A084230 -id:A084230 - cp's OEIS frontend

A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

0, 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369, 625, 629, 645, 661, 725, 741, 805, 869, 1125, 1141, 1205, 1269, 1525, 1589, 1845, 2101, 3125, 3129, 3145, 3161, 3225, 3241, 3305, 3369, 3625, 3641, 3705, 3769, 4025, 4089, 4345, 4601, 5625

Offset: 0

Roger L. Bagula, Mar 15 2006

Equivalently, a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(0)=0, a(1)=1, for (r,s) = (1,4). - N. J. A. Sloane, Feb 16 2016
A 5-divide version of A084230.
Zero together with the partial sums of A102376. - Omar E. Pol, May 05 2010
Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton in which A102376(n-1) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid, with n >= 1. - Omar E. Pol, Feb 13 2015
From Gary W. Adamson, Aug 27 2016: (Start)
The formula of Mar 26 2010 is equivalent to lim_{k->infinity} M^k of the following production matrix M:
  1, 0, 0, 0, 0, 0, ...
  5, 0, 0, 0, 0, 0, ...
  4, 1, 0, 0, 0, 0, ...
  0, 5, 0, 0, 0, 0, ...
  0, 4, 1, 0, 0, 0, ...
  0, 0, 5, 0, 0, 0, ...
  0, 0, 4, 1, 0, 0, ...
  0, 0, 0, 5, 0, 0, ...
  ...
The sequence with offset 1 divided by its aerated variant is (1, 5, 4, 0, 0, 0, ...). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 32.
D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Index entries for sequences related to cellular automata

Cf. A000120, A006046, A077465, A130665, A130667, A102376, A360189.

Sequences of the form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

import Data.List (transpose)
a116520 n = a116520_list !! n
a116520_list = 0 : zs where
   zs = 1 : (concat $ transpose
                      [zipWith (+) vs zs, zipWith (+) vs $ tail zs])
      where vs = map (* 4) zs
-- Reinhard Zumkeller, Apr 18 2012

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 5*a(n/2) else 4*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..52);

b[0] := 0 b[1] := 1 b[n_?EvenQ] := b[n] = 5*b[n/2] b[n_?OddQ] := b[n] = 4*b[(n - 1)/2] + b[(n + 1)/2] a = Table[b[n], {n, 1, 25}]

a(0) = 1, a(1) = 1; thereafter a(2n) = 5a(n) and a(2n+1) = 4a(n) + a(n+1).
Let r(x) = (1 + 5x + 4x^2). Then (1 + 5x + 9x^2 + 25x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... . - Gary W. Adamson, Mar 26 2010
a(n) = Sum_{k=0..n-1} 4^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 4^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2006, Jul 02 2008

A077465 Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.

Original entry on oeis.org

1, 3, 5, 11, 21, 43, 87, 173, 347, 693, 1387, 2775, 5549, 11099, 22197, 44395, 88789, 177579, 355159, 710317, 1420635, 2841269, 5682539, 11365079, 22730157, 45460315, 90920629, 181841259, 363682519, 727365037, 1454730075

Offset: 1

Eric W. Weisstein, Nov 05 2002

nonn

Harborth's recurrence can miss local minima that are 2 less than values in this sequence. A complete listing of cumulative minima is given by A084230.

H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A006046, A077464, A077466, A077467, A084230.

A116522 a(0)=1, a(1)=1, a(n)=7a(n/2) for n=2,4,6,..., a(n)=6a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

Original entry on oeis.org

0, 1, 7, 13, 49, 55, 91, 127, 343, 349, 385, 421, 637, 673, 889, 1105, 2401, 2407, 2443, 2479, 2695, 2731, 2947, 3163, 4459, 4495, 4711, 4927, 6223, 6439, 7735, 9031, 16807, 16813, 16849, 16885, 17101, 17137, 17353, 17569, 18865, 18901, 19117, 19333

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

A 7-divide version of A084230.
The Harborth: f(2^k) = 3^k suggests that a family of sequences of the form: f(2^k) = prime(n)^k.
From Gary W. Adamson, Aug 27 2016: (Start)
Let M = the production matrix below. Then lim_{k->infinity} M^k generates the sequence with offset 1 by extracting the left-shifted vector.
  1, 0, 0, 0, 0, ...
  7, 0, 0, 0, 0, ...
  6, 1, 0, 0, 0, ...
  0, 7, 0, 0, 0, ...
  0, 6, 1, 0, 0, ...
  0, 0, 7, 0, 0, ...
  0, 0, 6, 1, 0, ...
  ...
The sequence divided by its aerated variant is (1, 7, 6, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A130665, A116520, A130667, A161342, A360189.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 7*a(n/2) else 6*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..47);
# second Maple program:
b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=6, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 7*b[n/2]; b[n_?OddQ] := b[n] = 6*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + 7x + 6x^2).
a(n) = Sum_{k=0..n-1} 6^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 6^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2005

A116525 a(0)=1, a(1)=1, a(n) = 11a(n/2) for even n, and a(n) = 10a((n-1)/2) + a((n+1)/2) for odd n >= 3.

