A004126 -id:A004126 - cp's OEIS frontend

A063521 a(n) = n(7n^2-4)/3.

0, 1, 16, 59, 144, 285, 496, 791, 1184, 1689, 2320, 3091, 4016, 5109, 6384, 7855, 9536, 11441, 13584, 15979, 18640, 21581, 24816, 28359, 32224, 36425, 40976, 45891, 51184, 56869, 62960, 69471, 76416, 83809, 91664, 99995, 108816, 118141, 127984, 138359, 149280, 160761

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(14*n^3-8*n), n>0: structured heptagonal anti-diamond numbers (vertex structure 15) (Cf. A100186 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

A063521:=n->n*(7*n^2-4)/3; seq(A063521(k), k=0..100); # Wesley Ivan Hurt, Oct 24 2013

lst={};Do[AppendTo[lst, n*(7*n^2-4)/3], {n, 1, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[x*(1+12*x+x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)

a(n) = { n*(7*n^2 - 4)/3 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1+12*x+x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/3)*(3 + 21*x + 7*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A063522 a(n) = n(5n^2 - 3)/2.

Original entry on oeis.org

0, 1, 17, 63, 154, 305, 531, 847, 1268, 1809, 2485, 3311, 4302, 5473, 6839, 8415, 10216, 12257, 14553, 17119, 19970, 23121, 26587, 30383, 34524, 39025, 43901, 49167, 54838, 60929, 67455, 74431, 81872, 89793, 98209, 107135, 116586, 126577, 137123, 148239, 159940

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

(1/12)*t*(n^3 - n) + n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Bisections: A160674, A160699.

[n*(5*n^2 -3)/2: n in [0..30]]; // G. C. Greubel, May 02 2018

lst={};Do[AppendTo[lst, LegendreP[3, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[x*(1 + 13*x + x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)
LinearRecurrence[{4,-6,4,-1},{0,1,17,63},40] (* Harvey P. Dale, Sep 06 2023 *)

a(n) = { n*(5*n^2 - 3)/2 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1 + 13*x + x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/2)*(2 + 15*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A004467 a(n) = n(11n^2 - 5)/6.

Original entry on oeis.org

0, 1, 13, 47, 114, 225, 391, 623, 932, 1329, 1825, 2431, 3158, 4017, 5019, 6175, 7496, 8993, 10677, 12559, 14650, 16961, 19503, 22287, 25324, 28625, 32201, 36063, 40222, 44689, 49475, 54591, 60048

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers, that is: centered hendecagonal pyramidal numbers (see Deza paper in References).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

[n*(11*n^2-5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Table[n(11n^2-5)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,13,47},80] (* Harvey P. Dale, Sep 22 2013 *)

a(n)=n*(11*n^2-5)/6 \\ Charles R Greathouse IV, Sep 28 2011

G.f.: x*(1+9*x+x^2)/(1-x)^4. - Colin Barker, Jan 08 2012
a(0)=0, a(1)=1, a(2)=13, a(3)=47; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2013
E.g.f.: (x/6)*(6 + 33*x + 11*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A062025 a(n) = n(13n^2 - 7)/6.

Original entry on oeis.org

0, 1, 15, 55, 134, 265, 461, 735, 1100, 1569, 2155, 2871, 3730, 4745, 5929, 7295, 8856, 10625, 12615, 14839, 17310, 20041, 23045, 26335, 29924, 33825, 38051, 42615, 47530, 52809, 58465, 64511, 70960, 77825, 85119, 92855, 101046, 109705, 118845, 128479, 138620, 149281

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Table[n(13n^2-7)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n) = n*(13*n^2 - 7)/6 \\ Harry J. Smith, Jul 29 2009

From G. C. Greubel, Sep 01 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (x/6)*(6 + 39*x + 13*x^2)*exp(x). (End)

A063523 a(n) = n(8n^2 - 5)/3.

