Search: a101625 -id:a101625
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A101624
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Stern-Jacobsthal numbers.
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+10
9
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1, 1, 3, 1, 7, 5, 11, 1, 23, 21, 59, 17, 103, 69, 139, 1, 279, 277, 827, 273, 1895, 1349, 2955, 257, 5655, 5141, 14395, 4113, 24679, 16453, 32907, 1, 65815, 65813, 197435, 65809, 460647, 329029, 723851, 65793, 1512983, 1381397, 3881019, 1118225
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OFFSET
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0,3
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COMMENTS
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The Stern diatomic sequence A002487 could be called the Stern-Fibonacci sequence, since it is given by A002487(n) = Sum_{k=0..floor(n/2)} (binomial(n-k,k) mod 2), where F(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k). Now a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k,k) mod 2)*2^k, where J(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*2^k, with J(n) = A001045(n), the Jacobsthal numbers. - Paul Barry, Sep 16 2015
These numbers seem to encode Stern (0, 1)-polynomials in their binary expansion. See Dilcher & Ericksen paper, especially Table 1 on page 79, page 5 in PDF. See A125184 (A260443) for another kind of Stern-polynomials, and also A177219 for a reference to maybe a third kind. - Antti Karttunen, Nov 01 2016
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k, k) mod 2)*2^k.
a(2^n-1)=1, a(2*n) = 2*a(n-1) + a(n+1) = A099902(n); a(2*n+1) = A101625(n+1).
a(n) = Sum_{k=0..n} (binomial(k, n-k) mod 2)*2^(n-k). - Paul Barry, May 10 2005
a(0)=1, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*2), where XOR is the bitwise exclusive-OR operator. - Alex Ratushnyak, Apr 14 2012
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PROG
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(Python)
prpr = 1
prev = 1
print("1, 1", end=", ")
for i in range(99):
current = (prev)^(prpr*2)
print(current, end=", ")
prpr = prev
prev = current
(Python)
def A101624(n): return sum(int(not k & ~(n-k))*2**k for k in range(n//2+1)) # Chai Wah Wu, Jun 20 2022
(Haskell)
a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A204771
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a(n) = a(n-1) XOR (a(n-2)*3).
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+10
2
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0, 1, 1, 2, 1, 7, 4, 17, 29, 46, 121, 243, 408, 833, 1929, 3658, 6353, 12815, 30844, 61009, 100133, 216534, 514233, 930107, 1686288, 3352737, 8264081, 15163506, 27077825, 53153175, 133991380, 243114769, 428343405, 854649182, 2120804377, 3870970883, 6937439304
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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LINKS
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FORMULA
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a(0)=0, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*3), where XOR is the bitwise exclusive-OR operator.
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PROG
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(Python)
prpr = 0
prev = 1
for i in range(99):
current = (prev)^(prpr*3)
print(prpr, end=', ')
prpr = prev
prev = current
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CROSSREFS
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Cf. A101624: a(n) = a(n-1) XOR (a(n-2)*2).
Cf. A101625: a(n) = a(n-1) XOR (a(n-2)*4).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A260534
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Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).
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+10
1
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 7, 2, 1, 1, 1, 6, 1, 13, 5, 3, 1, 1, 1, 7, 1, 21, 10, 11, 1, 1, 1, 1, 8, 1, 31, 17, 31, 1, 4, 1, 1, 1, 9, 1, 43, 26, 69, 1, 23, 3, 1, 1, 1, 10, 1, 57, 37, 131, 1, 94, 21, 5, 1, 1, 1, 11
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table;
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listen;
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OFFSET
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0,9
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COMMENTS
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A parametrization of Stern's diatomic series (which is here T(1,k)). (For other generalizations of Dijkstra's fusc function see the Luschny link.)
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LINKS
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EXAMPLE
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Array starts:
n\k[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[1] 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, ... [A002487]
[2] 1, 1, 3, 1, 7, 5, 11, 1, 23, 21, 59, ... [A101624]
[3] 1, 1, 4, 1, 13, 10, 31, 1, 94, 91, 355, ...
[4] 1, 1, 5, 1, 21, 17, 69, 1, 277, 273, 1349, ... [A101625]
[5] 1, 1, 6, 1, 31, 26, 131, 1, 656, 651, 3881, ...
[6] 1, 1, 7, 1, 43, 37, 223, 1, 1339, 1333, 9295, ...
[7] 1, 1, 8, 1, 57, 50, 351, 1, 2458, 2451, 19559, ...
[8] 1, 1, 9, 1, 73, 65, 521, 1, 4169, 4161, 37385, ...
-,-,-,-,A002061,A002522,A071568,-,-,A059826,-,A002523,
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MAPLE
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T := (n, k) -> add(modp(binomial(k-j, j), 2)*n^j, j=0..k):
seq(lprint(seq(T(n, k), k=0..10)), n=0..5);
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MATHEMATICA
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Table[If[(n - k) == 0, 1, Sum[(n - k)^j Mod[Binomial[k - j, j], 2], {j, 0, k}]], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, Sep 21 2015 *)
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PROG
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(Python)
return sum(0 if ~(k-j) & j else n**j for j in range(k+1)) # Chai Wah Wu, Feb 08 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A284247
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Binary representation of generation n in the reversible cellular automaton RCA(1) when started with a single ON cell at generation 0.
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+10
1
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1, 1, 101, 1, 10101, 10001, 1000101, 1, 100010101, 100010001, 10101000101, 100000001, 1010000010101, 1000000010001, 100000001000101, 1, 10000000100010101, 10000000100010001, 1010000010101000101, 10000000100000001, 101010001010000010101, 100010001000000010001
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OFFSET
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0,3
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COMMENTS
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This sequence is the binary representation of A101625.
The count of ON cells at each generation is A002487.
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LINKS
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EXAMPLE
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The fourth generation (starting at 0) of RCA(1) is x.x.x where "x" is an ON cell and "." is OFF. Treating this as a binary number yields 10101. Thus a(4) = 10101.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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