cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A099901 Shifts left and divides by 2 under the XOR BINOMIAL transform (A099902).

Original entry on oeis.org

1, 2, 6, 14, 22, 46, 118, 206, 278, 558, 1654, 3790, 5910, 11310, 28790, 49358, 65814, 131630, 394870, 921294, 1447702, 3025966, 7762038, 13549774, 18284822, 36438574, 108004982, 247467726, 385881878, 738208814, 1879076982, 3221274830
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2004

Keywords

Comments

Equals the XOR BINOMIAL transform of A099902. Also, equals the leftmost column of the XOR difference triangle A099900, in which the central terms of the rows forms the powers of 2.

Crossrefs

Cf. A099884, A099900, A099902.

Programs

  • PARI
    {a(n)=local(B);B=0;if(n==0,B=1, for(k=0,n-1, B=bitxor(B,binomial(n-1-k+k\2,k\2)%2*2^(k+1))));B}

Formula

a(0)=1; a(n) = SumXOR_{k=0..n-1} (C(n-1-k+[k/2], [k/2])mod 2)*2^(k+1) for n>0. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A099902(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A101624 Stern-Jacobsthal numbers.

Original entry on oeis.org

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

Views

Author

Paul Barry, Dec 10 2004

Keywords

Comments

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

Links

Crossrefs

Cf. A002487, A011973, A000079, A006921.
Cf. A125184, A260443, A177219.

Programs

  • Haskell
    a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
-- Reinhard Zumkeller, Jul 14 2015
  • 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
# Alex Ratushnyak, Apr 14 2012
  • 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

Formula

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

A099900 XOR difference triangle, read by rows, of A099901 (in leftmost column) such that the main diagonal equals A099901 shift left and divided by 2.

Original entry on oeis.org

1, 2, 3, 6, 4, 7, 14, 8, 12, 11, 22, 24, 16, 28, 23, 46, 56, 32, 48, 44, 59, 118, 88, 96, 64, 112, 92, 103, 206, 184, 224, 128, 192, 176, 236, 139, 278, 472, 352, 384, 256, 448, 368, 412, 279, 558, 824, 736, 896, 512, 768, 704, 944, 556, 827, 1654, 1112, 1888, 1408
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2004

Keywords

Comments

Central terms of rows equal powers of 2: T(n,[n/2]) = 2^n for n>=0. The leftmost column is A099901. The diagonal forms A099902 and equals the XOR BINOMIAL transform of A099901.

Examples

			Rows begin:
[_1],
[_2,3],
[6,_4,7],
[14,_8,12,11],
[22,24,_16,28,23],
[46,56,_32,48,44,59],
[118,88,96,_64,112,92,103],
[206,184,224,_128,192,176,236,139],
[278,472,352,384,_256,448,368,412,279],
[558,824,736,896,_512,768,704,944,556,827],
[1654,1112,1888,1408,1536,_1024,1792,1472,1648,1116,1895],...
notice that the column terms equal twice the diagonal (with offset), and that the central terms in the rows form the powers of 2.

Crossrefs

Cf. A099884, A099901, A099902.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n, [n/2]) = 2^n. T(n+1, 0) = 2*T(n, n) (n>=0); T(0, 0)=1; T(n, k) = T(n, k-1) XOR T(n-1, k-1) for n>k>0. T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*T(n-i, 0), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A101680 A modular binomial transform of 10^n.

Original entry on oeis.org

1, 11, 111, 1011, 10111, 111011, 1100111, 10001011, 100010111, 1100111011, 11101100111, 101110001011, 1011000010111, 11100000111011, 110000001100111, 1000000010001011, 10000000100010111, 110000001100111011, 1110000011101100111, 10110000101110001011, 101110001011000010111
Offset: 0

Views

Author

Paul Barry, Dec 11 2004

Keywords

Crossrefs

A099902 in binary. Bisection of A101623.

Programs

  • PARI
    a(n) = sum(k=0, n, lift(Mod(binomial(2*n-k, k), 2))*10^k); \\ Michel Marcus, Jul 31 2025
  • Python
    def A101680(n): return sum(int(not ~((n<<1)-k)&k)*10**k for k in range(n+1)) # Chai Wah Wu, Jul 30 2025

Formula

a(n) = Sum{k=0..n} mod(binomial(2n-k, k), 2)*10^k<;

Extensions

More terms from Michel Marcus, Jul 31 2025
Showing 1-4 of 4 results.

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