A060037 -id:A060037 - cp's OEIS frontend

A060036 Triangular array T read by rows: T(n,k) = k^2 mod n, for k = 1,2,...,n-1, n = 2,3,...

1, 1, 1, 1, 0, 1, 1, 4, 4, 1, 1, 4, 3, 4, 1, 1, 4, 2, 2, 4, 1, 1, 4, 1, 0, 1, 4, 1, 1, 4, 0, 7, 7, 0, 4, 1, 1, 4, 9, 6, 5, 6, 9, 4, 1, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, 1, 4, 9, 2, 11, 8, 7, 8, 11, 2, 9, 4, 1, 1, 4, 9, 1, 10, 6, 4, 4, 6, 10

Offset: 2

N. J. A. Sloane, Mar 17 2001

T(n,k) = A048152(n-1,k), 1 <= k < n; T(2*n-1,n-1) = A123684(n-1) = A225126(n-1). - Reinhard Zumkeller, Apr 29 2013

			The triangle T(n,k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 ...
-------------------------------
2:  1
3:  1 1
4:  1 0 1
5:  1 4 4 1
6:  1 4 3 4 1
7:  1 4 2 2 4 1
6:  1 4 1 0 1 4 1
9:  1 4 0 7 7 0 4 1
10: 1 4 9 6 5 6 9 4 1
11: 1 4 9 5 3 3 5 9 4  1
12: 1 4 9 4 1 0 1 4 9  4  1
...  reformatted by - _Wolfdieter Lang_, Dec 17 2018

T. D. Noe, Rows n = 2..100 of triangle, flattened

Cf. A048152, A060037.

Cf. A048153 (row sums).

a060036 n k = a060036_tabl !! (n-2) !! (k-1)
a060036_row n = a060036_tabl !! (n-2)
a060036_tabl = map init $ tail a048152_tabl
-- Reinhard Zumkeller, Apr 29 2013

Flatten[Table[PowerMod[k,2,n],{n,2,20},{k,n-1}]] (* Harvey P. Dale, Feb 27 2012 *)

{ n=1; for (m=2, 46, for (k=1, m-1, write("b060036.txt", n++, " ", k^2 % m)); ) } \\ Harry J. Smith, Jul 01 2009

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2001

A030068 The "semi-Fibonacci numbers": a(n) = A030067(2n - 1), where A030067 is the semi-Fibonacci sequence.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 16, 17, 23, 26, 35, 37, 48, 53, 69, 70, 87, 93, 116, 119, 145, 154, 189, 191, 228, 239, 287, 292, 345, 361, 430, 431, 501, 518, 605, 611, 704, 727, 843, 846, 965, 991, 1136, 1145, 1299, 1334, 1523, 1525, 1716, 1753, 1981, 1992, 2231, 2279, 2566

Offset: 1

David W. Wilson

Also the unique values of A030067 sorted. - Ralf Stephan, Oct 28 2013
Also, the subsequence of record values of the semi-Fibonacci sequence A030067.
The first differences of this sequence give back A030067. - It is more natural to use offset 1 and a(n) = A060037(2n-1), rather than 0 and a(n) = A060037(2n+1): First, a set should have this offset, and this is indeed the set of values or the range of A030067, i.e., the set of semi-Fibonacci numbers. Second, A060037 also starts at index 1. Third, the sequence A284282(n) = (k such that A030067(2k-1)=n or 0 if there's no such k) is then the characteristic function of this sequence, with nonzero values read as 1. - M. F. Hasler, Mar 24 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..10001

Cf. A030067. Bisections: A169739, A169740.

f[1] = 1; f[n_?EvenQ] := f[n] = f[n/2]; f[n_?OddQ] := f[n] = f[n-1] + f[n-2]; a[n_] := f[2*n+1]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jul 16 2015 *)

A030068_vec=[1,2,3]; A030068(n)={n>#A030068_vec&&for(n=#A030068_vec,-1+#A030068_vec=concat(A030068_vec,vector(n-#A030068_vec)),A030068_vec[n+1]=A030068_vec[n]+A030067(n));A030068_vec[n]} \\ M. F. Hasler, Mar 24 2017

Vec(prod(k=0,5,1/Ser(x^2^k)+x^2^k)) \\ Correct for n < 2*2^5. - M. F. Hasler, Mar 27 2017

G.f.: x*(r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) is (1 + 2x + x^2 + x^3 + x^4 + x^5 + ...). - Gary W. Adamson, Sep 02 2016

a(n+1) = a(n) + A060037(n). The above g.f. can be written as x*Product_{k=0,oo} (1/(1-x^2^k)+x^2^k). - M. F. Hasler, Mar 27 2017

Offset changed to 1 by N. J. A. Sloane, Mar 27 2017

cp's OEIS Frontend

A060036 Triangular array T read by rows: T(n,k) = k^2 mod n, for k = 1,2,...,n-1, n = 2,3,...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A030068 The "semi-Fibonacci numbers": a(n) = A030067(2n - 1), where A030067 is the semi-Fibonacci sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions