A159797 -id:A159797 - cp's OEIS frontend

A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

1, 1, 1, 1, 2, 3, 1, 3, 5, 7, 1, 4, 7, 10, 13, 1, 5, 9, 13, 17, 21, 1, 6, 11, 16, 21, 26, 31, 1, 7, 13, 19, 25, 31, 37, 43, 1, 8, 15, 22, 29, 36, 43, 50, 57, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81

Offset: 1

Omar E. Pol, Jul 09 2009

Equals A081493 when first column is removed. - Georg Fischer, Jul 25 2023

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  5,  7;
  1,  4,  7, 10, 13;
  1,  5,  9, 13, 17, 21;
  1,  6, 11, 16, 21, 26, 31;

Cf. A002061, A159797, A159798, A162610, A161611, A162612, A162613.

Cf. A060354 (row sums), A081493 (without first column).

Table[NestList[#+(n-2)&,1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Oct 23 2017 *)

T(n,n) = A002061(n-1).

T(n,k) = A076110(n-1,k) = 1+(n-2)*(k-1). - R. J. Mathar, Mar 30 2023

A162622 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^4 - 1.

Original entry on oeis.org

0, 1, 1, 2, 17, 32, 3, 83, 163, 243, 4, 259, 514, 769, 1024, 5, 629, 1253, 1877, 2501, 3125, 6, 1301, 2596, 3891, 5186, 6481, 7776, 7, 2407, 4807, 7207, 9607, 12007, 14407, 16807, 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768, 9, 6569, 13129

Offset: 0

Omar E. Pol, Jul 15 2009

Note that the last term of the n-th row is the 5th power of n, A000584(n).

See also the triangles of A162623 and A162624.

			Triangle begins:
  0;
  1,    1;
  2,   17,    32;
  3,   83,   163,   243;
  4,  259,   514,   769,  1024;
  5,  629,  1253,  1877,  2501,  3125;
  6, 1301,  2596,  3891,  5186,  6481,  7776;
  7, 2407,  4807,  7207,  9607, 12007, 14407, 16807;
  8, 4103,  8198, 12293, 16388, 20483, 24578, 28673, 32768;
  9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 59049; etc.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162623, A162624.

/* Triangle: */ [[n+k*(n^4-1): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Dec 14 2012

A162622 := proc(n,k) n+k*(n^4-1) ; end proc: seq(seq( A162622(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 11 2010

Flatten[Table[NestList[#+n^4-1&,n,n],{n,0,9}]] (* Harvey P. Dale, Jun 23 2013 *)

Sum_{k=0..n} T(n,k) = n*(n+1)*(1+n^4)/2 (row sums). [R. J. Mathar, Jul 20 2009]

7th and later rows from R. J. Mathar, Feb 11 2010

A162612 Triangle read by rows in which row n lists n terms, starting with n^2+n-1, with gaps = n^2-1 between successive terms.

Original entry on oeis.org

1, 5, 8, 11, 19, 27, 19, 34, 49, 64, 29, 53, 77, 101, 125, 41, 76, 111, 146, 181, 216, 55, 103, 151, 199, 247, 295, 343, 71, 134, 197, 260, 323, 386, 449, 512, 89, 169, 249, 329, 409, 489, 569, 649, 729, 109, 208, 307, 406, 505, 604, 703, 802, 901, 1000, 131, 251

Offset: 1

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th cube A000578(n).

			Triangle begins:
   1;
   5,   8;
  11,  19,  27;
  19,  34,  49,  64;
  29,  53,  77, 101, 125;
  41,  76, 111, 146, 181, 216;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000578, A028387, A159797, A159798, A162610, A162611, A162613.

