A002817 -id:A002817 - cp's OEIS frontend

A302695 Number of 6-cycles in the (n+5)-path complement graph.

0, 5, 50, 265, 996, 2985, 7610, 17185, 35320, 67341, 120770, 205865, 336220, 529425, 807786, 1199105, 1737520, 2464405, 3429330, 4691081, 6318740, 8392825, 11006490, 14266785, 18295976, 23232925, 29234530, 36477225, 45158540, 55498721, 67742410, 82160385, 99051360

Offset: 0

Eric W. Weisstein, Apr 11 2018

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Path Complement Graph
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000292 (3-cycles of \bar P_{n+4}), A002817 (4-cycles of \bar P_{n+4}), A060446 (5-cycles of \bar P_{n+3}).

Table[n (4 + 22 n + 17 n^2 + 13 n^3 + 3 n^4 + n^5)/12, {n, 0, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {5, 50, 265, 996, 2985, 7610, 17185}, {0, 20}]
CoefficientList[Series[x (-5 - 15 x - 20 x^2 - 16 x^3 - 3 x^4 - x^5)/(-1 + x)^7, {x, 0, 20}], x]

a(n) = n*(4+22*n+17*n^2+13*n^3+3*n^4+n^5)/12; \\ Altug Alkan, Apr 12 2018

G.f.: x*(-5 - 15*x - 20*x^2 - 16*x^3 - 3*x^4 - x^5)/(-1 + x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
a(n) = n*(4 + 22*n + 17*n^2 + 13*n^3 + 3*n^4 + n^5)/12.

A321244 Non-isomorphic proper colorings of the 3 X 3 grid graph using at most n colors under rotational and reflectional symmetries.

Original entry on oeis.org

0, 2, 69, 1572, 19865, 153480, 830802, 3476144, 12003462, 35757630, 94780235, 228579252, 509929719, 1065625652, 2106541920, 3969848640, 7176749852, 12509692794, 21113614017, 34626453860, 55344881445, 86431928352, 132174030494, 198295824432, 292341936450, 424135940150, 606327641127, 855040875444, 1190635082147, 1638595028940

Offset: 1

Marko Riedel, Nov 01 2018

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.

Colin Barker, Table of n, a(n) for n = 1..1000
Marko Riedel et al., Tree graphs colorings, Math StackExchange, December 2017.
Marko Riedel et al., 3-colourings of a 3×3 table with one of 3 colors up to symmetries, Math StackExchange, October 2018.
Marko Riedel, Maple code for OCP computation by Burnside.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A002817, A321245, A321246.

[(1/8)*n^9-(3/2)*n^8+(33/4)*n^7-(53/2)*n^6+(217/4)*n^5-(291/4)*n^4 +(507/8)*n^3-(133/4)*n^2+8*n: n in [1..30]]; // Vincenzo Librandi, Nov 04 2018

CoefficientList[Series[x (2 + 49 x + 972 x^2 + 7010 x^3 + 17710 x^4 + 15273 x^5 + 4076 x^6 + 268 x^7) / (1 - x)^10, {x, 0, 30}], x] (* Vincenzo Librandi Nov 04 2018 *)

concat(0, Vec(x^2*(2 + 49*x + 972*x^2 + 7010*x^3 + 17710*x^4 + 15273*x^5 + 4076*x^6 + 268*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Nov 01 2018

a(n) = (1/8)*n^9 - (3/2)*n^8 + (33/4)*n^7 - (53/2)*n^6 + (217/4)*n^5 - (291/4)*n^4 + (507/8)*n^3 - (133/4)*n^2 + 8*n.
From Colin Barker, Nov 01 2018: (Start)
G.f.: x^2*(2 + 49*x + 972*x^2 + 7010*x^3 + 17710*x^4 + 15273*x^5 + 4076*x^6 + 268*x^7) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>10.
(End)

A047489 Numbers that are congruent to {1, 2, 3, 5, 7} mod 8.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 63, 65, 66, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 87, 89

Offset: 1

N. J. A. Sloane

Numbers n such that A002817(n) is divisible by n. [Bruno Berselli, Dec 11 2013]

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

Cf. A002817.

