A092498 -id:A092498 - cp's OEIS frontend

A134507 Number of rectangles in a pyramid built with squares. The squares counted in A092498 are excluded.

0, 4, 19, 57, 134, 269, 486, 813, 1281, 1926, 2788, 3910, 5340, 7130, 9335, 12015, 15234, 19059, 23562, 28819, 34909, 41916, 49928, 59036, 69336, 80928, 93915, 108405, 124510, 142345, 162030, 183689, 207449, 233442, 261804, 292674, 326196

Offset: 1

Philippe Lallouet (philip.lallouet(AT)orange.fr), Jan 19 2008

At the first step, the pyramid contains only one unitary square. At each step of rank n we add a row of 2*n-1 squares below the previous pyramid. The sequence is the number of rectangles of any size which can be seen in this pyramid of height n.
..........._.
|..|.........|..|
||......_|__|
..........|..|..|..|
..0.......||__|| 3 rectangles 2X1, 1 rectangle 3X1

			G.f. = 4*x^2 + 19*x^3 + 57*x^4 + 134*x^5 + 269*x^6 + 486*x^7 + 813*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Cf. A092498.

I:=[0,4,19,57,134,269,486]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Mar 01 2014

a[ n_] := SeriesCoefficient[ x^2 (4 + 3 x + 5 x^2) / ((1 - x)^5 (1 + x + x^2)), {x, 0, n}]; (* Michael Somos, Feb 25 2014 *)
a[ n_] := Quotient[ 3 n^4 + 5 n^3 - 3 n^2 - 3 n + 2, 18]; (* Michael Somos, Feb 25 2014 *)
CoefficientList[Series[-x (5 x^2 + 3 x + 4)/((x - 1)^5 (x^2 + x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi Mar 01 2014 *)

{a(n) = (3*n^4 + 5*n^3 - 3*n^2 - 3*n + 2) \ 18}; /* Michael Somos, Feb 17 2008 */

For n == 0 mod 3, a(n) = n*(3*n^3+5*n^2-3*n-3)/18; for n == 1 mod 3, a(n) = (n-1)*(3*n^3+8*n^2+5*n+2)/18; for n == 2 mod 3, a(n) = (3*n^4+5*n^3-3*n^2-3*n+2)/18. [corrected and edited by Michel Marcus, Apr 09 2024]
G.f.: -x^2*(5*x^2+3*x+4)/((x-1)^5*(x^2+x+1)). [Colin Barker, Nov 16 2012]
a(n) = (3*n^4+5*n^3-3*n^2-5*n+6*floor((n+1)/3))/18. - Luce ETIENNE, Jul 31 2015

A260918 Number of squares of all sizes in polyominoes obtained by union of two pyramidal figures (A092498) with intersection equals A002623.

Original entry on oeis.org

0, 1, 5, 15, 33, 60, 100, 154, 224, 313, 423, 555, 713, 898, 1112, 1358, 1638, 1953, 2307, 2701, 3137, 3618, 4146, 4722, 5350, 6031, 6767, 7561, 8415, 9330, 10310, 11356, 12470, 13655, 14913, 16245, 17655, 19144, 20714, 22368, 24108, 25935, 27853, 29863

Offset: 0

Luce ETIENNE, Aug 04 2015

The resulting polyforms are n*(3*n-1)/2-polyominoes.
Also they are 6*n-gons with n>1.
Schäfli's notation for figure corresponding to a(1): 4.

			a(1)=1, a(2)=5, a(3)=12+3=15, a(4)=22+9+2=33, a(5)=35+18+7=60, a(6)=51+30+15+4=100.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A002623, A092498, A173196, A000212, A258440.

[(52*n^3+90*n^2+20*n-3*(32*Floor((n+1)/3)+3*(1-(-1)^n)))/144: n in [0..50]]; // Vincenzo Librandi, Aug 12 2015

Table[(52 n^3 + 90 n^2 + 20 n - 3 (32 Floor[(n + 1) / 3] + 3 (1 - (-1)^n))) / 144, {n, 0, 45}] (* Vincenzo Librandi, Aug 12 2015 *)

concat(0, Vec(x*(4*x^3+5*x^2+3*x+1)/((x-1)^4*(x+1)*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Aug 08 2015

a(n) = A258440(n) - A000212(n+1).
a(n) = (1/8)*((Sum_{i=0..floor(2*n/3)} (4*n+1-6*i-(-1)^i)*(4*n-1-6*i+(-1)^i)) - (Sum_{j=0..(2*n-1+(-1)^n)/4} (2*n+1-(-1)^n-4*j)*(2*n+1+(-1)^n-4*j))).
a(n) = (52*n^3+90*n^2+20*n-3*(32*floor((n+1)/3)+3*(1-(-1)^n)))/144.
G.f.: x*(4*x^3+5*x^2+3*x+1) / ((x-1)^4*(x+1)*(x^2+x+1)). - Colin Barker, Aug 08 2015
E.g.f.: (3*exp(x)*x*(65 + x*(123 + 26*x)) + 32*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2) - 27*sinh(x))/216. - Stefano Spezia, Nov 15 2024

Two repeated terms deleted by Colin Barker, Aug 08 2015

A131177 Duplicate of A092498.