Table[NestList[#+n^2-1&,n^2+n-1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Mar 02 2024 *)

T(n,k)=n+k*(n^2-1) \\ Franklin T. Adams-Watters, Aug 06 2009

Sum_{k=1..n} T(n,k)= n*(n^3 + n^2 + n - 1)/2 (row sums). - R. J. Mathar, Jul 20 2009

More terms from Franklin T. Adams-Watters, Aug 06 2009

A162613 Triangle read by rows in which row n lists n terms, starting with n, with gaps = n^2-1 between successive terms.

Original entry on oeis.org

1, 2, 5, 3, 11, 19, 4, 19, 34, 49, 5, 29, 53, 77, 101, 6, 41, 76, 111, 146, 181, 7, 55, 103, 151, 199, 247, 295, 8, 71, 134, 197, 260, 323, 386, 449, 9, 89, 169, 249, 329, 409, 489, 569, 649, 10, 109, 208, 307, 406, 505, 604, 703, 802, 901, 11, 131, 251, 371, 491, 611

Offset: 1

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is A100104(n).

			Triangle begins:
  1;
  2,   5;
  3,  11,  19;
  4,  19,  34,  49;
  5,  29,  53,  77, 101;
  6,  41,  76, 111, 146, 181;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000578, A100104, A159797, A159798, A162610, A162611, A162612.

Cf. A100855 (row sums). - R. J. Mathar, Jul 20 2009

Table[NestList[#+n^2-1&,n,n-1],{n,11}]//Flatten (* Harvey P. Dale, Feb 24 2016 *)

More terms from Vincenzo Librandi, Aug 02 2010

A190717 Triplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 10, 10, 10, 20, 20, 20, 35, 35, 35, 56, 56, 56, 84, 84, 84, 120, 120, 120, 165, 165, 165, 220, 220, 220, 286, 286, 286, 364, 364, 364, 455, 455, 455, 560, 560, 560, 680, 680, 680, 816, 816, 816, 969, 969, 969

Offset: 0

Johannes W. Meijer, May 18 2011

The Ca1 and Ze3 triangle sums, see A180662 for their definitions, of the triangle A159797 are linear sums of shifted versions of the triplicated tetrahedral numbers, e.g. Ca1(n) = a(n-1) + a(n-2) + 2*a(n-3) + a(n-6).

The Ca1, Ca2, Ze3 and Ze4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), this sequence (triplicated), A190718 (quadruplicated), A049347, A144677.

A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A190717(n),n=0..50);

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,1,1,4,4,4,10,10,10,20},60] (* Harvey P. Dale, Mar 09 2018 *)

a(n) = binomial(floor(n/3)+3,3).
a(n) + a(n-1) + a(n-2) = A144677(n).
a(n) = Sum_{k=0..n} (A144677(n-k)*A049347(k)).
G.f.: 1/((x-1)^4*(x^2+x+1)^3).
Sum_{n>=0} 1/a(n) = 9/2. - Amiram Eldar, Aug 18 2022

A190718 Quadruplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20, 20, 20, 20, 35, 35, 35, 35, 56, 56, 56, 56, 84, 84, 84, 84, 120, 120, 120, 120, 165, 165, 165, 165, 220, 220, 220, 220, 286, 286, 286, 286, 364, 364, 364, 364, 455, 455, 455, 455

Offset: 0

Johannes W. Meijer, May 18 2011

The Gi1 triangle sums, for the definitions of these and other triangle sums see A180662, of the triangle A159797 are linear sums of shifted versions of the quadruplicated tetrahedral numbers A000292, i.e., Gi1(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-8).

The Gi1 and Gi2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,3,-3,0,0,-3,3,0,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), A190717 (triplicated).

