027383, A173862.

for i=0 to 12 : for j=1 to 3 : s=s+2^i : print s : next j : next i

I:=[1,2,3,5]; [n le 4 select I[n] else Self(n-1) + 2*Self(n-3)- 2*Self(n-4): n in [1..50]]; // Vincenzo Librandi, Nov 16 2018

CoefficientList[Series[(1 + x + x^2) / ((x - 1) (2 x^3 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 16 2018 *)
Accumulate[Flatten[Table[PadRight[{},3,2^n],{n,0,20}]]] (* or *) LinearRecurrence[ {1,0,2,-2},{1,2,3,5},60] (* Harvey P. Dale, Jul 12 2022 *)

G.f.: x*(1+x+x^2) / ( (x-1)*(2*x^3-1) ). - R. J. Mathar, Nov 28 2011
a(3*n) = 3*(2^n-1) = 3*A000225(n). - Philippe Deléham, Mar 13 2013
a(n) = 2*a(n-3) + 3 for n > 3. - Yuchun Ji, Nov 16 2018

A216218 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=2 or if k-n>=2, T(1,0) = T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 13 2013

With zeros omitted, this is A173862.

			Square array begins:
1, 1, 0, 0,  0,  0,  0, 0, ... row n=0
1, 2, 2, 0,  0,  0,  0, 0, ... row n=1
0, 2, 4, 4,  0,  0,  0, 0, ... row n=2
0, 0, 4, 8,  8,  0,  0, 0, ... row n=3
0, 0, 0, 8, 16, 16,  0, 0, ... row n=4
0, 0, 0, 0, 16, 32, 32, 0, ... row n=5
...

Cf. A000079, A016116, A216216, A216210, A068913

T(n,n) = T(n+1,n) = T(n,n+1) = 2^n = A000079(n).

Sum_{k, 0<=k<=n} T(n-k,k) = A016116(n+1) = A163403(n+1).

A200675 Powers of 2 repeated 4 times.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 32, 32, 32, 32, 64, 64, 64, 64, 128, 128, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096

Offset: 0

Jeremy Gardiner, Nov 20 2011

Run lengths in A173922.

			a(4) = 2^1 = 2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2)

Cf. A000079, A016116, A173862, A108105, A173922, A200678.

CoefficientList[Series[x*(1 + x)*(1 + x^2)/(1 - 2*x^4), {x,0,50}],x] (* or *) Table[2^(Floor[n/4]), {n,0,50}] (* G. C. Greubel, Apr 30 2017 *)

a(n)=2^(n\4) \\ Charles R Greathouse IV, Oct 03 2016

def A200675(n): return 1<<(n>>2) # Chai Wah Wu, Jan 30 2023

a(n) = 2^floor(n/4).

G.f.: x*(1+x)*(1+x^2)/(1-2*x^4 ). - R. J. Mathar, Nov 21 2011

Offset corrected by Charles R Greathouse IV, Oct 03 2016

A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 9, 9, 12, 12, 15, 16, 17, 21, 21, 27, 28, 32, 37, 38, 48, 49, 59, 65, 70, 85, 87, 107, 114, 129, 150, 157, 192, 201, 236, 264, 286, 342, 358, 428, 465, 522, 606, 644, 770, 823, 950, 1071, 1166, 1376

Offset: 1

Vicente Jesús Maniega Granado, Oct 03 2016

Generalized Fibonacci growth sequence using i = 2 as maturity period, j = 5 as conception period, and k = 2 as growth factor.

Maturity period is the number of periods that a Fibonacci tree node needs for being able to start developing branches. Conception period is the number of periods in a Fibonacci tree node needed to develop new branches since its maturity. Growth factor is the number of additional branches developed by a Fibonacci tree node, plus 1, and equals the base of the exponential series related to the given tree if maturity factor would be zero. Standard Fibonacci would use 1 as maturity period, 1 as conception period, and 2 as growth factor as the series becomes equal to 2^n with a maturity period of 0. Related to Lucas sequences.

			For n = 13 the a(13) = a(8) + a(6) = 2 + 1 = 3.

Colin Barker, Table of n, a(n) for n = 1..1000
Julia Collins, Fibonacci Tree
Fractal Foundation, Fibonacci Fractals
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,1).

