1,1

R. J. Mathar, Table of n, a(n) for n = 1..1000

Index entries for related sequences

Sequence in context: A072935 A049564 A072936 this_sequence A045382 A049560 A049588

Adjacent sequences: A049581 A049582 A049583 this_sequence A049585 A049586 A049587

nonn

N. J. A. Sloane (njas(AT)research.att.com).

0,7

G=product(1-x^(2^j), j=1..infinity)/product(1-x^i, i=2..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2006

a(12) = #{3+3+3+3, 6+3+3, 6+6, 7+5, 9+3, 12} = 6.

g:=product(1-x^(2^j), j=0..15)/product(1-x^i, i=1..75): gser:=series(g, x=0, 62): seq(coeff(gser, x, n), n=0..59); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2006

Cf. A000041, A018819, A000123.

Sequence in context: A127838 A017817 A053268 this_sequence A035636 A104554 A152414

Adjacent sequences: A101414 A101415 A101416 this_sequence A101418 A101419 A101420

nonn

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 16 2005

0,2

Diagonal sums of number triangle A110330.

G.f.: (1-2x-4x^2+2x^3+x^4)/((1+x)^3(1-x)^3); a(n)=3a(n-2)-3a(n-4)+a(n-6); a(n)=-n(n+6)/8-(n^2-2n-8)(-1)^n/8.

Sequence in context: A144774 A074720 A058359 this_sequence A159287 A052947 A021992

Adjacent sequences: A110329 A110330 A110331 this_sequence A110333 A110334 A110335

easy,sign

Paul Barry (pbarry(AT)wit.ie), Jul 20 2005

1,1

Sequence in context: A074072 A037115 A139921 this_sequence A023325 A094200 A003705

Adjacent sequences: A142506 A142507 A142508 this_sequence A142510 A142511 A142512

nonn

N. J. A. Sloane (njas(AT)research.att.com), Jul 11 2008

1,1

Coincides with the sequence of "primes p such that x^16 = 2 has a solution mod p" for first 58 terms (and then diverges).

A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und -2, Deutsche Math. 4 (1939), 44-52; FdM 65 - I (1939), 112.

H. Hasse, Der 2^n-te Potenzcharakter von 2 im Koerper der 2^n-ten Einheitswurzeln, Rend. Circ. Matem. Palermo (2), 7 (1958), 185-243.

A. L. Whiteman, The sixteenth power residue character of 2, Canad. J. Math. 6 (1954), 364-373; Zbl 55.27102

T. D. Noe, Table of n, a(n) for n=1..1000

Author?, More information

Index entries for related sequences

Cf. A001132, A040028, A040098, A045315.

Sequence in context: A045381 A042145 A040098 this_sequence A072935 A049564 A072936

Adjacent sequences: A045312 A045313 A045314 this_sequence A045316 A045317 A045318

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

0,2

At the same time that I introduced the polynomials P(n,x) defined by P(0,x)=1 and for n>0, P(n,x)=((-1)^n)/(n+1) + x*Sum_{ i=0..n-1 } [(((-1)^i)/(i+1))*P(n-1-i,x)] (Gazette des Mathematiciens 1992), I gave the generalization P(0,x)=u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } u(i)*P(n-1-i,x).

For u(n), n>=0, = 1 1 1 2 3 4 5 6 7 8 ... the array of coefficients of the polynomials P(n,x) is:

1

1 1

1 2 1

2 3 3 1

3 6 6 4 1

4 11 13 10 5 1

5 18 27 24 15 6 1

6 28 51 55 40 21 7 1

whose row sums are the present sequence.

The alternating row sums are are 1 0 0 1 0 0 0 -1 ...

The antidiagonal sums are : 1 1 2 4 7 13 23 41 73 ...

The first column of the inverse matrix is 1 -1 1 -2 5 -11 25 -63 ...

P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

G.f.: -(x^3-x+1)/(x^4-2*x^2+3*x-1). [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009]

a:= n-> (Matrix([1, 1, 0, 1]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..50); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009]

Sums of coefficients of polynomials defined in A140530.

Cf. A129841, A129696, A130620.

Sequence in context: A123720 A034007 A109975 this_sequence A130587 A129988 A035530

Adjacent sequences: A129888 A129889 A129890 this_sequence A129892 A129893 A129894

nonn

Paul Curtz (bpcrtz(AT)free.fr), Jun 04 2007

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 05 2007

More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009

0,5

This means the description of A038200 is slightly incorrect and ought be: "Row sums of triangle K(m,n), inverse to a triangle obtained from A020921 after eliminating the leftmost column."