G.f.: x*(x+1)*(x^4+x^3+x^2+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). [Colin Barker, Jun 22 2012]

A125777 Moessner triangle based on A000217.

Original entry on oeis.org

1, 3, 6, 13, 28, 21, 69, 161, 137, 55, 433, 1078, 1017, 477, 120, 3141, 8245, 8437, 4460, 1337, 231, 25873, 71008, 77620, 45058, 15415, 3220, 406, 238629, 680451, 786012, 492264, 186729, 44955, 6930, 666, 2436673, 7184170, 8699205, 5804448, 2394150

Offset: 1

Gary W. Adamson, Dec 07 2006

Begin with the triangular numbers A000217 and circle every T(k)-th term, getting the doubly triangular numbers, A002817. Per instructions shown in A125714, take partial sums of the uncircled terms in row 1, denoting this as row 2. Circle the row 2 terms which are one place to the left of row 1 terms. Take partial sums again in analogous operations for subsequent rows.

Left border = A104989: (1, 3, 13, 69, 433...). Right border = the doubly triangular numbers starting (1, 6, 21...): A002817.

			First few rows of the triangle are as follows:
    1;
    3,    6;
   13,   28,   21;
   69,  161,  137,  55;
  433, 1078, 1017, 477, 120;
  ...

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.

Joshua Zucker, Table of n, a(n) for n = 1..55
G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
M. Niqui and J. J. M. M. Rutten, A proof of Moessner's theorem by coinduction, High.-Order Symb. Comput. 24(3) (2011), 191-206.
Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Cf. A125714, A002817, A104989, A000217.

More terms from Joshua Zucker, Jun 17 2007

A178238 Triangle read by rows: partial column sums of the triangle of natural numbers (written sequentially by rows).

Original entry on oeis.org

1, 3, 3, 7, 8, 6, 14, 16, 15, 10, 25, 28, 28, 24, 15, 41, 45, 46, 43, 35, 21, 63, 68, 70, 68, 61, 48, 28, 92, 98, 101, 100, 94, 82, 63, 36, 129, 136, 140, 140, 135, 124, 106, 80, 45, 175, 183, 188, 189, 185, 175, 158, 133, 99, 55, 231, 240, 246, 248, 245, 236, 220, 196, 163, 120, 66

Offset: 1

Gary W. Adamson, May 23 2010

T(n,k) is the n-th partial sum of the k-th column of the triangle of natural numbers.

			First few rows of the triangle:
    1;
    3,   3;
    7,   8,   6;
   14,  16,  15,  10;
   25,  28,  28,  24,  15;
   41,  45,  46,  43,  35,  21;
   63,  68,  70,  68,  61,  48,  28;
   92,  98, 101, 100,  94,  82,  63,  36;
  129, 136, 140, 140, 135, 124, 106,  80,  45;
  175, 183, 188, 189, 185, 175, 158, 133,  99,  55;
  231, 240, 246, 248, 245, 236, 220, 196, 163, 120,  66;
  298, 308, 314, 318, 316, 308, 293, 270, 238, 196, 143, 78;
  ...
These are the partial sums of the columns of the triangle:
  1;
  2, 3;
  4, 5, 6;
  7, 8, 9, 10;
  ...
For example, T(4,2) = 3 + 5 + 8 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A004006.
Main diagonal is A000217.
Row sums are A002817.
Cf. A000012, A000027.

T(n,k) = {binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1)}
{ for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) } \\ Andrew Howroyd, Apr 18 2021

As infinite lower triangular matrices, A000012 * A000027.
From Andrew Howroyd, Apr 18 2021: (Start)
T(n,k) = Sum_{j=k..n} (k + j*(j-1)/2).
T(n,k) = binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1).
T(2*n, n) = A255211(n).
(End)

Name changed and terms a(56) and beyond from Andrew Howroyd, Apr 18 2021

A259473 Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 14, 87, 148, 87, 14, 1, 1, 103, 4306, 63110, 388615, 1115068, 1575669, 1115068, 388615, 63110, 4306, 103, 1, 1, 694, 184015, 15902580, 567296265, 9816969306, 91422589980, 490333468494, 1583419977390, 3166404385990, 3982599815746, 3166404385990

Offset: 1

N. J. A. Sloane, Jul 03 2015

The n-th row of A257493 is a polynomial of degree (n-1)^2. This triangle gives the coefficients of the numerator of the generating functions for A257493 with denominators being (1-x)^(1+(n-1)^2). - Andrew Howroyd, Apr 11 2020

			Triangle begins:
  1;
  1;
  1,1,1;
  1,14,87,148,87,14,1;
  1,103,4306,63110,388615,1115068,1575669,1115068,388615,63110,4306,103,1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..177 (rows 1..9)
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477, doi:10.1137/0204040.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)

Row sums are A037302.