Original entry on oeis.org

0, 1, 11, 21, 121, 131, 231, 331, 1331, 1341, 1441, 1541, 2541, 2641, 3641, 4641, 14641, 14651, 14751, 14851, 15851, 15951, 16951, 17951, 27951, 28051, 29051, 30051, 40051, 41051, 51051, 61051, 161051, 161061, 161161, 161261, 162261, 162361, 163361, 164361

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
    1,  0,  0, 0, 0, ...
   11,  0,  0, 0, 0, ...
   10,  1,  0, 0, 0, ...
    0, 11,  0, 0, 0, ...
    0, 10,  1, 0, 0, ...
    0,  0, 11, 0, 0, ...
    0,  0, 10, 1, 0, ...
  ...
Then lim_{k->infinity} M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 11, 10, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A084230.

Cf. A130665, A116520, A130667, A116522, A161342, A116526, A360189.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 11*a(n/2) else 10*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..42);

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 11*b[n/2]; b[n_?OddQ] := b[n] = 10*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

Let r(x) = (1 + 11x + 10x^2).  The sequence is r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Aug 30 2016
a(n) = Sum_{k=0..n-1} 10^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 10^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2005

A116526 a(0)=1, a(1)=1, a(n) = 9a(n/2) for even n >= 2, and a(n) = 8a((n-1)/2) + a((n+1)/2) for odd n >= 3.

Original entry on oeis.org

0, 1, 9, 17, 81, 89, 153, 217, 729, 737, 801, 865, 1377, 1441, 1953, 2465, 6561, 6569, 6633, 6697, 7209, 7273, 7785, 8297, 12393, 12457, 12969, 13481, 17577, 18089, 22185, 26281, 59049, 59057, 59121, 59185, 59697, 59761, 60273, 60785, 64881, 64945, 65457, 65969

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

A 9-divide version of A084230.
The interest this one has is in the prime form of even odd 2^n+1, 2^n.
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  9, 0, 0, 0, 0, ...
  8, 1, 0, 0, 0, ...
  0, 9, 0, 0, 0, ...
  0, 8, 1, 0, 0, ...
  0, 0, 9, 0, 0, ...
  0, 0, 8, 1, 0, ...
  ...
Then M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 9, 8, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A130665, A116520, A130667, A116522, A161342, A116525, A360189.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 9*a(n/2) else 8*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..45);

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 9*b[n/2]; b[n_?OddQ] := b[n] = 8*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

a(n) = Sum_{k=0..n-1} 8^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

a(n) = Sum_{k=0..floor(log_2(n))} 8^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2006

A116524 a(0)=1, a(1)=1, a(n) = 13a(n/2) for n=2,4,6,..., a(n) = 12a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

Original entry on oeis.org

0, 1, 13, 25, 169, 181, 325, 469, 2197, 2209, 2353, 2497, 4225, 4369, 6097, 7825, 28561, 28573, 28717, 28861, 30589, 30733, 32461, 34189, 54925, 55069, 56797, 58525, 79261, 80989, 101725, 122461, 371293, 371305, 371449, 371593, 373321

Offset: 0

Roger L. Bagula, Mar 15 2006

A 13-divide version of A084230.

The Harborth : f(2^k)=3^k suggests that a family of sequences of the form: f(2^k)=Prime[n]^k There does indeed seem to be an infinite family of such functions.

G. C. Greubel, Table of n, a(n) for n = 0..2500
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A084230.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 13*a(n/2) else 12*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..40);

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 13*b[n/2]; b[n_?OddQ] := b[n] = 12*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

a(n) = Sum_{k=0..n-1} 12^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

Edited by N. J. A. Sloane, Apr 16 2005

A116523 a(0)=1, a(1)=1, a(n) = 17a(n/2) for n=2,4,6,..., a(n) = 16a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

Original entry on oeis.org

0, 1, 17, 33, 289, 305, 561, 817, 4913, 4929, 5185, 5441, 9537, 9793, 13889, 17985, 83521, 83537, 83793, 84049, 88145, 88401, 92497, 96593, 162129, 162385, 166481, 170577, 236113, 240209, 305745, 371281, 1419857, 1419873, 1420129, 1420385

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

A 17-divide version of A084230.