G.f.: (1/(1+x))*Sum(x^(k-1)/((1+x)^k-y*x^k),k=1..infinity). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 26 2008

If the leftmost column of the triangle in A020921 is deleted we get

1

1 1

2 3 1

2 5 4 1

4 10 10 5 1

2 11 19 15 6 1

6 21 35 35 21 7 1

4 22 52 69 56 28 8 1

6 33 83 126 126 84 36 9 1

The present triangle is the inverse of this, namely

1

-1 1

1 -3 1

-1 7 -4 1

1 -15 10 -5 1

-1 31 -19 15 -6 1

1 -63 28 -35 21 -7 1

-1 127 -28 71 -56 28 -8 1

with row sums 1,0,-1,3,-8,21,-54,134,-318,720 of A038200.

A020921 := proc(n, k) option remember; local divs; if n <= 0 then 1; elif k > n then 0; else divs := numtheory[divisors](n); add(numtheory[mobius](op(i, divs))*binomial(n/op(i, divs), k), i=1..nops(divs)); fi; end: A020921t := proc(n, k) option remember; A020921(n+1, k+1); end: TriLInv := proc(nmax) local a, row, col; a := array(0..nmax, 0..nmax); for row from 0 to nmax do for col from row+1 to nmax do a[row, col] := 0; od; od; for row from 0 to nmax do for col from row to 0 by -1 do if row <> col then a[row, col] := -add(a[row, c]*A020921t(c, col), c=col+1..row)/A020921t(col, col); else a[row, col] := (1-add(a[row, c]*A020921t(c, col), c=col+1..row))/A020921t(col, col); fi; od; od; RETURN(a); end: nmax := 12 : a := TriLInv(nmax) : for row from 0 to nmax do for col from 0 to row do printf("%d, ", a[row, col]); od; od:

Cf. A039912.

Sequence in context: A116407 A135288 A078026 this_sequence A140068 A121300 A128119

Adjacent sequences: A126710 A126711 A126712 this_sequence A126714 A126715 A126716

sign,tabl

R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 12 2007

1,1

If n = 1, then

r(r(r(r(r(r(r(1)+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(0+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(0+1)+1)+1)+1)+1)+1 = r(r(r(r(r(1)+1)+1)+1)+1)+1 = r(r(r(r(0+1)+1)+1)+1)+1 = r(r(r(r(1)+1)+1)+1)+1 = r(r(r(0+1)+1)+1)+1 = r(r(r(1)+1)+1)+1 = r(r(0+1)+1)+1 = r(r(1)+1)+1 = r(0+1)+1 = r(1)+1 = 0+1 = 1(nonprime).

If n = 2, then

r(r(r(r(r(r(r(2)+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(1+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(2)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(1+1)+1)+1)+1)+1)+1 = r(r(r(r(r(2)+1)+1)+1)+1)+1 = r(r(r(r(1+1)+1)+1)+1)+1 = r(r(r(r(2)+1)+1)+1)+1 = r(r(r(1+1)+1)+1)+1 = r(r(r(2)+1)+1)+1 = r(r(1+1)+1)+1 = r(r(2)+1)+1 = r(1+1)+1 = r(2)+1 = 1+1 = 2 = a(1).

If n = 3, then

r(r(r(r(r(r(r(3)+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(4+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(5)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(8+1)+1)+1)+1)+1)+1 = r(r(r(r(r(9)+1)+1)+1)+1)+1 = r(r(r(r(14+1)+1)+1)+1)+1 = r(r(r(r(15)+1)+1)+1)+1 = r(r(r(22+1)+1)+1)+1 = r(r(r(23)+1)+1)+1 = r(r(33+1)+1)+1 = r(r(34)+1)+1 = r(48+1)+1 = r(49)+1 = 66+1 = 67 = a(2).

If n = 4, then

r(r(r(r(r(r(r(4)+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(6+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(7)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(10+1)+1)+1)+1)+1)+1 = r(r(r(r(r(11)+1)+1)+1)+1)+1 = r(r(r(r(16+1)+1)+1)+1)+1 = r(r(r(r(17)+1)+1)+1)+1 = r(r(r(25+1)+1)+1)+1 = r(r(r(26)+1)+1)+1 = r(r(36+1)+1)+1 = r(r(37)+1)+1 = r(51+1)+1 = r(52)+1 = 70+1 = 71 = a(3).