Cf. A005466, A005467, A257493.

T(n,k) = Sum_{i=0..k} A257493(n, k-i)*(-1)^i*binomial(1+(n-1)^2,i). - Andrew Howroyd, Apr 11 2020

a(1)=1 prepended and terms a(26) and beyond from Andrew Howroyd, Apr 11 2020

A264891 a(n) = n(5n - 3)(25n^2 - 15*n - 6)/8.

Original entry on oeis.org

0, 1, 112, 783, 2839, 7480, 16281, 31192, 54538, 89019, 137710, 204061, 291897, 405418, 549199, 728190, 947716, 1213477, 1531548, 1908379, 2350795, 2865996, 3461557, 4145428, 4925934, 5811775, 6812026, 7936137, 9193933, 10595614, 12151755, 13873306

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly heptagonal numbers.

G. C. Greubel, Table of n, a(n) for n = 0..1200
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000566, A002817, A000583, A232713, A063249.

[n*(5*n-3)*(25*n^2-15*n-6)/8: n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

seq(n*(5*n - 3)*(25*n^2 - 15*n - 6)/8, n=0..100); # Robert Israel, Dec 02 2015

Table[n (5 n - 3) (25 n^2 - 15 n - 6)/8, {n, 0, 35}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,112,783,2839},40] (* Harvey P. Dale, Nov 19 2019 *)

vector(100, n, n--; n*(5*n-3)*(25*n^2-15*n-6)/8) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 107*x + 233*x^2 + 34*x^3)/(1 - x)^5.
a(n) = A000566(A000566(n)).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 28 2015
Sum_{n>0} 1/a(n) = (4*(sqrt(33)*gamma + sqrt(33)*polygamma(0, 2/5) - 3*polygamma(0, (1/10)*(7 - sqrt(33))) + 3 polygamma(0, (1/10)* (7 + sqrt(33)))))/(9*sqrt(33)) = 1.0108420043...., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.
E.g.f.: x*(8 + 440*x + 600*x^2 + 125*x^3)*exp(x)/8, - Robert Israel, Dec 02 2015

A264892 a(n) = n(3n - 2)(9n^2 - 6*n - 2).

Original entry on oeis.org

0, 1, 176, 1281, 4720, 12545, 27456, 52801, 92576, 151425, 234640, 348161, 498576, 693121, 939680, 1246785, 1623616, 2080001, 2626416, 3273985, 4034480, 4920321, 5944576, 7120961, 8463840, 9988225, 11709776, 13644801, 15810256, 18223745, 20903520

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly octagonal numbers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Octagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000567, A002817, A000583, A232713, A063249.

[n*(3*n-2)*(9*n^2-6*n-2): n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Table[n (3 n - 2) (9 n^2 - 6 n - 2), {n, 0, 30}]

concat(0, Vec(x*(1+171*x+411*x^2+65*x^3)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 171*x + 411*x^2 + 65*x^3)/(1 - x)^5.
a(n) = A000567(A000567(n)).
Sum_{n>0} 1/a(n) = (sqrt(3)*gamma + sqrt(3)*polygamma(0, 1/3) - polygamma(0, (1/3)*(2 - sqrt(3))) + polygamma(0, (1/3)*(2 + sqrt(3))))/(4*sqrt(3)) = 1.006842786293...,where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

A294259 a(n) = n(n^3 + 2n^2 - 5*n + 10)/8.

Original entry on oeis.org

0, 1, 4, 15, 43, 100, 201, 364, 610, 963, 1450, 2101, 2949, 4030, 5383, 7050, 9076, 11509, 14400, 17803, 21775, 26376, 31669, 37720, 44598, 52375, 61126, 70929, 81865, 94018, 107475, 122326, 138664, 156585, 176188, 197575, 220851, 246124, 273505, 303108, 335050, 369451

Offset: 0

Bruno Berselli, Oct 30 2017

a(n) is even for n in A047481.

