George E. Antoniou - cp's OEIS frontend

A330987 Alternatively add and half-multiply pairs of the nonnegative integers.

1, 3, 9, 21, 17, 55, 25, 105, 33, 171, 41, 253, 49, 351, 57, 465, 65, 595, 73, 741, 81, 903, 89, 1081, 97, 1275, 105, 1485, 113, 1711, 121, 1953, 129, 2211, 137, 2485, 145, 2775, 153, 3081, 161, 3403, 169, 3741, 177, 4095, 185, 4465, 193, 4851, 201, 5253, 209

Offset: 1

George E. Antoniou, Jan 05 2020

In groups of two, add and half-multiply the integers: 0+1, (2*3)/2, 4+5, (6*7)/2, ....
From Bernard Schott, Jan 06 2020: (Start)
The bisection of this sequence gives:
For n odd = 2*k+1, k >= 0: a(2*k+1) = 8*k+1 = A017077(k),
For n even = 2*k, k >= 1: a(2*k) = T(4*k-2) = A000217(4*k-2) = (2*k-1)*(4*k-1) = A033567(k) where T(j) is the j-th triangular number. (End)

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A330983.

Interspersion of A017077 and A033567 (excluding first term). - Michel Marcus, Jan 06 2020

a[n_]:=If[OddQ[n],4n-3,(n-1)(2n-1)]; Array[a,53] (* Stefano Spezia, Jan 05 2020 *)

Vec(x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jan 06 2020

From Colin Barker, Jan 05 2020: (Start)
G.f.: x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = -1 + 2*(-1)^n - (1/2)*(-1+7*(-1)^n)*n + (1+(-1)^n)*n^2.
(End)
E.g.f.: (1 + 4*x + 2*x^2)*cosh(x) - (3 + x)*sinh(x) - 1. - Stefano Spezia, Jan 05 2020 after Colin Barker

A330983 Alternatively add and multiply pairs of the nonnegative integers.

Original entry on oeis.org

1, 6, 9, 42, 17, 110, 25, 210, 33, 342, 41, 506, 49, 702, 57, 930, 65, 1190, 73, 1482, 81, 1806, 89, 2162, 97, 2550, 105, 2970, 113, 3422, 121, 3906, 129, 4422, 137, 4970, 145, 5550, 153, 6162, 161, 6806, 169, 7482, 177, 8190, 185, 8930, 193, 9702, 201, 10506, 209

Offset: 1

George E. Antoniou, Jan 05 2020

In groups of two, add and multiply the integers: 0+1, 2*3, 4+5, 6*7, ....

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A330987.

Interspersion of A017077 and A256833. - Michel Marcus, Jan 06 2020

a[n_]:=If[OddQ[n],4n-3,2(n-1)(2n-1)]; Array[a,53] (* Stefano Spezia, Jan 05 2020 *)

Vec(x*(1 + 6*x + 6*x^2 + 24*x^3 - 7*x^4 + 2*x^5) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jan 07 2020

From Colin Barker, Jan 05 2020: (Start)
G.f.: x*(1 + 6*x + 6*x^2 + 24*x^3 - 7*x^4 + 2*x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = (1/2)*(-1 + 5*(-1)^n - 2*(1 + 5*(-1)^n)*n + 4*(1+(-1)^n)*n^2).
(End)
E.g.f.: (2 + 4*x*(1 + x))*cosh(x) - (3 + 2*x)*sinh(x) - 2. - Stefano Spezia, Jan 05 2020 after Colin Barker

A097804 a(n) = 3(25^n + 1).

Original entry on oeis.org

9, 33, 153, 753, 3753, 18753, 93753, 468753, 2343753, 11718753, 58593753, 292968753, 1464843753, 7324218753, 36621093753, 183105468753, 915527343753, 4577636718753, 22888183593753, 114440917968753, 572204589843753, 2861022949218753, 14305114746093753

Offset: 0

George E. Antoniou, Aug 25 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -5).

Cf. A097802, A097803.

