A372352 -id:A372352 - cp's OEIS frontend

A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 4, 0, 0, 0, 256, 32, 6, 0, 0, 0, 0, 6144, 16, 0, 0, 0, 0, 0, 16777216, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 896, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6144, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16777216, 0, 56, 4

Offset: 1

Antti Karttunen, Apr 29 2024

Zeros occur in the same locations where 1's occur in array A372287.

			Array begins:
n\k| 1  2  3    4         5   6  7    8  9        10 11  12                 13
---+---------------------------------------------------------------------------
1  | 0, 0, 0,   2,        4,  6, 0,   2, 4,        6, 0,  2,                 4,
2  | 0, 0, 0,  24,       32, 16, 0,   8, 0,       32, 0, 56,                96,
3  | 0, 0, 0, 256,     6144,  0, 0, 896, 0,     6144, 0,  0,              8192,
4  | 0, 0, 0,   0, 16777216,  0, 0,   0, 0, 16777216, 0,  0,         402653184,
5  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0, 72057594037927936,
6  | 0, 0, 0,   0,        0,  0, 0,   0, 0,        0, 0,  0,                 0,

Cf. A086893, A096773, A372352, A372282, A372287.

Cf. also A372285 and A372355 (columnwise first differences).

up_to = 91;
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372352(n) = { my(k); for(i=1,oo,k=A086893(i); if(k>n, return(n-A086893(i-1)))); };
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372353sq(n,k) = A372352(A372282sq(n,k));
A372353list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372353sq((a-(col-1)),col))); (v); };
v372353 = A372353list(up_to);
A372353(n) = v372353[n];

A086893 a(n) is the index of F(n+1) at the unique occurrence of the ordered pair of reversed consecutive terms (F(n+1),F(n)) in Stern's diatomic sequence A002487, where F(k) denotes the k-th term of the Fibonacci sequence A000045.

Original entry on oeis.org

1, 3, 5, 13, 21, 53, 85, 213, 341, 853, 1365, 3413, 5461, 13653, 21845, 54613, 87381, 218453, 349525, 873813, 1398101, 3495253, 5592405, 13981013, 22369621, 55924053, 89478485, 223696213, 357913941, 894784853, 1431655765, 3579139413

Offset: 1

John W. Layman, Sep 18 2003

If the Fibonacci pairs are kept in the natural order (F(n),F(n+1)), it appears that the first term of the pair occurs in A002487 at the index given by A061547(n).
Equals row sums of triangle A177954. - Gary W. Adamson, May 15 2010
Starting at n=3, begin subtracting from (2^(n-1)-1)/2^(n-1): 3/4 - 1/2 = 1/4 with 1+4=5=a(3); 7/8 - 1/4 = 5/8 with 5+8=13=a(4); 15/16 - 5/8 = 5/16 with 5+16=21= a(5); 31/32 - 5/16 = 21/32 with 21+32=53=a(6); 63/64 - 21/32 = 21/64 with 21+64=85=a(7) and so on. For n odd in the first fraction (2^(n-1)-1)/2^(n-1), the result approaches 1/3, and for n even in the first fraction, the result approaches 2/3. - J. M. Bergot, May 08 2015
Also, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283641. - Robert Price, Mar 12 2017

			A002487 begins 0,1,1,2,1,3,2,... with offset 0. Thus a(1)=1 since (F(2),F(1)) = (1,1) occurs at term 1 of A002487. Similarly, a(2)=3 and a(3)=5, since (F(3),F(2))=(2,1) occurs at term 3 and (F(4),F(3))=(3,2) at term 5 of A002487.

A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig. 12.

Cf. A000045, A002487, A061547, A077954, A283641, A372286, A372352.
Interleaving of A002450\{0} and A072197.
Positive terms of A096773 in ascending order.
Partial sums of A158302.

[2^(n-1)*(3-(-1)^n/3)-1/3: n in [0..35]]; // Vincenzo Librandi, May 09 2015

f[n_] := Module[{a = 1, b = 0, m = n}, While[m > 0, If[OddQ@ m, b = a + b, a = a + b]; m = Floor[m/2]]; b]; a = Table[f[n], {n, 0, 10^6}]; b = Reverse /@ Partition[Map[Fibonacci, Range[Ceiling@ Log[GoldenRatio, Max@ a] + 1]], 2, 1]; Map[If[Length@ # > 0, #[[1, 1]] - 1, 0] &@ SequencePosition[a, #] &, b] (* Michael De Vlieger, Mar 15 2017, Version 10.1, after Jean-François Alcover at A002487 *)

a(n)=if(n%2,2^(n+1),2^(n+1)+2^(n-1))\3 \\ Charles R Greathouse IV, May 08 2015

def A086893(n): return (1<Chai Wah Wu, Apr 29 2024

It appears that a(n)=(4^((n+1)/2)-1)/3 if n is odd and a(n)=(a(n-1)+a(n+1))/2 if n is even.
G.f.: (1+2*x-2*x^2)/((1-x)*(1-4*x^2)); a(n) = 2^(n-1)(3-(-1)^n/3)-1/3 (offset 0); a(n) = Sum{k=0..n+1, 4^floor(k/2)/2} (offset 0); a(2n) = A002450(n+1) (offset 0); a(2n+1) = A072197(n) (offset 0). - Paul Barry, May 21 2004
a(n+2) = 4*a(n) + 1, a(1) = 1, a(2) = 3, n > 0. - Yosu Yurramendi, Mar 07 2017
a(n+1) = a(n) + A158302(n), a(1) = 1, n > 0. - Yosu Yurramendi, Mar 07 2017

More terms from Paul Barry, May 21 2004

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A372353 Array read by upward antidiagonals: A(n, k) = A372352(A372282(n, k)), n,k >= 1.

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A086893 a(n) is the index of F(n+1) at the unique occurrence of the ordered pair of reversed consecutive terms (F(n+1),F(n)) in Stern's diatomic sequence A002487, where F(k) denotes the k-th term of the Fibonacci sequence A000045.

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