Original entry on oeis.org

1, 4, 11, 23, 41, 67, 102, 147, 204, 274, 358, 458, 575, 710, 865, 1041, 1239, 1461

Offset: 1

dead

A000969 Expansion of g.f. (1 + x + 2x^2)/((1 - x)^2(1 - x^3)).

Original entry on oeis.org

1, 3, 7, 12, 18, 26, 35, 45, 57, 70, 84, 100, 117, 135, 155, 176, 198, 222, 247, 273, 301, 330, 360, 392, 425, 459, 495, 532, 570, 610, 651, 693, 737, 782, 828, 876, 925, 975, 1027, 1080, 1134, 1190, 1247, 1305, 1365, 1426, 1488, 1552, 1617, 1683, 1751, 1820, 1890

Offset: 0

N. J. A. Sloane

From Paul Curtz, Oct 07 2018: (Start)
Terms that are on the x-axis of the following spiral (without 0):
     28--29--29--30--31--31--32
      |
     27  13--14--15--15--16--17
      |   |                   |
     27  13   4---5---5---6  17
      |   |   |           |   |
     26  12   3   0---1   7  18
      |   |   |       |   |   |
     25  11   3---2---1   7  19
      |   |               |   |
     25  11--10---9---9---8  19
      |                       |
     24--23--23--22--21--21--20  (End)
Diagonal 1, 4, 8, 13, 20, 28, ... (without 0) is A143978. - Bruno Berselli, Oct 08 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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.]
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A004773 (first differences), A092498 (partial sums).

Cf. A014105, A139250, A143978, A160165, A258708.

a000969 = flip div 3 . a014105 . (+ 1)  -- Reinhard Zumkeller, Jun 23 2015

[Floor(Binomial(2*n+3,2)/3): n in [0..60]]; // G. C. Greubel, Apr 18 2023

A000969:=-(1+z+2*z**2)/(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

f[x_, y_]:= Floor[Abs[y/x -x/y]]; Table[f[3, 2n^2+n+2], {n,53}] (* Robert G. Wilson v, Aug 11 2010 *)
CoefficientList[Series[(1+x+2*x^2)/((1-x)^2*(1-x^3)), {x, 0, 50}], x] (* Stefano Spezia, Oct 08 2018 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-2,1,-1,2]^n*[1;3;7;12;18])[1,1] \\ Charles R Greathouse IV, May 10 2016

[(binomial(2*n+3,2)//3) for n in range(61)] # G. C. Greubel, Apr 18 2023

a(n) = floor( (2*n+3)*(n+1)/3 ). Or, a(n) = (2*n+3)*(n+1)/3 but subtract 1/3 if n == 1 mod 3. - N. J. A. Sloane, May 05 2010
a(2^k-2) = A139250(2^k-1), k >= 1. - Omar E. Pol, Feb 13 2010
a(n) = Sum_{i=0..n} floor(4*i/3). - Enrique Pérez Herrero, Apr 21 2012
a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-3) -2*a(n-4) +1*a(n-5). - Joerg Arndt, Apr 22 2012
a(n) = A014105(n+1) = A258708(n+3,n). - Reinhard Zumkeller, Jun 23 2015
Sum_{n>=0} 1/a(n) = 6 - Pi/sqrt(3) - 10*log(2)/3. - Amiram Eldar, Oct 01 2022
E.g.f.: (exp(x)*(8 + 21*x + 6*x^2) + exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 05 2023

A014126 Number of partitions of 2*n into at most 4 parts.

Original entry on oeis.org

1, 2, 5, 9, 15, 23, 34, 47, 64, 84, 108, 136, 169, 206, 249, 297, 351, 411, 478, 551, 632, 720, 816, 920, 1033, 1154, 1285, 1425, 1575, 1735, 1906, 2087, 2280, 2484, 2700, 2928, 3169, 3422, 3689, 3969, 4263, 4571, 4894, 5231, 5584, 5952, 6336, 6736, 7153, 7586

Offset: 0

N. J. A. Sloane

Bisection of A001400.
Molien series for 4-dimensional group of structure S_4 X C_2 and order 48, arising from complete weight enumerators of even trace-Hermitian self-dual additive codes over GF(4) containing the all-ones vector.
Partial sums of A156040. - Bob Selcoe, Feb 08 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A000631, A001400, A014125, A049347, A092498, A156040.

with(combstruct): seq(count(Partition((2*n+4)), size=4), n=0..50); # Zerinvary Lajos, Mar 28 2008