[3*(2*5^n+1) : n in [0..30]]; // Wesley Ivan Hurt, Aug 16 2016

A097804:=n->3*(2*5^n+1): seq(A097804(n), n=0..30); # Wesley Ivan Hurt, Aug 16 2016

Table[3(2*5^n + 1), {n, 0, 20}] (* Robert G. Wilson v, Aug 26 2004 *)
LinearRecurrence[{6,-5}, {9,33}, 30] (* Harvey P. Dale, Dec 17 2012 *)
6*5^Range[0, 30] + 3 (* Wesley Ivan Hurt, Aug 16 2016 *)

a(0)=9, a(1)=33, a(n) = 6*a(n-1) - 5*a(n-2) for n > 1. - Harvey P. Dale, Dec 17 2012

G.f.: 3*(3-7*x)/((1-x)*(1-5*x)). - Wesley Ivan Hurt, Aug 16 2016

More terms from Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 26 2004

A097803 a(n) = 3(2n^2 + 1).

Original entry on oeis.org

3, 9, 27, 57, 99, 153, 219, 297, 387, 489, 603, 729, 867, 1017, 1179, 1353, 1539, 1737, 1947, 2169, 2403, 2649, 2907, 3177, 3459, 3753, 4059, 4377, 4707, 5049, 5403, 5769, 6147, 6537, 6939, 7353, 7779, 8217, 8667, 9129, 9603, 10089, 10587, 11097, 11619, 12153, 12699

Offset: 0

George E. Antoniou, Aug 25 2004

a(n) is also the number of Arnoux-Rauzy factors of length (n+1) over a 3-letter alphabet. - Genevieve Paquin (genevieve.paquin(AT)univ-savoie.fr), Nov 07 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
F. Mignosi and L. Q. Zamboni, On the number of Arnoux-Rauzy words, Acta arith., 101 (2002), no. 2, 121-129. [From Genevieve Paquin (genevieve.paquin(AT)univ-savoie.fr), Nov 07 2008]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A058331, A097802.

Table[ 3(2*n^2 + 1), {n, 0, 44}] (* Robert G. Wilson v, Aug 26 2004 *)
3(2Range[0,50]^2+1) (* or *) LinearRecurrence[{3,-3,1},{3,9,27},50] (* Harvey P. Dale, Dec 29 2011 *)

a(n)=3*(2*n^2+1) \\ Charles R Greathouse IV, Jun 17 2017

From Harvey P. Dale, Dec 29 2011: (Start)
a(0)=3, a(1)=9, a(2)=27, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -3*(3*x^2+1)/(x-1)^3. (End)
From Elmo R. Oliveira, Feb 18 2025: (Start)
E.g.f.: 3*exp(x)*(1 + 2*x + 2*x^2).
a(n) = 3*A058331(n). (End)

More terms from Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 26 2004

A097802 a(n) = 3(25n + 1).

Original entry on oeis.org

3, 78, 153, 228, 303, 378, 453, 528, 603, 678, 753, 828, 903, 978, 1053, 1128, 1203, 1278, 1353, 1428, 1503, 1578, 1653, 1728, 1803, 1878, 1953, 2028, 2103, 2178, 2253, 2328, 2403, 2478, 2553, 2628, 2703, 2778, 2853, 2928, 3003, 3078, 3153, 3228, 3303, 3378, 3453, 3528

Offset: 0

George E. Antoniou, Aug 25 2004

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A158060.

75*Range[0, 50]+3 (* Paolo Xausa, Feb 16 2024 *)

a(n)=3*(25*n+1) \\ Charles R Greathouse IV, Jul 10 2016

my(x='x+O('x^48)); Vec(3*(1+24*x)/(1-x)^2) \\ Elmo R. Oliveira, May 25 2025

From Elmo R. Oliveira, May 25 2025: (Start)
G.f.: 3*(1 + 24*x)/(1-x)^2.
E.g.f.: 3*exp(x)*(1 + 25*x).
a(n) = 3*A158060(n) for n > 0.
a(n) = 2*a(n-1) - a(n-2). (End)

More terms from Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 26 2004

a(45)-a(47) from Elmo R. Oliveira, May 25 2025

A066335 Binary string which equals n when 1's and 2's bits have negative weights.

Original entry on oeis.org

0, 111, 110, 101, 100, 1011, 1010, 1001, 1000, 1111, 1110, 1101, 1100, 10011, 10010, 10001, 10000, 10111, 10110, 10101, 10100, 11011, 11010, 11001, 11000, 11111, 11110, 11101, 11100, 100011, 100010, 100001, 100000, 100111, 100110, 100101

Offset: 0

George E. Antoniou, Dec 15 2001

			For example: 4-2-1 = 1, so a(1) = 111; 4-2+0 = 2 so a(2) = 110; 4+0-1 = 3 so a(3) = 101; 4+0+0 = 4 so a(4) = 100, etc.

Morris M. Mano, Digital Design, Prentice Hall, 2002. p. 20.

