A006495 -id:A006495 - cp's OEIS frontend

A116484 Expansion of (-1+3x)/(5x^2 + 1 - 2*x).

-1, 1, 7, 9, -17, -79, -73, 249, 863, 481, -3353, -9111, -1457, 42641, 92567, -28071, -518977, -897599, 799687, 6087369, 8176303, -14084239, -69049993, -67678791, 209892383, 758178721, 466895527, -2857102551, -8048682737, -1811852719

Offset: 0

Creighton Dement, Feb 17 2006

Binomial transform of signed powers of 2: (-1, 2, 4, -8, -16, 32, 64, -128, -256, 512, 1024). Inverse binomial transform of (-1, 0, 8, 32, 64, 0, -512, -2048, -4096, 0, 32768, 131072, 262144, 0, -2097152, -8388608). Compare with A116483.

Floretion Algebra Multiplication Program, FAMP Code: 2basekforseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' ; 1vesforseq = A000004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A000079, A006495, A116483.

CoefficientList[Series[(-1+3x)/(5x^2+1-2x),{x,0,40}],x] (* or *) LinearRecurrence[{2,-5},{-1,1},40] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 3*A045873(n) - A045873(n+1). - R. J. Mathar, Apr 23 2009
E.g.f.: exp(x)*(sin(2*x) - cos(2*x)). - Arkadiusz Wesolowski, Aug 31 2012
a(0)=-1, a(1)=1, a(n) = 2*a(n-1) - 5*a(n-2). - Harvey P. Dale, Jun 24 2013
a(n) = (1/2)*((-1 - i)*(1 + 2*i)^n - (1 - i)*(1 - 2*i)^n), n >= 0, where i=sqrt(-1). - Taras Goy, Apr 20 2019

A117411 Skew triangle associated to the Euler numbers.

Original entry on oeis.org

1, 0, 1, 0, -4, 1, 0, 0, -12, 1, 0, 0, 16, -24, 1, 0, 0, 0, 80, -40, 1, 0, 0, 0, -64, 240, -60, 1, 0, 0, 0, 0, -448, 560, -84, 1, 0, 0, 0, 0, 256, -1792, 1120, -112, 1, 0, 0, 0, 0, 0, 2304, -5376, 2016, -144, 1, 0, 0, 0, 0, 0, -1024, 11520, -13440, 3360, -180, 1, 0, 0, 0, 0, 0, 0, -11264, 42240, -29568, 5280, -220, 1

Offset: 0

Paul Barry, Mar 13 2006

Inverse is A117414. Row sums of the inverse are the Euler numbers A000364.

Triangle, read by rows, given by [0,-4,4,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2009

			Triangle begins
  1;
  0,  1;
  0, -4,   1;
  0,  0, -12,   1;
  0,  0,  16, -24,    1;
  0,  0,   0,  80,  -40,     1;
  0,  0,   0, -64,  240,   -60,      1;
  0,  0,   0,   0, -448,   560,    -84,      1;
  0,  0,   0,   0,  256, -1792,   1120,   -112,      1;
  0,  0,   0,   0,    0,  2304,  -5376,   2016,   -144,      1;
  0,  0,   0,   0,    0, -1024,  11520, -13440,   3360,   -180,    1;
  0,  0,   0,   0,    0,     0, -11264,  42240, -29568,   5280, -220,    1;
  0,  0,   0,   0,    0,     0,   4096, -67584, 126720, -59136, 7920, -264, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000364, A006495 (row sums), A098158, A117413, A117414.

Cf. A000217, A000332, A000579, A000581, A262710.