Original entry on oeis.org

0, 1, 18, 67, 164, 325, 566, 903, 1352, 1929, 2650, 3531, 4588, 5837, 7294, 8975, 10896, 13073, 15522, 18259, 21300, 24661, 28358, 32407, 36824, 41625, 46826, 52443, 58492, 64989, 71950, 79391, 87328, 95777, 104754, 114275, 124356, 135013, 146262, 158119, 170600

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(16*n^3-10*n), n>0: structured octagonal anti-diamond numbers (vertex structure 17) (Cf. A100187 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Table[n(8n^2-5)/3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,67},81] (* or *) CoefficientList[ Series[ (x+14 x^2+x^3)/(x-1)^4,{x,0,80}],x] (* Harvey P. Dale, Jul 11 2011 *)

a(n) = n*(8*n^2 - 5)/3 \\ Harry J. Smith, Aug 25 2009

a(0)=0, a(1)=1, a(2)=18, a(3)=67, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 11 2011
G.f.: (x+14*x^2+x^3)/(x-1)^4. - Harvey P. Dale, Jul 11 2011
E.g.f.: (x/3)*(3 + 24*x + 8*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A322640 Numbers that are sums of consecutive centered heptagonal numbers (A069099).

Original entry on oeis.org

0, 1, 8, 9, 22, 30, 31, 43, 65, 71, 73, 74, 106, 114, 136, 144, 145, 148, 177, 197, 220, 242, 250, 251, 253, 254, 316, 325, 345, 368, 386, 390, 398, 399, 450, 451, 463, 522, 547, 565, 569, 587, 595, 596, 598, 638, 702, 704, 736, 766, 775, 818, 840, 841, 848, 849, 914, 953

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Wikipedia, Centered heptagonal number

Cf. A004126, A069099, A152043, A322610, A322611, A322636, A322638, A320728.

terms = 58;
nmax = 17; kmax =  8; (* empirical *)
T = Table[(7 n^2 - 7 n + 2)/2, {n, 1, nmax}];
Union[{0}, T, Table[k MovingAverage[T, k], {k, 2, kmax}] // Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 27 2018 *)

A100179 Structured heptagonal diamond numbers (vertex structure 5).

Original entry on oeis.org

1, 9, 34, 86, 175, 311, 504, 764, 1101, 1525, 2046, 2674, 3419, 4291, 5300, 6456, 7769, 9249, 10906, 12750, 14791, 17039, 19504, 22196, 25125, 28301, 31734, 35434, 39411, 43675, 48236, 53104

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A004126 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.

[(1/6)*(10*n^3-9*n^2+5*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(10n^3-9n^2+5n)/6,{n,40}] (* Harvey P. Dale, Oct 28 2018 *)

vector(40, n, (10*n^3-9*n^2+5*n)/6) \\ G. C. Greubel, Nov 08 2018

a(n) = (1/6)*(10*n^3 - 9*n^2 + 5*n).
G.f.: x*(1 + 5*x + 4*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Oct 28 2018
E.g.f.: (6*x + 21*x^2 + 10*x^3)*exp(x)/6. - G. C. Greubel, Nov 08 2018

A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

Original entry on oeis.org

6, 6, 3, 30, 48, 57, 54, 45, 30, 12, 72, 126, 165, 186, 195, 186, 168, 138, 102, 66, 27, 132, 240, 327, 390, 435, 456, 462, 444, 414, 366, 309, 246, 177, 114, 48, 210, 390, 543, 666, 765, 834, 882, 900, 900, 870, 825, 756, 675, 582, 480, 378, 270, 174, 75

Offset: 1

Emeric Deutsch, Aug 31 2012

The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.
Row n contains 4n-1 entries.
T(n,1) = 9n^2-3n = A152743(n).
T(n,2) = 6n(3n-2)= A153796(n).
T(n,3) = 3(9n^2-9n+1)= 3*A069131(n) (for n>5 this is a conjecture).
T(n,2n) = n(7n^2-1) = 6*A004126(n) (for n>5 this is a conjecture).
T(n,4n-2) = 6(n^2+n-1) = 6*A028387(n-1) (for n>5 this is a conjecture).
T(n,4n-1) = 3n^2 = A033428(n) (for n>5 this is a conjecture).
Sum(k*T(n,k), k>=1) = A143366(n).

S. Klavzar, A bird's eye view of the cut method and a survey of its applications in chemical graph theory, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.
Bo-Yin Yang and Yeong-Nan Yeh, A Crowning Moment for Wiener Indices, Studies in Appl. Math., 112 (2004), 333-340.
Bo-Yin Yang and Yeong-Nan Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, 99 (2004), 80-91, 2004.
P. Zigert, S. Klavzar, and I. Gutman, Calculating the hyper-Wiener index of benzenoid hydrocarbons, ACH Models Chem., 137, 2000, 83-94.

Cf. A143366, A214580, A152743, A153796, A069131, A004126, A028387, A033428.