A190718:= proc(n) binomial(floor(n/4)+3,3) end:
seq(A190718(n),n=0..52);

LinearRecurrence[{1,0,0,3,-3,0,0,-3,3,0,0,1,-1},{1,1,1,1,4,4,4,4,10,10,10,10,20},60] (* Harvey P. Dale, Oct 20 2012 *)

a(n) = binomial(floor(n/4)+3,3).
a(n-3) + a(n-2) + a(n-1) + a(n) = A144678(n).
a(n) = +a(n-1) +3*a(n-4) -3*a(n-5) -3*a(n-8) +3*a(n-9) +a(n-12) -a(n-13).
G.f.: 1 / ( (1+x)^3*(1+x^2)^3*(x-1)^4 ).
Sum_{n>=0} 1/a(n) = 6. - Amiram Eldar, Aug 18 2022

A144678 Related to enumeration of quantum states (see reference for precise definition).

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 95, 110, 125, 140, 161, 182, 203, 224, 252, 280, 308, 336, 372, 408, 444, 480, 525, 570, 615, 660, 715, 770, 825, 880, 946, 1012, 1078, 1144, 1222, 1300, 1378, 1456, 1547, 1638, 1729, 1820, 1925, 2030, 2135

Offset: 0

N. J. A. Sloane, Feb 06 2009

The Gi2 triangle sums of the triangle A159797 are linear sums of shifted versions of the sequence given above, i.e., Gi2(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). For the definitions of the Gi2 and other triangle sums see A180662. [Johannes W. Meijer, May 20 2011]
Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - R. J. Mathar, Aug 25 2013
Number of partitions of n into two different parts of size 4 and two different parts of size 1. a(4) = 7: 4, 4', 1111, 1111', 111'1', 11'1'1', 1'1'1'1'. - Alois P. Heinz, Dec 22 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=4]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Cf. A000292, A006918, A144677, A144679, A190718.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021

n:=80; lambda:=4; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b),snc];
od:
S10b;
A144678 := proc(n) option remember;
   local k;
   sum(A190718(n-k),k=0..3)
end:
A190718:= proc(n)
   binomial(floor(n/4)+3,3)
end:
seq(A144678(n),n=0..54); # Johannes W. Meijer, May 20 2011

a[n_] = (r = Mod[n, 4]; (4+n-r)(8+n-r)(3+n+2r)/96); Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Sep 02 2011 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1}, {1,2,3,4,7,10,13,16,22,28}, 60] (* G. C. Greubel, Oct 18 2021 *)

Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013

def A144678_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^4))^2 ).list()
A144678_list(60) # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n).
a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3).
G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End)
a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - R. J. Mathar, Jun 20 2013
a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014
From Alois P. Heinz, Dec 22 2021: (Start)
G.f.: 1/((1-x)*(1-x^4))^2.
a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)

A159787 a(n) = A159786(n+1)*3/4.

Original entry on oeis.org

0, 0, 3, 6, 6, 9, 24, 36, 36, 39, 48, 54, 57, 78, 132, 168, 168, 171, 180, 186, 189, 210, 252, 276, 279, 294, 318, 333, 360, 456, 648, 720, 720, 723, 732, 738, 741, 762, 804, 828, 831, 846, 870, 885, 912, 1008, 1152, 1224, 1227, 1242

Offset: 0

Omar E. Pol, Apr 28 2009, May 02 2009

nonn

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Toothpick sequence: A139250.

Cf. A139251, A153003, A159786, A159788, A159789, A159797.

a(11)-a(49) from Robert Price, May 10 2019

A050187 a(n) = n * floor((n-1)/2).

Original entry on oeis.org

0, 0, 0, 3, 4, 10, 12, 21, 24, 36, 40, 55, 60, 78, 84, 105, 112, 136, 144, 171, 180, 210, 220, 253, 264, 300, 312, 351, 364, 406, 420, 465, 480, 528, 544, 595, 612, 666, 684, 741, 760, 820, 840, 903, 924, 990, 1012, 1081, 1104, 1176