Also, a(n) is divisible by 5 if and only if n belongs to A047218.

			After 0:
1   =                     -(0) + (1);
4   =                 -(0 + 1) + (2 + 2*3/2);
15  =             -(0 + 1 + 2) + (3 + 4 + 5 + 3*4/2);
43  =         -(0 + 1 + 2 + 3) + (4 + 5 + 6 + 7 + 8 + 9 + 4*5/2);
100 =     -(0 + 1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + ... + 14 + 5*6/2);
201 = -(0 + 1 + 2 + 3 + 4 + 5) + (6 + 7 + 8 + 9 + ... + 20 + 6*7/2), etc.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A002817, A176145.
Cf. A101374: the sums in the Example section end in squares.
Subsequence of A047207.

List([0..50], n -> n*(n^3+2*n^2-5*n+10)/8);

[n*(n^3+2*n^2-5*n+10)/8: n in [0..50]];

a := n -> n*(n*(n*(n+2)-5)+10)/8: seq(a(n),n=0..41); # Peter Luschny, Nov 06 2017

Table[n (n^3 + 2 n^2 - 5 n + 10)/8, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,4,15,43},50] (* Harvey P. Dale, Jan 08 2024 *)

makelist(n*(n^3+2*n^2-5*n+10)/8, n, 0, 50);

vector(50, n, n--; n*(n^3+2*n^2-5*n+10)/8)

[n*(n^3+2*n^2-5*n+10)/8 for n in range(50)]

O.g.f.: x*(1 - x + 5*x^2 - 2*x^3)/(1 - x)^5.
E.g.f.: x*(8 + 8*x + 8*x^2 + x^3)*exp(x)/8.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
a(n) = 2*n + Sum_{i=0..n} i*(i^2 - 3)/2.

A321245 Non-isomorphic proper colorings of the 4 X 4 grid graph using at most n colors under rotational and reflectional symmetries.

Original entry on oeis.org

0, 1, 1155, 759759, 103786510, 4767856260, 107118740001, 1465350136810, 13956101513964, 100946621623995, 588405869207695, 2882842751900001, 12245455022841690, 46164185630256694, 157281327978056205, 491245336843482180, 1422828159652548376, 3857444027819847045, 9864873410828916699, 23951146853875652515, 55509091777214287590

Offset: 1

Marko Riedel, Nov 01 2018

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Tree graphs colorings, Math StackExchange, December 2017.
Marko Riedel et al., 3-colourings of a 3×3 table with one of 3 colors up to symmetries, Math StackExchange, October 2018.

Cf. A002817, A321244, A321246.

a(n) = (1/8)*n^16 - 3*n^15 + (69/2)*n^14 - (2015/8)*n^13 + (10437/8)*n^12 - (20307/4)*n^11 + 15333*n^10 - (292907/8)*n^9 + (557915/8)*n^8 - (848501/8)*n^7 + (1023195/8)*n^6 - (240539/2)*n^5 + (683997/8)*n^4 - (347485/8)*n^3 + (112831/8)*n^2 - (8807/4)*n.

cp's OEIS Frontend

A302695 Number of 6-cycles in the (n+5)-path complement graph.

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A321244 Non-isomorphic proper colorings of the 3 X 3 grid graph using at most n colors under rotational and reflectional symmetries.

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A047489 Numbers that are congruent to {1, 2, 3, 5, 7} mod 8.

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A125777 Moessner triangle based on A000217.

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A178238 Triangle read by rows: partial column sums of the triangle of natural numbers (written sequentially by rows).

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A259473 Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).

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A264891 a(n) = n*(5*n - 3)*(25*n^2 - 15*n - 6)/8.

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A264892 a(n) = n*(3*n - 2)*(9*n^2 - 6*n - 2).

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A264891 a(n) = n(5n - 3)(25n^2 - 15*n - 6)/8.

A264892 a(n) = n(3n - 2)(9n^2 - 6*n - 2).

A294259 a(n) = n(n^3 + 2n^2 - 5*n + 10)/8.