CoefficientList[Series[(1 + x^2) / ((1 - x)^2 (1 - x^2) (1 - x^3)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,5,9,15,23,34},50] (* Harvey P. Dale, Aug 31 2015 *)

a(n)=(4*n^3+30*n^2+72*n+55+8*[1,-1,0][(n%3)+1]+9*(-1)^n)/72

G.f.: (1+x^2)/((1-x)^2*(1-x^2)*(1-x^3)). - James Sellers
a(n) = (1/72) * (4*n^3 + 30*n^2 + 72*n + 55 + 8*A049347(n) + 9*(-1)^n ). - Ralf Stephan, Aug 15 2013
E.g.f.: exp(-x)*(27 + 3*exp(2*x)*(55 + 106*x + 42*x^2 + 4*x^3) + 8*exp(x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/216. - Stefano Spezia, Apr 05 2023

A241526 Number of different positions in which a square with side length k, 1 <= k <= n - floor(n/3), can be placed within a bi-symmetric triangle of 1 X 1 squares of height n.

Original entry on oeis.org

2, 7, 16, 31, 53, 83, 123, 174, 237, 314, 406, 514, 640, 785, 950, 1137, 1347, 1581, 1841, 2128, 2443, 2788, 3164, 3572, 4014, 4491, 5004, 5555, 6145, 6775, 7447, 8162, 8921, 9726, 10578, 11478, 12428, 13429, 14482, 15589, 16751, 17969, 19245, 20580, 21975

Offset: 1

Christopher Hunt Gribble and Luce ETIENNE, Apr 24 2014

			The bi-symmetric triangle of 1 X 1 squares of height 5 is:
                   ___
                 _|_|_|_
               _|_|_|_|_|_
             _|_|_|_|_|_|_|_
           _|_|_|_|_|_|_|_|_|_
          |_|_|_|_|_|_|_|_|_|_|
.
No. of positions in which a 1 X 1 square can be placed = 2 + 4 + 6 + 8 + 10 = 30.
No. of positions in which a 2 X 2 square can be placed = 1 + 3 + 5 + 7 = 16.
No. of positions in which a 3 X 3 square can be placed = 2 + 4 = 6.
No. of positions in which a 4 X 4 square can be placed = 1.
Thus, a(5) = 30 + 16 + 6 + 1 = 53.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A092498.

a := proc (n::integer)::integer;
       (2/9)*n^3+(5/6)*n^2+(17/18)*n-(1/3)*floor((1/3)*n)
     end proc:
seq(a(n), n = 1..60);

Vec(x*(x^2+x+2)/((x-1)^4*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Apr 26 2014

a(n) = sum_{j=0..n-1-floor(n/3)} ((4*n-6*j+1-(-1)^j)/4)*((4*n-6*j+3+(-1)^j)/4).
a(n) = (4*n^3+15*n^2+17*n-6*floor(n/3))/18.
G.f.: x*(x^2+x+2) / ((x-1)^4*(x^2+x+1)). - Colin Barker, Apr 26 2014

A231180 Let A={2,3,6,8,9,11,14,...} be the sequence of numbers k>=1 such that k+5 is odious (A000069). Let B be the complement of A. The sequence lists numbers for which the number of A-divisors equals the number of B-divisors.

Original entry on oeis.org

1, 4, 9, 16, 36, 121, 144, 289, 441, 484, 529, 1156, 1369, 1600, 1764, 2025, 2116, 2209, 3249, 3481, 4624, 5041, 5476, 6241, 6889, 7056, 7569, 7921, 8100, 8464, 8649, 8836, 11449, 12321, 12769, 12996, 13924, 14641, 15129, 16641, 20164, 24336, 24649, 24964

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Nov 05 2013

All terms are perfect squares.

			n=16 has 4 proper divisors {1,2,4,8} from which 2 from A {2,8} and 2 from B {1,4}. So 16 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, A set of sequences of perfect squares

Cf. A227891, A231175, A231176, A092498, A231178, A000005, A000069.

odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]];
Select[Range[200],0==Length[#]-2Length[Select[#,odiousQ[#+5]&]]&[Most[Divisors[#^2]]]&]^2 (* Peter J. C. Moses, Nov 08 2013 *)

cp's OEIS Frontend

A134507 Number of rectangles in a pyramid built with squares. The squares counted in A092498 are excluded.

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A260918 Number of squares of all sizes in polyominoes obtained by union of two pyramidal figures (A092498) with intersection equals A002623.

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A131177 Duplicate of A092498.

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A000969 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).

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A014126 Number of partitions of 2*n into at most 4 parts.

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A241526 Number of different positions in which a square with side length k, 1 <= k <= n - floor(n/3), can be placed within a bi-symmetric triangle of 1 X 1 squares of height n.

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A231180 Let A={2,3,6,8,9,11,14,...} be the sequence of numbers k>=1 such that k+5 is odious (A000069). Let B be the complement of A. The sequence lists numbers for which the number of A-divisors equals the number of B-divisors.

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A000969 Expansion of g.f. (1 + x + 2x^2)/((1 - x)^2(1 - x^3)).