The entries have been obtained by using the Maple Graph Theory package for finding the distance matrix of each of the five graphs H(n) (n=1,2,3,4,5). The given Maple program yields the Wiener polynomial of H(2) (having as coefficients the entries in row 2).

A270694 Alternating sum of centered heptagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 8, -23, 51, -94, 157, -242, 354, -495, 670, -881, 1133, -1428, 1771, -2164, 2612, -3117, 3684, -4315, 5015, -5786, 6633, -7558, 8566, -9659, 10842, -12117, 13489, -14960, 16535, -18216, 20008, -21913, 23936, -26079, 28347, -30742, 33269

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

More generally, the ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers is -x*(1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^4).

Ilya Gutkovskiy, Table of n, a(n) for n = 0..500
OEIS Wiki, Centered pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A004126 (centered heptagonal pyramidal numbers).

Cf. A000330, A006323 (partial sums of centered heptagonal pyramidal numbers), A019298, A232599.

[((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270694:= n-> ((-1)^n*(2*n+1)*(14*n^2+14*n-9) + 9)/48; seq(A270694(n), n=0..40); # G. C. Greubel, Apr 02 2021

LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -23, 51}, 39]
Table[((-1)^n (2 n + 1) (14 n^2 + 14 n - 9) + 9)/48, {n, 0, 38}]

my(x='x+O('x^50)); concat(0, Vec(-x*(1-5*x+x^2)/((1-x)*(1+x)^4))) \\ Altug Alkan, Mar 21 2016

[((-1)^n*(2*n+1)*(14*n^2+14*n-9) +9)/48 for n in (0..40)] # G. C. Greubel, Apr 02 2021

G.f.: -x*(1 - 5*x + x^2)/((1 - x)*(1 + x)^4).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = ((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48.
E.g.f.: (1/48)*(9*exp(x) - (9 + 66*x - 126*x^2 + 28*x^3)*exp(-x)). - G. C. Greubel, Mar 28 2016

A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).

Original entry on oeis.org

0, 2, 7, 18, 35, 62, 98, 148, 210, 290, 385, 502, 637, 798, 980, 1192, 1428, 1698, 1995, 2330, 2695, 3102, 3542, 4028, 4550, 5122, 5733, 6398, 7105, 7870, 8680, 9552, 10472, 11458, 12495, 13602, 14763, 15998, 17290, 18660, 20090, 21602, 23177, 24838, 26565, 28382, 30268, 32248, 34300, 36450

Offset: 0

Petros Hadjicostas, Mar 10 2021

nonn

One-half of the antidiagonal sums of array A220508.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A004126, A169607, A220508, A342462.

CoefficientList[Series[(2x^2+3x+2) x/((x+1)^2 (x-1)^4),{x,0,70}],x] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{0,2,7,18,35,62},70] (* Harvey P. Dale, Jul 08 2023 *)

/* First program */
seq1(n)={my(x='x+O('x^n)); Vec((2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4), -n)}
/* Second program (array M is A220508) */
seq2(nn) = {my(M=matrix(nn+1, nn+1)); my(a=vector(nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n, k]=if(k == n, n^2-n, if(k < n, n^2-2*n+k, k^2-n)))); for(n=1, nn+1, a[n] = sum(k=1, n, M[n-k+1,k])/2); a}

a(n) = (n+1)*(1 - (-1)^n)/16 + (7/4)*(binomial(n+3, 3) - binomial(n+2, 2)).
a(n) = (A342362(n) - (n + 1))/4.
a(2*n) = A169607(n) and a(2*n + 1) = 2*A004126(n + 1).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5. - Chai Wah Wu, Mar 11 2021

cp's OEIS Frontend

A063521 a(n) = n*(7*n^2-4)/3.

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A063522 a(n) = n*(5*n^2 - 3)/2.

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A004467 a(n) = n*(11*n^2 - 5)/6.

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A062025 a(n) = n*(13*n^2 - 7)/6.

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A063523 a(n) = n*(8*n^2 - 5)/3.

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A322640 Numbers that are sums of consecutive centered heptagonal numbers (A069099).

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Mathematica

A100179 Structured heptagonal diamond numbers (vertex structure 5).

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Magma

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A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

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A063521 a(n) = n(7n^2-4)/3.

A063522 a(n) = n(5n^2 - 3)/2.

A004467 a(n) = n(11n^2 - 5)/6.

A062025 a(n) = n(13n^2 - 7)/6.

A063523 a(n) = n(8n^2 - 5)/3.

A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).