A001753 -id:A001753 - cp's OEIS frontend

A175111 a(n) = ((2*n+1)^5+(-1)^n)/2.

1, 121, 1563, 8403, 29525, 80525, 185647, 379687, 709929, 1238049, 2042051, 3218171, 4882813, 7174453, 10255575, 14314575, 19567697, 26260937, 34671979, 45112099, 57928101, 73504221, 92264063, 114672503, 141237625, 172512625

Offset: 0

R. J. Mathar, Feb 13 2010

Partial sums of A175112.

Convolution of the finite sequence 1,116,967,1672,967,116,1 with A001753.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

I:=[1, 121, 1563, 8403, 29525, 80525, 185647]; [n le 7 select I[n] else 5*Self(n-1) - 9*Self(n-2) + 5*Self(n-3) + 5*Self(n-4) - 9*Self(n-5) + 5*Self(n-6) - Self(n-7): n in [1..40]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[(1 + 116*x + 967*x^2 + 1672*x^3 + 967*x^4 + 116*x^5 + x^6)/((1 + x)*(x - 1)^6), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,121,1563,8403,29525,80525,185647},50] (* Harvey P. Dale, May 30 2014 *)

a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7).

G.f: (1+116*x+967*x^2+1672*x^3+967*x^4+116*x^5+x^6)/((1+x)*(x-1)^6).

A176646 a(n) is the number of convex pentagons in an n-triangular net.

Original entry on oeis.org

0, 0, 3, 21, 78, 216, 498, 1014, 1884, 3264, 5349, 8379, 12642, 18480, 26292, 36540, 49752, 66528, 87543, 113553, 145398, 184008, 230406, 285714, 351156, 428064, 517881, 622167, 742602, 880992, 1039272, 1219512, 1423920, 1654848, 1914795, 2206413, 2532510, 2896056

Offset: 1

Jonathan Vos Post, Dec 21 2010

See P(n) in Theorem 2.1, p.2 of Zhu.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jun-Ming Zhu, The number of convex pentagons and hexagons in an n-triangular net, arXiv:1012.4058 [math.CO], 2010. See P(n), formula (1), page 2.
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Cf. A166189 (for the hexagons).

[(1/320)*(12*n^5 -10*n^4 -60*n^3 +40*n^2 +48*n -15 +15*(-1)^n): n in [1..40]]; // G. C. Greubel, Jul 02 2021

A176646:= n-> (12*n^5 -10*n^4 -60*n^3 +40*n^2 +48*n -15 +15*(-1)^n)/320;
seq(A176646(n), n=1..40); # R. J. Mathar, Dec 21 2010

LinearRecurrence[{5,-9,5,5,-9,5,-1}, {0,0,3,21,78,216,498}, 40] (* Harvey P. Dale, Jan 14 2015 *)

f(k) = (12*k^5 + 25*k^4 + 5*k^3 - 10*k^2 - 2*k)/10;
g(k) = (12*k^5 - 5*k^4 - 15*k^3 + 5*k^2 + 3*k)/10;
a(n) = if (n%2, f((n-1)/2), g(n/2)); \\ Michel Marcus, Jul 04 2021

[(1/320)*(12*n^5 -10*n^4 -60*n^3 +40*n^2 +48*n -15 +15*(-1)^n) for n in (1..40)] # G. C. Greubel, Jul 02 2021

From G. C. Greubel, Jul 03 2021: (Start)
a(n) = (1/320)*(12*n^5 - 10*n^4 - 60*n^3 + 40*n^2 + 48*n - 15 + 15*(-1)^n).
a(2*n+1) = n*(n+1)*(12*n^3 + 13*n^2 - 8*n - 2)/10.
a(2*n) = n*(4*n-3)*(3*n+1)*(n-1)*(n+1)/10.
G.f.: 3*x^3*(1 + 2*x)/((1 + x)*(1 - x)^6).
E.g.f.: (1/320)*(15*exp(-x) - (15 -30*x +30*x^2 -180*x^3 -110*x^4 -12*x^5)*exp(x)). (End)
a(n) = 3*A001753(n-3) + 6*A001753(n-4). - R. J. Mathar, Jul 04 2021