The Harborth : f(2^k)=3^k suggests that a family of sequences of the form: f(2^k)=Prime[n]^k There does indeed seem to be an infinite family of such functions.

G. C. Greubel, Table of n, a(n) for n = 0..2500
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A084230.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 17*a(n/2) else 16*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..38);

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 17*b[n/2]; b[n_?OddQ] := b[n] = 16*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

a(n) = Sum_{k=0..n-1} 16^wt(k), where wt = A000120. - Mike Warburton, Mar 22 2019

Edited by N. J. A. Sloane, Apr 16 2005

A116593 a(n) = b(n+2) + b(n), where b(n) = A006046(n) is the sequence defined by b(0)=0, b(1)=1, b(n) = 2b((n-1)/2) + b((n+1)/2) for n =3,5,7,... and b(n) = 3b(n/2) for n =2,4,6,....

Original entry on oeis.org

3, 6, 12, 16, 24, 30, 42, 48, 60, 66, 78, 86, 102, 114, 138, 148, 168, 174, 186, 194, 210, 222, 246, 258, 282, 294, 318, 334, 366, 390, 438, 456, 492, 498, 510, 518, 534, 546, 570, 582, 606, 618, 642, 658, 690, 714, 762, 782, 822, 834, 858, 874, 906, 930, 978

Offset: 0

Roger L. Bagula, Mar 27 2006

nonn

A similar definition applied to the Fibonacci sequence (A000045) leads to the Lucas sequence (A000032). b(n) in the definition is also the number of odd entries in the first n rows of the Pascal triangle.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000032, A084230.

Cf. A006046.

b:=proc(n) option remember: if n = 0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 3*b(n/2) else 2*b((n-1)/2)+b((n+1)/2) fi end: a:=n->b(n+2)+b(n): seq(a(n),n=0..60);

b[0] := 0 b[1] := 1; b[n_?EvenQ] := b[n] = 3*b[n/2]; b[n_?OddQ] := b[n] = 2*b[(n - 1)/2] + b[(n + 1)/2]; L[0] = 1; L[n_] := L[n] = b[n - 1] + b[n + 1]; Table[L[n], {n, 1, 200}]

a(n) = A006046(n+2) + A006046(n) for n>=1.

Edited by N. J. A. Sloane, Apr 15 2006

cp's OEIS Frontend

A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

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A077465 Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.

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A116522 a(0)=1, a(1)=1, a(n)=7*a(n/2) for n=2,4,6,..., a(n)=6*a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

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A116525 a(0)=1, a(1)=1, a(n) = 11*a(n/2) for even n, and a(n) = 10*a((n-1)/2) + a((n+1)/2) for odd n >= 3.

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A116526 a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3.

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A116524 a(0)=1, a(1)=1, a(n) = 13*a(n/2) for n=2,4,6,..., a(n) = 12*a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

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A116523 a(0)=1, a(1)=1, a(n) = 17*a(n/2) for n=2,4,6,..., a(n) = 16*a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

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A116522 a(0)=1, a(1)=1, a(n)=7a(n/2) for n=2,4,6,..., a(n)=6a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

A116525 a(0)=1, a(1)=1, a(n) = 11a(n/2) for even n, and a(n) = 10a((n-1)/2) + a((n+1)/2) for odd n >= 3.

A116526 a(0)=1, a(1)=1, a(n) = 9a(n/2) for even n >= 2, and a(n) = 8a((n-1)/2) + a((n+1)/2) for odd n >= 3.

A116524 a(0)=1, a(1)=1, a(n) = 13a(n/2) for n=2,4,6,..., a(n) = 12a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

A116523 a(0)=1, a(1)=1, a(n) = 17a(n/2) for n=2,4,6,..., a(n) = 16a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

A116593 a(n) = b(n+2) + b(n), where b(n) = A006046(n) is the sequence defined by b(0)=0, b(1)=1, b(n) = 2b((n-1)/2) + b((n+1)/2) for n =3,5,7,... and b(n) = 3b(n/2) for n =2,4,6,....