A101625 -id:A101625 - cp's OEIS frontend

A101624 Stern-Jacobsthal numbers.

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

Offset: 0

Paul Barry, Dec 10 2004

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000
K. Dilcher and L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, Volume 22/2014 , pp. 77-102.

Cf. A002487, A011973, A000079, A006921.

Cf. A125184, A260443, A177219.

a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
-- Reinhard Zumkeller, Jul 14 2015

prpr = 1
prev = 1
print("1, 1", end=", ")
for i in range(99):
    current = (prev)^(prpr*2)
    print(current, end=", ")
    prpr = prev
    prev = current
# Alex Ratushnyak, Apr 14 2012

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

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(n) = Sum_{k=0..n} A106344(n,k)*2^(n-k). - Philippe Deléham, Dec 18 2008
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
A000120(a(n-1)) = A002487(n). - Karl-Heinz Hofmann, Jun 18 2025

A166555 Triangle read by rows, T(n, k) = 2^k * A047999(n, k).

Original entry on oeis.org

1, 1, 2, 1, 0, 4, 1, 2, 4, 8, 1, 0, 0, 0, 16, 1, 2, 0, 0, 16, 32, 1, 0, 4, 0, 16, 0, 64, 1, 2, 4, 8, 16, 32, 64, 128, 1, 0, 0, 0, 0, 0, 0, 0, 256, 1, 2, 0, 0, 0, 0, 0, 0, 256, 512, 1, 0, 4, 0, 0, 0, 0, 0, 256, 0, 1024, 1, 2, 4, 8, 0, 0, 0, 0, 256, 512, 1024, 2048, 1, 0, 0, 0, 16, 0, 0, 0, 256, 0, 0, 0, 4096

Offset: 0

Gary W. Adamson, Oct 17 2009

Number of positive terms in n-th row (n>=0) equals to A000120(n). - Vladimir Shevelev, Oct 25 2010

Former name: Triangle read by rows, Sierpinski's gasket, A047999 * (1,2,4,8,...) diagonalized. - G. C. Greubel, Dec 02 2024

			First few rows of the triangle are:
  1;
  1, 2;
  1, 0, 4;
  1, 2, 4, 8;
  1, 0, 0, 0, 16;
  1, 2, 0, 0, 16,.32;
  1, 0, 4, 0, 16,..0,..64;
  1, 2, 4, 8, 16,.32,..64,..128;
  1, 0, 0, 0,..0,..0,...0,....0,..256;
  1, 2, 0, 0,..0,..0,...0,....0,..256,...512;
  1, 0, 4, 0,..0,..0,...0,....0,..256,.....0,...1024;
  1, 2, 4, 8,..0,..0,...0,....0,..256,...512,...1024,...2048;
  1, 0, 0, 0, 16,..0,...0,....0,..256,.....0,......0,......0,..4096;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A000007, A000079, A000120, A038183, A147999.

Sums include: A001317 (row), A101624 (diagonal), A101625 (odd rows of signed diagonal).

A166555:= func< n,k | 2^k*( Binomial(n,k) mod 2) >;
[A166555(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 01 2024

A166555[n_, k_]:= 2^k*Mod[Binomial[n, k], 2];
Table[A166555[n,k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 01 2024 *)

def A166555(n,k): return 2^k*int(not ~n & k) if kA166555(n,k) for k in range(n+1)] for n in range(15)])) # G. C. Greubel, Dec 01 2024

Triangle read by rows, A047999 * Q. A047999 = Sierpinski's gasket, Q = an infinite lower triangular matrix with (1,2,4,8,...) as the main diagonal and the rest zeros.
Sum_{k=0..n} T(n, k) = A001317(n).
From G. C. Greubel, Dec 02 2024: (Start)
T(n, k) = 2^k * (binomial(n,k) mod 2).
T(n, n) = A000079(n).
T(2*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*A038183(n/2) - (1-(-1)^n) *A038183((n-1)/2) ).
Sum_{k=0..floor(n/2)} T(n-k, k) = A101624(n).
Sum_{k=0..floor((2*m+1)/2)} T(2*m-k+1, k) = A101625(m+1), m >= 0. (End)

New name by G. C. Greubel, Dec 02 2024

A204771 a(n) = a(n-1) XOR (a(n-2)*3).

Original entry on oeis.org

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

Offset: 0

Alex Ratushnyak, May 07 2012

Ivan Panchenko, Table of n, a(n) for n = 0..200

Cf. A101624: a(n) = a(n-1) XOR (a(n-2)*2).

Cf. A101625: a(n) = a(n-1) XOR (a(n-2)*4).

prpr = 0
prev = 1
for i in range(99):
    current = (prev)^(prpr*3)
    print(prpr, end=',')
    prpr = prev
    prev = current

a(0)=0, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*3), where XOR is the bitwise exclusive-OR operator.

A260534 Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Sep 20 2015

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.)

			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,

Chai Wah Wu, Table of n, a(n) for n = 0..10010
Peter Luschny, Rational Trees and Binary Partitions.

Cf. A002061, A002487, A002522, A002523, A059826, A071568, A101624, A101625.

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);

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 *)

def A260534_T(n,k):
    return sum(0 if ~(k-j) & j else n**j for j in range(k+1)) # Chai Wah Wu, Feb 08 2016

A284247 Binary representation of generation n in the reversible cellular automaton RCA(1) when started with a single ON cell at generation 0.

Original entry on oeis.org

1, 1, 101, 1, 10101, 10001, 1000101, 1, 100010101, 100010001, 10101000101, 100000001, 1010000010101, 1000000010001, 100000001000101, 1, 10000000100010101, 10000000100010001, 1010000010101000101, 10000000100000001, 101010001010000010101, 100010001000000010001

Offset: 0

Robert Price, Mar 23 2017

nonn

This sequence is the binary representation of A101625.

The count of ON cells at each generation is A002487.

			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.

Robert Price, Table of n, a(n) for n = 0..999
Alan J. Macfarlane, Linear reversible second-order cellular automata and their first-order matrix equivalents, Journal of Physics A: Mathematical and General 37.45 (2004): 10791. See Fig. 3.
Robert Price, Diagram of the first 35 generations
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A002487, A101625.

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A101624 Stern-Jacobsthal numbers.

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A166555 Triangle read by rows, T(n, k) = 2^k * A047999(n, k).

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A204771 a(n) = a(n-1) XOR (a(n-2)*3).

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A260534 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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A284247 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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