A117411:= func< n,k | (-4)^(n-k)*(&+[Binomial(n,k-j)*Binomial(j,n-k): j in [0..n-k]]) >;
[A117411(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 07 2022

T[n_,k_]:= T[n,k]= (-4)^(n-k)*Sum[Binomial[n, k-j]*Binomial[j, n-k], {j,0,n-k}];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 07 2022 *)

def A117411(n,k): return (-4)^(n-k)*sum(binomial(n,k-j)*binomial(j,n-k) for j in (0..n-k))
flatten([[A117411(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 07 2022

Sum_{k=0..n} T(n, k) = A006495(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117413(n).
T(n, k) = (-4)^(n-k)*Sum_{j=0..n-k} C(n,k-j)*C(j,n-k).
G.f.: (1-x*y)/(1-2x*y+x^2*y(y+4)). - Paul Barry, Mar 14 2006
T(n, k) = (-4)^(n-k)*A098158(n,k). - Philippe Deléham, Nov 01 2009
T(n, k) = 2*T(n-1,k-1) - 4*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 31 2013
From G. C. Greubel, Sep 07 2022: (Start)
T(n, n) = 1.
T(n, n-1) = -4*A000217(n-1), n >= 1.
T(n, n-2) = (-4)^2 * A000332(n), n >= 2.
T(n, n-3) = (-4)^3 * A000579(n), n >= 3.
T(n, n-4) = (-4)^4 * A000581(n), n >= 4.
T(2*n, n) = A262710(n). (End)

A117435 Triangle related to exp(x)cos(2x).

Original entry on oeis.org

1, 0, 1, -4, 0, 1, 0, -12, 0, 1, 16, 0, -24, 0, 1, 0, 80, 0, -40, 0, 1, -64, 0, 240, 0, -60, 0, 1, 0, -448, 0, 560, 0, -84, 0, 1, 256, 0, -1792, 0, 1120, 0, -112, 0, 1, 0, 2304, 0, -5376, 0, 2016, 0, -144, 0, 1, -1024, 0, 11520, 0, -13440, 0, 3360, 0, -180, 0, 1

Offset: 0

Paul Barry, Mar 16 2006

Diagonals correspond to rows of A117438.

			Triangle begins:
    1;
    0,   1;
   -4,   0,   1;
    0, -12,   0,   1;
   16,   0, -24,   0,   1;
    0,  80,   0, -40,   0, 1;
  -64,   0, 240,   0, -60, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A006495 (row sums), A117411, A117436 (inverse), A117438.

T[n_,k_]:= Binomial[n,k]*(2*I)^(n-k)*(1+(-1)^(n+k))/2;
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)

flatten([[binomial(n,k)*(2*i)^(n-k)*(1+(-1)^(n+k))/2 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Number triangle whose k-th column has e.g.f. (x^k/k!)*cos(2x);
T(n, k) = binomial(n,k) * (-4)^((n-k)/2) * (1+(-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A006495(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = i^n * ((1+(-1)^n)/2) * (2*floor(n/2) + 1). - G. C. Greubel, Jun 01 2021

A116483 Expansion of (1 + x) / (5x^2 - 2x + 1).

Original entry on oeis.org

1, 3, 1, -13, -31, 3, 161, 307, -191, -1917, -2879, 3827, 22049, 24963, -60319, -245453, -189311, 848643, 2643841, 1044467, -11130271, -27482877, 685601, 138785587, 274143169, -145641597, -1661999039, -2595790093, 3118415009, 19215780483

Offset: 0

Creighton Dement, Feb 17 2006

Binomial transform of signed powers of 2: (1, 2, -4, -8, 16, 32, -64, -128, ...).
Inverse binonomial transform of (1, 4, 8, 0, -64, -256, -512, 0, 4096, 16384, 32768, 0, -262144, -1048576, -2097152, 0, ...).
G.f.*(1-x)/(1+x) (i.e, convolution with 1,-2,2,-2,2,-2, ... ) yields A006495.
Floretion Algebra Multiplication Program, FAMP Code: 2ibaseforseq[A*B] with A = - .5'i + .5'j - .5i' + .5j' + 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj' and B = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' ;

Colin Barker, Table of n, a(n) for n = 0..1000
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A000079, A006495, A116484.

