A139598 -id:A139598 - cp's OEIS frontend

A139592 A033585(n) followed by A139271(n+1).

0, 2, 10, 20, 36, 54, 78, 104, 136, 170, 210, 252, 300, 350, 406, 464, 528, 594, 666, 740, 820, 902, 990, 1080, 1176, 1274, 1378, 1484, 1596, 1710, 1830, 1952, 2080, 2210, 2346, 2484, 2628, 2774, 2926, 3080, 3240, 3402, 3570, 3740

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 2,... and the same line from 0, in the direction 0, 10,..., in the square spiral whose vertices are the triangular numbers A000217.

a(n) = 2*A006578(n) - A002378(n)/2 = 2*A035608(n). [From Reinhard Zumkeller, Feb 07 2010]

			Array begins:
0, 2
10, 20
36, 54
78, 104

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A033585, A046092, A139271, A077221, A139591, A139593, A139595, A139596, A139597, A139598.

Array read by rows: row n gives 8*n^2 + 2n, 8*(n+1)^2 - 6(n+1).
a(n) = 2*floor((n + 1/4)^2). [From Reinhard Zumkeller, Feb 07 2010]
G.f.: 2*x*(1+3*x)/((1-x)^3*(1+x)). [Colin Barker, Apr 26 2012]

A139593 A139276(n) followed by A139272(n+1).

Original entry on oeis.org

0, 3, 11, 22, 38, 57, 81, 108, 140, 175, 215, 258, 306, 357, 413, 472, 536, 603, 675, 750, 830, 913, 1001, 1092, 1188, 1287, 1391, 1498, 1610, 1725, 1845, 1968, 2096, 2227, 2363, 2502, 2646, 2793, 2945, 3100, 3260, 3423, 3591, 3762

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 3, ... and the same line from 0, in the direction 0, 11, ..., in the square spiral whose vertices are the triangular numbers A000217.

A139593 appears (both numerically and via back of an envelope algebra, but not a publishable proof) to be the cumulative sum of A047470. - Markus J. Q. Roberts, Jul 12 2009

			Array begins:
   0,   3;
  11,  22;
  38,  57;
  81, 108;

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A046092, A139272, A139276, A077221, A139591, A139592, A139594, A139595, A139596, A139597, A139598, A047470.

LinearRecurrence[{2,0,-2,1},{0,3,11,22},50] (* Harvey P. Dale, Feb 09 2019 *)

Array read by rows: row n gives 8*n^2 + 3n, 8*(n+1)^2 - 5(n+1).
From Colin Barker, Sep 15 2013: (Start)
a(n) = (-1 + (-1)^n + 6*n + 8*n^2)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -x*(5*x+3) / ((x-1)^3*(x+1)). (End)

Edited by Omar E. Pol, Jul 13 2009

A139596 A033587(n) followed by even hexagonal number A014635(n+1).

Original entry on oeis.org

0, 6, 14, 28, 44, 66, 90, 120, 152, 190, 230, 276, 324, 378, 434, 496, 560, 630, 702, 780, 860, 946, 1034, 1128, 1224, 1326, 1430, 1540, 1652, 1770, 1890, 2016, 2144, 2278, 2414, 2556, 2700, 2850, 3002, 3160, 3320, 3486, 3654, 3828

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 6,... and the same line from 0, in the direction 0, 14,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 6
14, 28
44, 66
90, 120

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A014635, A033587, A046092, A077221, A139591, A139592, A139593, A139595, A139597, A139598.

LinearRecurrence[{2,0,-2,1},{0,6,14,28},50] (* Harvey P. Dale, Jan 20 2024 *)

Array read by rows: row n gives 8*n^2 + 6*n, 8*(n+1)^2 - 2(n+1).
O.g.f.: -2*x*(x+3)/((x-1)^3*(1+x)). - R. J. Mathar, May 06 2008
a(n) = 2*A156859(n). - R. J. Mathar, Feb 28 2018

A139591 A139275(n) followed by 18-gonal number A051870(n+1).

Original entry on oeis.org

0, 1, 9, 18, 34, 51, 75, 100, 132, 165, 205, 246, 294, 343, 399, 456, 520, 585, 657, 730, 810, 891, 979, 1068, 1164, 1261, 1365, 1470, 1582, 1695, 1815, 1936, 2064, 2193, 2329, 2466, 2610, 2755, 2907, 3060, 3220, 3381, 3549, 3718, 3894, 4071, 4255, 4440, 4632

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 9, ... and the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
   0,   1;
   9,  18;
  34,  51;
  75, 100;
  ...

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000217, A046092, A051870, A139275, A077221, A139592, A139593, A139595, A139596, A139597, A139598.

Partial sums of A047393.

Array read by rows: row n gives 8*n^2 + n, 8*(n+1)^2 - 7*(n+1).
G.f.: -x*(7*x+1)/((x-1)^3*(x+1)). - Colin Barker, Oct 16 2012
a(n) = 2*n^2 + (7/2)*n + (3/4)*((-1)^n-1). - Sean A. Irvine, Jul 14 2022

A139595 A139277(n) followed by A139273(n+1).

