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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.

User: Yasir Karamelghani Gasmallah

Yasir Karamelghani Gasmallah's wiki page.

Yasir Karamelghani Gasmallah has authored 4 sequences.

A214393 Numbers of the form (4k+3)^2+4 or (4k+5)^2-8.

Original entry on oeis.org

13, 17, 53, 73, 125, 161, 229, 281, 365, 433, 533, 617, 733, 833, 965, 1081, 1229, 1361, 1525, 1673, 1853, 2017, 2213, 2393, 2605, 2801, 3029, 3241, 3485, 3713, 3973, 4217, 4493, 4753, 5045, 5321, 5629, 5921, 6245, 6553, 6893, 7217, 7573, 7913, 8285, 8641
Offset: 0

Views

Author

Yasir Karamelghani Gasmallah, Jul 15 2012

Keywords

Comments

For every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2.
In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2, e.g., (17^2, 53^2, 73^2).
The first differences of this sequence is the interleaved sequence 4,36,20,52,36,68,52,....

Examples

			a(5) = 2*a(4) - 2*a(2) + a(1) = 2*125 - 2*53 + 17 = 161.

Links

Crossrefs

Cf. A178218, A214345.

Programs

  • Magma
    I:=[13, 17, 53, 73]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
  • Mathematica
    A214393[n_] := 4*n*(n+3) + 6*(-1)^n + 7; Array[A214393, 50, 0] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {13, 17, 53, 73}, 50] (* Paolo Xausa, Feb 22 2024 *)
  • Maxima
    A214393(n):=4*n*(n+3)+6*(-1)^n+7$
makelist(A214393(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

Formula

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (13-9*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).
a(n) = 4*n*(n+3)+6*(-1)^n+7.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.

A214405 Numbers of the form (4k+3)^2-8 or (4k+5)^2+4.

Original entry on oeis.org

1, 29, 41, 85, 113, 173, 217, 293, 353, 445, 521, 629, 721, 845, 953, 1093, 1217, 1373, 1513, 1685, 1841, 2029, 2201, 2405, 2593, 2813, 3017, 3253, 3473, 3725, 3961, 4229, 4481, 4765, 5033, 5333, 5617, 5933, 6233, 6565, 6881, 7229, 7561, 7925, 8273, 8653
Offset: 1

Views

Author

Yasir Karamelghani Gasmallah, Jul 15 2012

Keywords

Comments

For every odd n the triple (a(n-1)^2, a(n)^2 , a(n+1)^2) is an arithmetic progression, i.e., 2*a(n)^2 = a(n-1)^2 + a(n+1)^2.
In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2.
The first differences of this sequence is the interleaved sequence 28,12,44,28,60,44....

Examples

			a(4) = 2*a(3) - 2*a(1) + a(0) = 2*85 - 2*29 + 1 = 113.

Crossrefs

Cf. A178218, A214345.

Programs

  • Magma
    I:=[1, 29, 41, 85]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
  • Maxima
    A214405(n):=4*n*(n+3)-6*(-1)^n+7$
makelist(A214405(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

Formula

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
O.G.f.: (1+27*x-17*x^2+5*x^3)/((1+x)*(1-x)^3).
a(n) = 4*n*(n+3)-6*(-1)^n+7.
2*a(2n+1)^2 = a(2n)^2 + a(2n+2)^2.

A214493 Numbers of the form ((6k+5)^2+9)/2 or 2(3k+4)^2-9.

Original entry on oeis.org

17, 23, 65, 89, 149, 191, 269, 329, 425, 503, 617, 713, 845, 959, 1109, 1241, 1409, 1559, 1745, 1913, 2117, 2303, 2525, 2729, 2969, 3191, 3449, 3689, 3965, 4223, 4517, 4793, 5105, 5399, 5729, 6041, 6389, 6719, 7085, 7433, 7817, 8183, 8585, 8969, 9389, 9791, 10229, 10649, 11105, 11543, 12017, 12473, 12965, 13439, 13949
Offset: 0

Views

Author

Yasir Karamelghani Gasmallah, Jul 19 2012

Keywords

Comments

For every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2.
In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2.
The first differences of this sequence is the interleaved sequence 6,42,24,60,42,78.... = 9*n*(39-27*(-1)^n)/2.

Examples

			For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*269-2*149+89=329.

Links

Crossrefs

Cf. A178218, A214345.

Programs

  • Magma
    I:=[17, 23, 65, 89]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
  • Mathematica
    LinearRecurrence[{2,0,-2,1},{17,23,65,89},60] (* Harvey P. Dale, Aug 07 2015 *)

Formula

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (17-11*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).
a(n) = (6*n*(3*n+10)+27*(-1)^n+41)/4.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.

A214345 Interleaved reading of A073577 and A053755.

Original entry on oeis.org

5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839, 901, 959, 1025, 1087, 1157, 1223, 1297, 1367, 1445, 1519, 1601, 1679, 1765, 1847, 1937, 2023, 2117, 2207, 2305, 2399, 2501
Offset: 0

Views

Author

Yasir Karamelghani Gasmallah, Jul 13 2012

Keywords

Comments

The elements of this sequence satisfy the property that for every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2. In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2 : in the case of this sequence 7^2, 17^2, and 23^2 is such a triple (i.e. 15-8 =7, 17, 8+15=23, and 8^2+15^2=17^2) .
The first differences of such a sequence is always an interleaved sequence; in this case the interleaved sequence is 2,10,6,14,10,... (A142954).

Examples

			For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79

Links

Crossrefs

Cf. A053755, A073577, A178218.
First differences: A142954; 2-element moving average (a(n-1) + a(n))/2: A002378. - Guenther Schrack, Oct 25 2018

Programs

  • GAP
    a:=[7,17];; for n in [3..50] do a[n]:=4*(n+1)+a[n-2]; od; Concatenation([5],a); # Muniru A Asiru, Oct 26 2018
  • Magma
    I:=[5, 7, 17, 23];[n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
  • Maple
    seq(coeff(series((x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 26 2018
  • Mathematica
    LinearRecurrence[{2,0,-2,1},{5,7,17,23},50] (* Harvey P. Dale, Apr 02 2018 *)
  • Maxima
    A214345(n):=(2*n*(n+4)+3*(-1)^n+7)/2$
makelist(A214345(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

Formula

a(2n+1) = A073577(n+1); a(2n) = A053755(n+1).
a(n+1)-a(n) = A142954(n+1).
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)).
a(n) = (2*n*(n+4)+3*(-1)^n+7)/2.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
a(n) = 4*(n+1) + a(n-2) for n > 1; a(-n) = a(n-4). - Guenther Schrack, Oct 24 2018
E.g.f.: (5 + 5*x + x^2)*cosh(x) + (2 + 5*x + x^2)*sinh(x). - Stefano Spezia, Feb 22 2024

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