Cf. A000079 (i = 0, j = 1, k = 2), A000244 (i = 0, j = 1, k = 3), A000302 (i = 0, j = 1, k = 4), A000351 (i = 0, j = 1, k = 5), A000400 (i = 0, j = 1, k = 6), A000420 (i = 0, j = 1, k = 7), A001018 (i = 0, j = 1, k = 8), A001019 (i = 0, j = 1, k = 9), A011557 (i = 0, j = 1, k = 10), A001020 (i = 0, j = 1, k = 11), A001021 (i = 0, j = 1, k = 12), A016116 (i = 0, j = 2, k = 2), A108411 (i = 0, j = 2, k = 3), A213173 (i = 0, j = 2, k = 4), A074872 (i = 0, j = 2, k = 5), A173862 (i = 0, j = 3, k = 2), A127975 (i = 0, j = 3, k = 3), A200675 (i = 0, j = 4, k = 2), A111575 (i = 0, j = 4, k = 3), A000045 (i = 1, j = 1, k = 2), A001045 (i = 1, j = 1, k = 3), A006130 (i = 1, j = 1, k = 4), A006131 (i = 1, j = 1, k = 5), A015440 (i = 1, j = 1, k = 6), A015441 (i = 1, j = 1, k = 7), A015442 (i = 1, j = 1, k = 8), A015443 (i = 1, j = 1, k = 9), A015445 (i = 1, j = 1, k = 10), A015446 (i = 1, j = 1, k = 11), A015447 (i = 1, j = 1, k = 12), A000931 (i = 1, j = 2, k = 2), A159284 (i = 1, j = 2, k = 3), A238389 (i = 1, j = 2, k = 4), A097041 (i = 1, j = 2, k = 10), A079398 (i = 1, j = 3, k = 2), A103372 (i = 1, j = 4, k = 2), A103373 (i = 1, j = 5, k = 2), A103374 (i = 1, j = 6, k = 2), A000930 (i = 2, j = 1, k = 2), A077949 (i = 2, j = 1, k = 3), A084386 (i = 2, j = 1, k = 4), A089977 (i = 2, j = 1, k = 5), A178205 (i = 2, j = 1, k = 11), A103609 (i = 2, j = 2, k = 2), A077953 (i = 2, j = 2, k = 3), A226503 (i = 2, j = 3, k = 2), A122521 (i = 2, j = 6, k = 2), A003269 (i = 3, j = 1, k = 2), A052942 (i = 3, j = 1, k = 3), A005686 (i = 3, j = 2, k = 2), A237714 (i = 3, j = 2, k = 3), A238391 (i = 3, j = 2, k = 4), A247049 (i = 3, j = 3, k = 2), A077886 (i = 3, j = 3, k = 3), A003520 (i = 4, j = 1, k = 2), A108104 (i = 4, j = 2, k = 2), A005708 (i = 5, j = 1, k = 2), A237716 (i = 5, j = 2, k = 3), A005709 (i = 6, j = 1, k = 2), A122522 (i = 6, j = 2, k = 2), A005710 (i = 7, j = 1, k = 2), A237718 (i = 7, j = 2, k = 3), A017903 (i = 8, j = 1, k = 2).

[n le 7 select 1 else Self(n-5)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Nov 30 2016

LinearRecurrence[{0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 70] (*  or *)
CoefficientList[ Series[(1+x+x^2+x^3+x^4)/(1-x^5-x^7), {x, 0, 70}], x] (* Robert G. Wilson v, Nov 25 2016 *)
nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,a+c}; NestList[nxt,{1,1,1,1,1,1,1},70][[;;,1]] (* Harvey P. Dale, Oct 22 2024 *)

Vec(x*(1+x+x^2+x^3+x^4)/((1-x+x^2)*(1+x-x^3-x^4-x^5)) + O(x^100)) \\ Colin Barker, Oct 27 2016

@CachedFunction # a = A242763
def a(n): return 1 if n<8 else a(n-5) +a(n-7)
[a(n) for n in range(1,76)] # G. C. Greubel, Oct 23 2024

Generic a(n) = 1 for n <= i+j; a(n) = a(n-j) + (k-1)*a(n-(i+j)) for n>i+j where i = maturity period, j = conception period, k = growth factor.
G.f.: x*(1+x+x^2+x^3+x^4) / ((1-x+x^2)*(1+x-x^3-x^4-x^5)). - Colin Barker, Oct 09 2016
Generic g.f.: x*(Sum_{l=0..j-1} x^l) / (1-x^j-(k-1)*x^(i+j)), with i > 0, j > 0 and k > 1.

A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)f(n+1)+1, and f(2n+1) = f(n)f(n+1)+1 for n > 2.

Original entry on oeis.org

1, 1, 3, 5, 7, 9, 19, 29, 43, 53, 79, 149, 187, 293, 583, 849, 1311, 1601, 2343, 3525, 4315, 8025, 12027, 15029, 28119, 44169, 55303, 109533, 171843, 249781, 495991, 766361, 1115087, 1361297, 2103007, 3075769, 3755239, 5651717, 8267267, 10118237

Offset: 1

William Phoenix Marcum, Nov 04 2020

			f(1) = 1/1, so a(1) = 1.
f(2) = 1/2, so a(2) = 1.
f(3) = f(1) * f(2) + 1 = 3/2, so a(3) = 3.
f(4) = f(1) * f(3) + 1 = 1 * 3/2 + 1 = 5/2, so a(4) = 5.
f(5) = f(2) * f(3) + 1 = 1/2 * 3/2 + 1 = 7/4, so a(5) = 7.
f(n) for n>=1: 1, 1/2, 3/2, 5/2, 7/4, 9/4, 19/4, 29/8, 43/8, 53/8, 79/16, 149/16, 187/16, 293/32, 583/32, 849/32 ...

Georg Fischer, Table of n, a(n) for n = 1..1000

Denominators are given in A173862.

f(n) = if (n<=2, 1/n, my(x=n\2); if (n%2, f(x)*f(x+1)+1, f(x-1)*f(x+1)+1));
a(n) = numerator(f(n)); \\ Michel Marcus, Nov 05 2020

cp's OEIS Frontend

A200672 Partial sums of A173862.

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A216218 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=2 or if k-n>=2, T(1,0) = T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A200675 Powers of 2 repeated 4 times.

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A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

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A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)*f(n+1)+1, and f(2n+1) = f(n)*f(n+1)+1 for n > 2.

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A338637 a(n) is the numerator of f(n) where f(n) = 1/n for n <= 2 and f(2n) = f(n-1)f(n+1)+1, and f(2n+1) = f(n)f(n+1)+1 for n > 2.