Original entry on oeis.org

0, 5, 13, 26, 42, 63, 87, 116, 148, 185, 225, 270, 318, 371, 427, 488, 552, 621, 693, 770, 850, 935, 1023, 1116, 1212, 1313, 1417, 1526, 1638, 1755, 1875, 2000, 2128, 2261, 2397, 2538, 2682, 2831, 2983, 3140, 3300, 3465, 3633, 3806

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 5,... and the same line from 0, in the direction 0, 13,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 5
13, 26
42, 63
87, 116

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1)

Cf. A000217, A046092, A139273, A139277, A077221, A139591, A139592, A139593, A139596, A139597, A139598.

LinearRecurrence[{2,0,-2,1},{0,5,13,26},50] (* Harvey P. Dale, Jul 31 2021 *)

Array read by rows: row n gives 8*n^2 + 5n, 8*(n+1)^2 - 3(n+1).

G.f.: -x*(5+3*x) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Feb 13 2011

A139597 A139278(n) followed by A139274(n+1).

Original entry on oeis.org

0, 7, 15, 30, 46, 69, 93, 124, 156, 195, 235, 282, 330, 385, 441, 504, 568, 639, 711, 790, 870, 957, 1045, 1140, 1236, 1339, 1443, 1554, 1666, 1785, 1905, 2032, 2160, 2295, 2431, 2574, 2718, 2869, 3021, 3180, 3340, 3507, 3675, 3850

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 7,... and the line from 15, in the direction 15, 46,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 7
15, 30
46, 69
93, 124

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A046092, A139274, A139278, A077221, A139591, A139592, A139593, A139595, A139596, A139598.

Array read by rows: row n gives 8*n^2 + 7n, 8*(n+1)^2 - (n+1).

a(n) = (3-3*(-1)^n+14*n+8*n^2)/4. a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). G.f.: x*(7+x)/((1-x)^3*(1+x)). [Colin Barker, Jul 22 2012]

A241684 The total number of rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 0, 4, 8, 32, 120, 464, 1848, 7312, 29240, 116624, 466488, 1864592, 7458360, 29827984, 119311928, 477225872, 1908903480, 7635526544, 30542106168, 122168075152, 488672300600, 1954687804304, 7818751217208, 31274999276432, 125099997105720, 500399966053264, 2001599864213048

Offset: 0

Kival Ngaokrajang, Apr 27 2014

a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 1 column or 1 row, 2 columns) that appear as rectangles in the Thue-Morse logical matrices after n stages. See links for more details.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4,5,-20,-4,16)

Cf. A010060.

[(8+3*2^n+2*4^n+(-1)^n*(24-2^n))/18: n in [0..30]]; // Vincenzo Librandi, Sep 29 2017

CoefficientList[Series[-4*x^2*(8*x^3 - 5*x^2 - 2*x + 1)/((x - 1)*(x + 1)*(2*x - 1)*(2*x + 1)*(4*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)

x='x+O('x^50); concat([0,0], Vec(-4*x^2*(8*x^3-5*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)))) \\ G. C. Greubel, Sep 28 2017

a(n) = A007590(A005578(n+1)) - (A139598(A000975(n-2)) + A007590(A000975(n-1))).
G.f.: -4*x^2*(8*x^3-5*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014
a(n) = (8 + 3*2^n + 2*4^n + (-1)^n*(24 - 2^n))/18, n>0. - R. J. Mathar, May 04 2014

Terms a(21) onward added by G. C. Greubel, Sep 28 2017

A241682 Total number of unit squares appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 2, 0, 8, 16, 72, 240, 968, 3696, 14792, 58480, 233928, 932976, 3731912, 14916720, 59666888, 238623856, 954495432, 3817806960, 15271227848, 61084212336, 244336849352, 977344601200, 3909378404808

Offset: 0

Kival Ngaokrajang, Apr 27 2014

nonn

a(n) is the total number of isolated "1s" (no adjacent 1s on horizontal and vertical) which appear as unit squares in the Thue-Morse logical matrices after n stages. See links for more details.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4, 5, -20, -4, 16).

Cf. A010060.

CoefficientList[Series[2*x*(12*x^4 - 12*x^3 + x^2 + 4*x - 1)/((x - 1)*(x + 1)*(2*x - 1)*(2*x + 1)*(4*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Sep 29 2017 *)

{a0=0;a1=2;print1(a0,", ",a1,", "); for (n=0,50, b=ceil(2*(2^n-1)/3);a=1- (-1)^b+4*b+2*b^2; print1(a,", "))}

x='x+O('x^50); concat(0, Vec(2*x*(12*x^4-12*x^3+x^2+4*x-1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)))) \\ G. C. Greubel, Sep 29 2017

a(n) = A139598(A000975(n - 2)).
G.f.: 2*x*(12*x^4-12*x^3+x^2+4*x-1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014
a(n) = 16*a(n-5) -4*a(n-4) -20*a(n-3) +5*a(n-2) +4*a(n-1), n>=5. (Also valid for A241683, A241684 and A241685.) - Fung Lam, May 02 2014

cp's OEIS Frontend

A139592 A033585(n) followed by A139271(n+1).

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A139593 A139276(n) followed by A139272(n+1).

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A139596 A033587(n) followed by even hexagonal number A014635(n+1).

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A139591 A139275(n) followed by 18-gonal number A051870(n+1).

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A139595 A139277(n) followed by A139273(n+1).

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A139597 A139278(n) followed by A139274(n+1).

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A241684 The total number of rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

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Magma

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A241682 Total number of unit squares appearing in the Thue-Morse sequence logical matrices after n stages.

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