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])+imag(v[k]));}
/* cf. A138749 */ /* Joerg Arndt, Jul 06 2011 */

Vec((1 + x) / (5*x^2 - 2*x + 1) + O(x^50)) \\ Colin Barker, Aug 25 2017

a(n) = 2*a(n-1) -5*a(n-2). - Paul Curtz, Apr 18 2011

a(n) = (1/2 + i/2)*((1 - 2*i)^n - i*(1 + 2*i)^n) where i=sqrt(-1). - Colin Barker, Aug 25 2017

A120743 a(n) = (1/2)(1 + 3i)^n + (1/2)(1 - 3i)^n where i = sqrt(-1).

Original entry on oeis.org

1, -8, -26, 28, 316, 352, -2456, -8432, 7696, 99712, 122464, -752192, -2729024, 2063872, 31417984, 42197248, -229785344, -881543168, 534767104, 9884965888, 14422260736, -70005137408, -284232882176, 131585609728, 3105500041216

Offset: 1

Creighton Dement, Jun 11 2007

From R. J. Mathar, Jun 15 2007: (Start)
These are the row sums of the triangle A013610 after every 2nd column is deleted, then every 2nd column reversed in sign, creating an intermediate irregular triangle with entries C(n,2*k)*(-9)^k, k = 0..floor(n/2):
  1;
  1,   -9;
  1,  -27;
  1,  -54,    81;
  1,  -90,   405;
  1, -135,  1215,    -729;
  1, -189,  2835,   -5103;
  1, -252,  5670,  -20412,   6561;
  1, -324, 10206,  -61236,  59049;
  1, -405, 17010, -153090, 295245, -59049; (End)
Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[A*B] with A = + 1.5i' + .5j' + .5k' + .5e and B = 'ji' + e

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (2, -10).

Cf. A006495

[ n eq 1 select 1 else n eq 2 select -8 else 2*Self(n-1) -10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 24 2011

LinearRecurrence[{2,-10}, {1,-8}, 30] (* G. C. Greubel, Nov 09 2018 *)

x='x+O('x^30); Vec((1-10*x)/(1-2*x+10*x^2)) \\ G. C. Greubel, Nov 09 2018

a(n) = 2*a(n-1) - 10*a(n-2).
G.f.: x*(1-10*x)/(10*x^2 - 2*x + 1).
a(n) mod 9 = 1. - Paul Curtz, Apr 20 2011
G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(9*k+1)/(x*(9*k+10) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
E.g.f.: exp(x)*cos(3*x). - Sergei N. Gladkovskii, May 29 2013
a(n) = A190958(n)-10*A190958(n-1). - R. J. Mathar, Dec 13 2022

a(0)=1: a(n) is main diagonal of A009116(n). - Paul Curtz, Jul 22 2011

Edited by Jon E. Schoenfield, Nov 09 2018

A120905 Real part of (1 + 2i)^(2^n) where i is sqrt(-1).

Original entry on oeis.org

1, -3, -7, -527, 164833, -98248054847, -3977703802948722503807, -510456831154766758152181998159655209453904127

Offset: 0

Gary W. Adamson, Jul 14 2006

sign

The next term a(8) has 90 digits and is too large to be displayed here.

			a(3) = -527 since (1 + 2i)^8 = (-527 + 336i) = A006495(8).

Cf. A006495.

a:= n-> Re((1 + 2*I)^(2^n));
seq (a(n), n=0..10);

Table[Re[(1 + 2I)^(2^n)], {n, 0, 8}] (* Stefan Steinerberger, Jul 23 2006 *)

a(n) = A006495(2^n).

a(n) = real part of (2 + sqrt(-1))^(2^n) for n >= 0.

