L. Van Warren - cp's OEIS frontend

A262710 Powers of -4.

1, -4, 16, -64, 256, -1024, 4096, -16384, 65536, -262144, 1048576, -4194304, 16777216, -67108864, 268435456, -1073741824, 4294967296, -17179869184, 68719476736, -274877906944, 1099511627776, -4398046511104, 17592186044416, -70368744177664, 281474976710656

Offset: 0

L. Van Warren, Sep 28 2015

Purely real values from the sequence generated by (1 + i)^k where i = sqrt(-1) and k is a real nonnegative integer.

This sequence gives the values of (1 + i)^k when k is a multiple of 4. When k = 2 mod 4, (1 + i)^k is purely imaginary, and when k is odd, (1 + i)^k has both a real and an imaginary part, and abs(Re((1 + i)^k)) = abs(Im((1 + i)^k)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Caroline Nunn, A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021).
Index entries for linear recurrences with constant coefficients, signature (-4).

Cf. A000302, A016825, A005408, A122803.

[(-1)^n*4^n: n in [0..30]]; // Vincenzo Librandi, Oct 06 2015

(-4)^Range[0, 15] (* Alonso del Arte, Mar 16 2016 *)

vector(100, n, n--; (-4)^n) \\ Altug Alkan, Oct 05 2015

Vec(1/(1+4*x) + O(x^30)) \\ Michel Marcus, Oct 06 2015

lista(nn) = for (n=0, nn, z = (1+I)^n; if (imag(z)==0, print1(real(z), ", "))); \\ Michel Marcus, Nov 01 2015

a(n) = (-4)^n.
G.f.: 1/(1 + 4 * x).
E.g.f.: exp(-4*x). - Alejandro J. Becerra Jr., Jan 28 2021
a(n) = Sum_{k=0..2*n} (-1)^k*binomial(4*n, 2*k) (see Nunn, p. 9). - Stefano Spezia, Dec 28 2021

A124264 Hilbert-Warren Add Sequence.

Original entry on oeis.org

2, 5, 20, 51, 312, 740, 15000, 34125, 160720, 4988592, 180774720, 844074000, 27991656000, 66264953040

Offset: 1

L. Van Warren (van(AT)wdv.com), Oct 23 2006

Variant of A124261.

Cf. A124261.

/* [wxMaxima: input start ] */ load("eigen"); order : 4; X : columnvector(makelist(concat(x,i), i, 1, order)); h[i,j] := (i + j)/(i + j - 1); Unity[i,j] := 1; A : genmatrix(h, order, order); A . X; B : genmatrix(Unity, 1, order); A . X = B; Ap : triangularize(A); Ap . X = B; App : invert(Ap); App . B; /* [wxMaxima: input end ] */

Terms generated by solving n x n linear system [H']x = b where b is unity and elements of H' are given by H'[i,j] = (i + j)/(i + j - 1). Sequence term is the denominator of last unknown xn. Numerator is unity for last term for all problem orders examined.

The author said that this version of the submission was unsatisfactory and set me a better version in email, which may replace this. - N. J. A. Sloane, Oct 24 2006

A124265 Variant sequence generated by solving the order n x n linear problem [H]x = b where b is the unit vector and the sequence term is given by the denominator of the last unknown xn.

Original entry on oeis.org

3, 3, 11, 27, 162, 380, 7650, 17325, 81340, 2518992, 91128240, 424947600, 14078156400, 33300661680, 424624548348

Offset: 1

L. Van Warren (van(AT)wdv.com), Oct 23 2006

HilbertWarren(fun, order) := ( Unity[i,j] := 1, A : genmatrix(fun, order, order), B : genmatrix(Unity, 1, order), App : invert(triangularize(A)), Xp : App . B, 1/Xp[order] ); findWarrenSequenceTerms(fun, a, b) := ( L : append(), for order: a next order+1 through b do L: cons(first(HilbertWarren(fun,order)), L), S : reverse(L) ); k : 15; hilbert[i,j] := 1/(i + j - 1); findWarrenSequenceTerms(hilbert, 1, k); hilbertA0[i,j] := (i + j + 0)/(i + j - 1); /* sum 1 */ findWarrenSequenceTerms(hilbertA0, 1, k); hilbertA1[i,j] := (i + j + 1)/(i + j - 1); /* sum 2: there are lots of these, increment numerator */ findWarrenSequenceTerms(hilbertA1, 1, k); hilbertD1[i,j] := (i - j + 1)/(i + j - 1); /* difference 1 */ findWarrenSequenceTerms(hilbertD1, 1, k); hilbertP1[i,j] := (i * j + 0)/(i + j - 1); /* product 1 */ findWarrenSequenceTerms(hilbertP1, 1, k); hilbertQ1[i,j] := (i / j)/(i + j - 1); /* quotient 1 */ findWarrenSequenceTerms(hilbertQ1, 1, k);

[H] is defined by hilbertWarrenA1[i,j]:=(1+j+i)/(-1+j+i) where numbering starts at 1.