Offset: 0

Clark Kimberling, Dec 11 1999

T(n,2), array T as in A050186; a count of aperiodic binary words.
The Row2 triangle sums A159797 lead to the sequence given above for n >= 1 with a(1)=0. For the definitions of the Row2 and other triangle sums see A180662. - Johannes W. Meijer, May 20 2011
The number of chords joining n equally distributed points on a circle with a length less than the diameter. - Wesley Ivan Hurt, Nov 23 2013
a(n) is the maximum possible length of a circuit in the complete graph on n vertices. - Geoffrey Critzer, May 23 2014
For n > 0, a(n) is half the sum of the perimeters of the distinct rectangles that can be made with positive integer sides such that L + W = n, W < L. For example, a(14) = 84; the rectangles are 1 X 13, 2 X 12, 3 X 11, 4 X 10, 5 X 9, 6 X 8 (the 7 X 7 rectangle is not considered since we have W < L). The sum of the perimeters gives 28 + 28 + 28 + 28 + 28 + 28 = 168, half of which is 84. - Wesley Ivan Hurt, Nov 23 2017
Sum of the middle side lengths of all integer-sided triangles with perimeter 3n whose side lengths are in arithmetic progression (For example, when n=5 there are two triangles with perimeter 3(5) = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(5) = 5+5 = 10). - Wesley Ivan Hurt, Nov 01 2020

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

Cf. A050186, A159797, A180662.

[n*Floor((n-1)/2): n in [0..50]]; // Wesley Ivan Hurt, May 24 2014

A050187:=n->n*floor((n-1)/2); seq(A050187(n), n=0..100); # Wesley Ivan Hurt, Nov 23 2013

Table[n*Floor[(n-1)/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 23 2013 *)

a(n) = n * floor((n-1)/2).
From R. J. Mathar, Aug 08 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x^3*(3+x) / ((1+x)^2*(1-x)^3). (End)
a(n) = binomial(n,2) - (n/2) * ((n+1) mod 2). - Wesley Ivan Hurt, Nov 23 2013
E.g.f.: x*(x*cosh(x) + sinh(x)*(x - 1))/2. - Stefano Spezia, Nov 02 2020

Name change by Wesley Ivan Hurt, Nov 23 2013

A162615 Triangle read by rows in which row n lists n terms, starting with n, such that the difference between successive terms is equal to n^3 - 1 = A068601(n).

Original entry on oeis.org

1, 2, 9, 3, 29, 55, 4, 67, 130, 193, 5, 129, 253, 377, 501, 6, 221, 436, 651, 866, 1081, 7, 349, 691, 1033, 1375, 1717, 2059, 8, 519, 1030, 1541, 2052, 2563, 3074, 3585, 9, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833, 10, 1009, 2008, 3007, 4006, 5005, 6004

Offset: 1

Omar E. Pol, Jul 12 2009

See also the triangles of A162614 and A162616.

			Triangle begins:
  1;
  2,   9;
  3,  29,  55;
  4,  67, 130, 193;
  5, 129, 253, 377, 501;
  6, 221, 436, 651, 866, 1081;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162616.

A162615 := proc(n,k) n+(k-1)*(n^3-1) ; end proc: seq(seq(A162615(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2010

Flatten[Table[c=n^3-1;NestList[#+c&,n,n-1],{n,10}]] (* Harvey P. Dale, Nov 13 2011 *)

Row sums: n*(n^4 - n^3 + n + 1)/2. - R. J. Mathar, Jul 20 2009

Terms beyond the 6th row from R. J. Mathar and Max Alekseyev, Feb 05 2010

cp's OEIS Frontend

A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

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A162622 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^4 - 1.

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A162612 Triangle read by rows in which row n lists n terms, starting with n^2+n-1, with gaps = n^2-1 between successive terms.

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A162613 Triangle read by rows in which row n lists n terms, starting with n, with gaps = n^2-1 between successive terms.

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A190717 Triplicated tetrahedral numbers A000292.

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A190718 Quadruplicated tetrahedral numbers A000292.

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A144678 Related to enumeration of quantum states (see reference for precise definition).

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