Corrected and extended by Stefan Steinerberger and Emeric Deutsch, Jul 23 2006

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

Original entry on oeis.org

0, 1, 1, 0, -2, -4, -1, -1, 3, 11, 0, 2, 4, -2, -24, 1, 1, -3, -11, -7, 41, 0, -2, -4, 2, 24, 38, -44, -1, -1, 3, 11, 7, -41, -117, -29, 0, 2, 4, -2, -24, -38, 44, 278, 336, 1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 0, -2, -4, 2, 24, 38, -44, -278, -336, 718

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
   0;
   1,  1;
   0, -2, -4;
  -1, -1,  3,  11;
   0,  2,  4,  -2, -24;
   1,  1, -3, -11,  -7,  41;
   0, -2, -4,   2,  24,  38,  -44;
  -1, -1,  3,  11,   7, -41, -117, -29;
   0,  2,  4,  -2, -24, -38,   44, 278, 336.

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is related to A099456.
The next diagonal of the triangle is related to A139011.
Cf. A006495, A006496.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), this sequence (b=-2), A117918 and A228405 (b=1), A227418 (b=2), A292466 (b=4).

T(n+1,n)^2 + T(n,n)^2 = 5^n.

A349195 a(n) is the X-coordinate of the n-th point of the R5 dragon curve; A349196 gives Y-coordinates.

Original entry on oeis.org

0, 1, 1, 0, 0, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, -2, -2, -3, -3, -4, -4, -3, -3, -4, -4, -3, -3, -4, -4, -5, -5, -6, -6, -5, -5, -6, -6, -5, -5, -4, -4, -5, -5, -4, -4, -5, -5, -6, -6, -7, -7, -8, -8, -7, -7, -8, -8, -7, -7, -6, -6, -5, -5, -6, -6, -5, -5

Offset: 0

Rémy Sigrist, Nov 10 2021

sign

The R5 dragon curve can be represented using an L-system.

			The R5 dragon curve starts as follows:
         +-----+
       24|   25
         |
         |
         +-----+     +-----+     +-----+
       23    22|   11|   10|    7|    6|
               |     |     |     |     |
             21|   12|    9|    8|     |
         +-----+-----+-----+-----+-----+
       20|   17|   16|   13|    4|    5
         |     |     |     |     |
         |     |     |     |     |
         +-----+     +-----+     +-----+
       19    18    15    14     3     2|
                                       |
                                       |
                                 +-----+
                                0     1
- so a(0) = a(3) = a(4) = a(7) = a(8) = 0,
     a(1) = a(2) = a(5) = a(6) = 1,
     a(9) = a(10) = a(13) = a(14) = -1.

Rémy Sigrist, Table of n, a(n) for n = 0..3125
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.5 Dragon curves based on radix-R counting.
Kevin Ryde, Iterations of the R5 Dragon Curve, see index "point".
Rémy Sigrist, Colored representation of the first 1 + 5^9 points of the R5 dragon curve (where the hue is function of the number of steps from the origin)
Rémy Sigrist, PARI program for A349195
Index entries for sequences related to coordinates of 2D curves

Cf. A006495, A349008, A349009, A349010, A349196.

PARI
```
See Links section.
```

a(5^k) = A006495(k) for any k >= 0.

A179087 Triangle T(n,k) read by rows: the real part of the coefficient [x^k] of (1-x)^(n+1) * Sum_{s>=0} ((2s + 1 + 2i)^n)*x^s, where i is the imaginary unit.

Original entry on oeis.org

1, 1, 1, -3, 14, -3, -11, 35, 35, -11, -7, -84, 566, -84, -7, 41, -843, 2722, 2722, -843, 41, 117, -2854, 763, 50028, 763, -2854, 117, 29, -4681, -80211, 407423, 407423, -80211, -4681, 29, -527, 4504, -720740, 1560616, 8634214, 1560616, -720740, 4504, -527, -1199, 68393, -4275340, -6925948, 104031374, 104031374, -6925948, -4275340, 68393, -1199, 237, 338918, -19903639, -195090616, 799237802, 2546725796, 799237802, -195090616

Offset: 0

Roger L. Bagula, Jun 28 2010

			Triangle begins
    1;
    1,     1;
   -3,    14,     -3;
  -11,    35,     35,    -11;
   -7,   -84,    566,    -84,     -7;
   41,  -843,   2722,   2722,   -843,     41;
  117, -2854,    763,  50028,    763,  -2854,   117;
   29, -4681, -80211, 407423, 407423, -80211, -4681, 29;

Cf. A000165 (row sums), A006495 (column k=0).