If n = 5, then

r(r(r(r(r(r(r(5)+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(8+1)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(r(9)+1)+1)+1)+1)+1)+1 = r(r(r(r(r(14+1)+1)+1)+1)+1)+1 = r(r(r(r(r(15)+1)+1)+1)+1)+1 = r(r(r(r(22+1)+1)+1)+1)+1 = r(r(r(r(23)+1)+1)+1)+1 = r(r(r(33+1)+1)+1)+1 = r(r(r(34)+1)+1)+1 = r(r(48+1)+1)+1 = r(r(49)+1)+1 = r(66+1)+1 = r(67)+1 = 90+1 = 91(nonprime),

etc.

Cf. A000040, A141468.

Sequence in context: A016535 A139864 A107993 this_sequence A139861 A065721 A030472

Adjacent sequences: A131601 A131602 A131603 this_sequence A131605 A131606 A131607

nonn

Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Aug 25 2008

211 removed, 355 replaced by 359 and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 05 2008

1,1

Row sums are:{-2, -2, 0, 0, 7, 52, 162, 546, 1730, 5291}.

Reasoning behind the sequence is:

Suppose we have n affine transforms that form a group:

g={ a(1)*x+b(1),a(2)*x+b(2),...,a(n)*x+b(n)}

on the sequences a(n) and b(n).

We form rational projections as Moebius / bilinear transforms:

g(projection)={( a(1)*x+b(1))/(a(n)*x+b(n)),( a(2)*x+b(2))/(a(n)*x+b(n)),...,( a(n-1)*x+b(n-1))/(a(n)*x+b(n))

With determinants:

g_det={a(1)*b(n)-b(1)*a(n),a(2)*b(n)-b(2)*a(n),...,a(n-1)*b(n)-b(n-1)*a(n)}

So that we have the triangular sequences:

t(n,m)=a(m)*b(n)-b(m)*a(n)

a(n)=A000045(n); b(n)=b(n-1)+b(n-2)+b(n-3) :2 start;A141036; t(n,m)=t(n,m)=a(m)*b(n)-b(m)*a(n).

{-2},

{-2, 0},

{-4, 2, 2},

{-6, 3, 3, 0},

{-10, 6, 6, 2, 3},

{-16, 13, 13, 10, 15, 17},

{-26, 25, 25, 24, 36, 47, 31},

{-42, 49, 49, 56, 84, 119, 119, 112},

{-68, 95, 95, 122, 183, 271, 318, 385, 329},

{-110, 182, 182, 254, 381, 580, 741, 991, 1127, 963}

Clear[a, b, t, n, m] a[n_] := Fibonacci[n]; b[0] = 2; b[1] = 1; b[2] = 1; b[n_] := b[n] = b[n - 1] + b[n - 2] + b[n - 3]; t[n_, m_] := a[m]*b[n] - b[m]*a[n]; Table[Table[t[n, m], {m, 0, n - 1}], {n, 1, 10}]; Flatten[%]

Cf. A141036, A000045.

Sequence in context: A138094 A060821 A005881 this_sequence A098268 A128585 A141333

Adjacent sequences: A144455 A144456 A144457 this_sequence A144459 A144460 A144461

uned,sign

Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 07 2008

0,1

"Version: basek" in the name field is a reference to the floretion k'. It means that in order to calculate a(n), the rule given for A108618: "a(n) is given by twice the coefficient of e (the unit) in y from step 4 inside of the n-th loop." should be replaced by "a(n) is given by 4 times the coefficient of k' in y from step 4 inside of the n-th loop." This sequence appears to be unbounded. Moreover, (a(n)) produces a "spiral" when plotted against sequences from the same batch (i.e. against versions: tes, ves etc.). Ray-traced plots similar to the one given in the link can be formed using this sequence (for example). (a(n)) appears to become more "predictable" with increasing n. For n in the range from 987 to 1000 we have: a(987) = -59, a(988) = 112, a(989) = -52, a(990) = -55, a(991) = 108, a(992) = -51, a(993) = -53, a(994) = 109, a(995) = -55, a(996) = -51, a(997) = 107, a(998) = -54, a(999) = -50, a(1000) = 109

C. Dement, Floretion Online Multiplier. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 10 2009]

Floretion Algebra Multiplication Program, FAMP Code: 4baseksumseq[(+ .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e)(+ .5'i + .5j' + .5'ij' + .5e)] Sumtype is set to: sum[Y[15]] = sum[ * ]

Cf. A108618.

Sequence in context: A099475 A120569 A128113 this_sequence A059682 A156548 A112883

Adjacent sequences: A108927 A108928 A108929 this_sequence A108931 A108932 A108933

easy,sign

Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jul 26 2005