Definition corrected and edited by Michel Marcus and G. C. Greubel, Jul 03 2021

A054497 Number of symmetric nonnegative integer 7 X 7 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

Original entry on oeis.org

1, 7, 31, 105, 300, 756, 1732, 3676, 7330, 13870, 25102, 43714, 73612, 120340, 191620, 298012, 453739, 677677, 994565, 1436435, 2044328, 2870296, 3979768, 5454280, 7394660, 9924668, 13195196, 17389028, 22726280, 29470520, 37935704, 48493928, 61584149

Offset: 0

Vladeta Jovovic, May 14 2000

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,14,25,-63,36,36,-63,25,14,-18,7,-1).

Cf. A001753, A038163.

CoefficientList[Series[1/((1-x)^7(1-x^2)^3),{x,0,30}],x] (* Harvey P. Dale, Feb 03 2012 *)

Vec(1 / ((1-x)^7*(1-x^2)^3) + O(x^40)) \\ Colin Barker, Jan 15 2017

G.f.: 1 / ((1-x)^7 * (1-x^2)^3).
a(0)=1, a(1)=7, a(2)=31, a(3)=105, a(4)=300, a(5)=756, a(6)=1732, a(7)=3676, a(8)=7330, a(9)=13870, a(10)=25102, a(11)=43714, a(12)=73612, a(n) = 7*a(n-1) - 18*a(n-2) + 14*a(n-3) + 25*a(n-4) - 63*a(n-5) + 36*a(n-6) + 36*a(n-7) - 63*a(n-8) + 25*a(n-9) + 14*a(n-10) - 18*a(n-11) + 7*a(n-12) - a(n-13). - Harvey P. Dale, Feb 03 2012
a(n) = ((2835*(4017+79*(-1)^n) + 18*(1513217+4095*(-1)^n)*n + 90*(284609+63*(-1)^n)*n^2 + 12771104*n^3 + 3787056*n^4 + 701400*n^5 + 81900*n^6 + 5856*n^7 + 234*n^8 + 4*n^9)) / 11612160. - Colin Barker, Jan 15 2017

A054549 Number of symmetric nonnegative integer 9 X 9 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

Original entry on oeis.org

1, 9, 51, 219, 786, 2466, 6974, 18126, 43929, 100321, 217683, 451707, 901128, 1735752, 3239928, 5878328, 10393902, 17950878, 30340682, 50273658, 81787476, 130811124, 205935756, 319456044, 488764246, 738197766, 1101468114, 1624826306, 2371158504, 3425244456

Offset: 0

Vladeta Jovovic, May 14 2000

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-30,30,75,-243,152,360,-690,130,780,-780,-130,690,-360,-152,243,-75,-30,30,-9,1).

Cf. A001753, A038163.

Vec(1 / ((1-x)^9*(1-x^2)^6) + O(x^40)) \\ Colin Barker, Jan 15 2017

G.f.: 1 / ((1-x)^9 * (1-x^2)^6).

a(n) = ((85135050*(256719+5425*(-1)^n) + 1890*(30592018355+141137997*(-1)^n)*n + 9*(7020005494399+6456074625*(-1)^n)*n^2 + 15288*(2534162507+393525*(-1)^n)*n^3 + 455*(33274853083+654885*(-1)^n)*n^4 + 126126*(31995443+45*(-1)^n)*n^5 + 763179408992*n^6 + 104737240608*n^7 + 10538322080*n^8 + 777020244*n^9 + 41479438*n^10 + 1559376*n^11 + 39130*n^12 + 588*n^13 + 4*n^14)) / 22317642547200. - Colin Barker, Jan 15 2017

cp's OEIS Frontend

A175111 a(n) = ((2*n+1)^5+(-1)^n)/2.

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A176646 a(n) is the number of convex pentagons in an n-triangular net.

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A054497 Number of symmetric nonnegative integer 7 X 7 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

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A054549 Number of symmetric nonnegative integer 9 X 9 matrices with sum of elements equal to 4*n, under action of dihedral group D_4.

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