A179087 := proc(n,k)
        (1-x)^(n+1)*add( (2*s+1+2*I)^n*x^s,s=0..k) ;
        expand(%) ;
        coeftayl(%,x=0,k) ;
        Re(%) ;
end proc: # R. J. Mathar, Oct 06 2011

Sequence replaced with one that is more likely to occur in practice by R. J. Mathar, Oct 06 2011

A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -4, 1, 1, 1, -3, -8, -7, -4, 1, 1, 1, -4, -11, -8, 1, 0, 1, 1, 1, -5, -14, -7, 16, 23, 8, 1, 1, 1, -6, -17, -4, 41, 64, 43, 16, 1, 1, 1, -7, -20, 1, 76, 117, 64, 17, 16, 1, 1, 1, -8, -23, 8, 121, 176, 29, -128, -95, 0, 1

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com) and Robert G. Wilson v, Jan 02 2013

.j\k.........0..1...2....3...4....5....6......7.......8......9......10
.0: A000012..1..1...1....1...1....1....1......1.......1......1.......1
-1: A146559..1..1...0...-2..-4...-4....0......8......16.....16.......0
-2: A087455..1..1..-1...-5..-7....1...23.....43......17....-95....-241
-3: A138230..1..1..-2...-8..-8...16...64.....64....-128...-512....-512
-4: A006495..1..1..-3..-11..-7...41..117.....29....-527..-1199.....237
-5: A138229..1..1..-4..-14..-4...76..176...-104...-1264..-1904....3776
-6: A090592..1..1..-5..-17...1..121..235...-377...-2399..-2159...12475
-7: A090590..1..1..-6..-20...8..176..288...-832...-3968..-1280...29184
-8: A025172..1..1..-7..-23..17..241..329..-1511...-5983...1633...57113
-9: A120743..1..1..-8..-26..28..316..352..-2456...-8432...7696...99712
-10: ........1..1..-9..-29..41..401..351..-3709..-11279..18241..160551

Cf. A084097,
Rows: A000012, A146559, A087455, A138230, A006495, A138229, A090592, A090590, A025172, A120743.
Columns: A000012, A000012, A024000, 1-3n, A028884, A134593.

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; Table[ T[ -j + k, k], {j, 0, 11}, {k, 0, j}] // Flatten

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A116484 Expansion of (-1+3*x)/(5*x^2 + 1 - 2*x).

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A117411 Skew triangle associated to the Euler numbers.

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A117435 Triangle related to exp(x)*cos(2*x).

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A116483 Expansion of (1 + x) / (5*x^2 - 2*x + 1).

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A120743 a(n) = (1/2)*(1 + 3*i)^n + (1/2)*(1 - 3*i)^n where i = sqrt(-1).

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A120905 Real part of (1 + 2i)^(2^n) where i is sqrt(-1).

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A292495 Triangle read by rows: T(n,k) = (-2)*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = (-1)^m.

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Formula

A116484 Expansion of (-1+3x)/(5x^2 + 1 - 2*x).

A117435 Triangle related to exp(x)cos(2x).

A116483 Expansion of (1 + x) / (5x^2 - 2x + 1).

A120743 a(n) = (1/2)(1 + 3i)^n + (1/2)(1 - 3i)^n where i = sqrt(-1).

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

A179087 Triangle T(n,k) read by rows: the real part of the coefficient [x^k] of (1-x)^(n+1) * Sum_{s>=0} ((2s + 1 + 2i)^n)*x^s, where i is the imaginary unit.