A124261 Sequence terms are generated by solving the n x n linear algebra problem [H]x = b, where b is the unit vector. Only xn, the last unknown is used.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 300, 525, 1960, 49392, 1481760, 5821200, 164656800, 336370320, 3741185448

Offset: 1

L. Van Warren (van(AT)wdv.com), Oct 23 2006

These matrices are notorious for being ill-conditioned, they are solved using rational arithmetic. To reproduce these numbers, or larger ones, one can edit the program above with a text editor and then drag it into the Maxima window. The last term required approx. 10 minutes to generate on an Intel dual-core 6600 clocked at 2.4 GHz with 2 Gig of RAM. To change the order of the Hilbert matrix, just change the order variable at the top of the file. It is currently 4, which runs quickly. So a point of curiosity really. The numbers become extremely expensive to find, growing at least as the cube of n times a large constant. So to me they are a kind of symbolic gold. The 100th number for example, is probably not knowable at the current time, but that is speculation on my part. Perhaps you will observe some functional relationship that allows their simple generation thus "Cracking the Hilbert-Warren Sequence Code". For example 12 = 4 * 4 - 4, 20 = 5 * 5 - 5, 300 = 20 * 20 - 100, 525 = 25 * 25 - 100 and so on.

			Triangularization of [H] is performed, followed by inversion or back substitution. The terms are the denominators of the last solution xn to the Hilbert matrices of order 1, 2, ... 15, respectively. The numerator is unity in all cases.
For order 3 problem the solutions are: ([1/6],[ -1/3],[1/2]), thus a() = 2, since we just take the denominator.

William Schelter, DOE Maxima - A Symbolic Algebra Program
Gilbert Strang, Linear Algebra and Its Applications
L. V. Warren, Cracking the Hilbert-Warren Code

Cf. A124264.

load("eigen"); order : 4; X : columnvector(makelist(concat(x,i), i, 1, order)); h[i,j] := 1/(i + j -1); Unity[i,j] := 1; A : genmatrix(h, order, order); A . X; B : genmatrix(Unity, 1, order); A . X = B; Ap : triangularize(A); Ap . X = B; App : invert(Ap); App . B;

Solve the n x n linear algebra problem [H]x = b, where b is the unit vector and n is the order. Only xn, the last unknown is used.

The author said that this version of the submission was unsatisfactory and set me a better version in email, which may replace this. - N. J. A. Sloane, Oct 24 2006

A124266 Variant sequence generated by solving the order n X n linear problem [H]x = b where b is the unit vector and the sequence term is given by the denominator of the last unknown xn.

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 150, 525, 980, 24696, 740880, 2910600, 82328400, 168185160, 1870592724

Offset: 1

L. Van Warren (van(AT)wdv.com), Oct 23 2006

HilbertWarren(fun, order) := ( Unity[i,j] := 1, A : genmatrix(fun, order, order), B : genmatrix(Unity, 1, order), App : invert(triangularize(A)), Xp : App . B, 1/Xp[order] ); findWarrenSequenceTerms(fun, a, b) := ( L : append(), for order: a next order+1 thru b do L: cons(first(HilbertWarren(fun,order)), L), S : reverse(L) ); k : 15; hilbert[i,j] := 1/(i + j - 1); findWarrenSequenceTerms(hilbert, 1, k); hilbertA0[i,j] := (i + j + 0)/(i + j - 1); /* sum 1 */ findWarrenSequenceTerms(hilbertA0, 1, k); hilbertA1[i,j] := (i + j + 1)/(i + j - 1); /* sum 2: there are lots of these, increment numerator */ findWarrenSequenceTerms(hilbertA1, 1, k); hilbertD1[i,j] := (i - j + 1)/(i + j - 1); /* difference 1 */ findWarrenSequenceTerms(hilbertD1, 1, k); hilbertP1[i,j] := (i * j + 0)/(i + j - 1); /* product 1 */ findWarrenSequenceTerms(hilbertP1, 1, k); hilbertQ1[i,j] := (i / j)/(i + j - 1); /* quotient 1 */ findWarrenSequenceTerms(hilbertQ1, 1, k); /* Program does not display the correct sequence. Robert C. Lyons, Jul 30 2025 */

[H] is defined by hilbertWarrenA1[i,j]:=(1-j+i)/(-1+j+i) where numbering starts at 1.

cp's OEIS Frontend

User: L. Van Warren

A262710 Powers of -4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A124264 Hilbert-Warren Add Sequence.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maxima

Formula

Extensions

A124265 Variant sequence generated by solving the order n x n linear problem [H]x = b where b is the unit vector and the sequence term is given by the denominator of the last unknown xn.

Views

Author

Keywords

Programs

Maxima

Formula

A124261 Sequence terms are generated by solving the n x n linear algebra problem [H]x = b, where b is the unit vector. Only xn, the last unknown is used.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

Formula

Extensions

A124266 Variant sequence generated by solving the order n X n linear problem [H]x = b where b is the unit vector and the sequence term is given by the denominator of the last unknown xn.

Views

Author

Keywords

Programs

Maxima

Formula