A120634 is the decimal equivalent of these numbers in binary.

More terms from Sascha Kurz, Jan 28 2003

Corrected and extended by Joshua Zucker, Jun 21 2006

A066104 a(2n) = 2n, a(2n+1) = 4(n+1).

Original entry on oeis.org

0, 4, 2, 8, 4, 12, 6, 16, 8, 20, 10, 24, 12, 28, 14, 32, 16, 36, 18, 40, 20, 44, 22, 48, 24, 52, 26, 56, 28, 60, 30, 64, 32, 68, 34, 72, 36, 76, 38, 80, 40, 84, 42, 88, 44, 92, 46, 96, 48, 100, 50, 104, 52, 108, 54, 112, 56, 116, 58, 120, 60, 124, 62, 128, 64, 132, 66, 136

Offset: 0

George E. Antoniou, Dec 04 2001

Fourth column of table A210530 for n>0. - Boris Putievskiy, Jan 29 2013

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A065423, A210530.

Table[(3*n+2-(n+2)*(-1)^n)/2, {n,0,50}] (* or *) LinearRecurrence[{0, 2, 0, -1}, {0, 4, 2, 8}, 50] (* G. C. Greubel, Dec 24 2016 *)

{ for (n=0, 1000, if(n%2, a=2*n + 2, a=n); write("b066104.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 14 2009

concat([0], Vec(2*x*(x+2)/(1-x^2)^2 + O(x^50))) \\ G. C. Greubel, Dec 24 2016

a(n) = 2*A065423(n+1).
O.g.f.: 2*x(2+x)/(1-x^2)^2. - Len Smiley, Dec 06 2001
a(n) = (3*n+2-(n+2)*(-1)^n)/2. - Boris Putievskiy, Jan 29 2013

A066336 Decimal equivalent of A066334.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 14, 15, -1, -1, -1, -1, -1, -1, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, -1, -1, -1, -1, -1, -1, 32, 33, 34, 35, 36, 37, 38, 39, 46, 47, -1, -1, -1, -1, -1, -1, 48, 49, 50, 51, 52, 53, 54, 55, 62, 63, -1, -1, -1, -1, -1, -1, 64, 65, 66, 67, 68, 69, 70, 71, 78, 79, -1, -1, -1, -1, -1, -1, 80, 81, 82, 83, 84, 85, 86

Offset: 0

George E. Antoniou, Dec 15 2001

Corrected and extended by Joshua Zucker, Jun 21 2006

A066338 Decimal equivalent of A066330.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 13, 14, 15, -1, -1, -1, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 30, 31, -1, -1, -1, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 46, 47, -1, -1, -1, 48, 49, 50, 51, 52, 56, 57, 58, 59, 60, 61, 62, 63, -1, -1, -1, 64, 65, 66, 67, 68, 72, 73, 74, 75, 76, 77, 78, 79, -1, -1, -1, 80, 81, 82, 83, 84, 88, 89

Offset: 0

George E. Antoniou, Dec 15 2001

Corrected by Sascha Kurz, Feb 02 2003

More terms from Joshua Zucker, Jun 21 2006

A066334 Binary string which equals n when 1's, 2's, 4's and 8's bits have weights 1, 2, 4, 2 respectively, while the other bits have their usual weights. -1 if no such string exists.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1110, 1111, -1, -1, -1, -1, -1, -1, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11110, 11111, -1, -1, -1, -1, -1, -1, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111, 101110, 101111, -1, -1, -1, -1, -1, -1, 110000, 110001, 110010, 110011

Offset: 0

George E. Antoniou, Dec 15 2001

Morris M. Mano, Digital Design, Prentice Hall, 2002. p. 20.

Cf. A066335.

More terms from Joshua Zucker, Jun 21 2006

cp's OEIS Frontend

User: George E. Antoniou

A330987 Alternatively add and half-multiply pairs of the nonnegative integers.

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A330983 Alternatively add and multiply pairs of the nonnegative integers.

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A097804 a(n) = 3*(2*5^n + 1).

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A097803 a(n) = 3*(2*n^2 + 1).

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A097802 a(n) = 3*(25*n + 1).

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A066335 Binary string which equals n when 1's and 2's bits have negative weights.

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A066104 a(2n) = 2n, a(2n+1) = 4(n+1).

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A097804 a(n) = 3(25^n + 1).

A097803 a(n) = 3(2n^2 + 1).

A097802 a(n) = 3(25n + 1).