A002417 -id:A002417 - cp's OEIS frontend

A076455 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly two ways.

5, 57, 246, 710, 1635, 3255, 5852, 9756, 15345, 23045, 33330, 46722, 63791, 85155, 111480, 143480, 181917, 227601, 281390, 344190, 416955, 500687, 596436, 705300, 828425, 967005, 1122282, 1295546, 1488135, 1701435, 1936880, 2195952, 2480181

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

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

Cf. A002417, A076454-A076459.

[n*(n+1)*(4*n^2+2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

Maple
```
seq(1/2*n*(n+1)*(4*n^2+2*n-1),n=1..40);
```

CoefficientList[Series[(5 + 32 x + 11 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

G.f.: x*(5 + 32*x + 11*x^2)/(1 - x)^5.

a(n) = n*(n + 1)*(4*n^2 + 2*n - 1)/2.

A213840 a(n) = n(1 + n)(3 - 4n + 4n^2)/6.

Original entry on oeis.org

1, 11, 54, 170, 415, 861, 1596, 2724, 4365, 6655, 9746, 13806, 19019, 25585, 33720, 43656, 55641, 69939, 86830, 106610, 129591, 156101, 186484, 221100, 260325, 304551, 354186, 409654, 471395, 539865, 615536, 698896, 790449, 890715, 1000230, 1119546, 1249231

Offset: 1

Clark Kimberling, Jul 05 2012

Antidiagonal sums of the convolution array A213838.
The sequence is the binomial transform of (1, 10, 33, 40, 16, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
From Mircea Dan Rus, Jul 11 2020: (Start)
a(n) is also the number of rectangles in a square biscuit of order n, which is obtained by stacking 2n-1 rows with their centers vertically aligned which consist successively of 1, 3, ..., 2n-3, 2n-1, 2n-3, ..., 3, 1 consecutive unit lattice squares. The order 2 and 3 square biscuits are shown below which contain 11 and 54 rectangles respectively.
                        
                   |__|
     |__|     |__||__|
    ||__||   ||__||__||
       ||         ||__||
                       ||
(End)

Clark Kimberling, Table of n, a(n) for n = 1..200
Teofil Bogdan and Mircea Rus, Numărând dreptunghiuri pe foaia de matematică (in Romanian). Gazeta Matematică, seria B, 2020 (6-7-8), pp. 281-288.
Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A002417, A213838.

First differences of A271870. - J. M. Bergot, Aug 29 2016

[n*(1+n)*(3-4*n+4*n^2)/6: n in [1..60]]; // Vincenzo Librandi, Aug 01 2015

A213840:=n->n*(1 + n)*(3 - 4*n + 4*n^2)/6: seq(A213840(n), n=1..50); # Wesley Ivan Hurt, Sep 16 2017

Table[n (1 + n) (3 - 4 n + 4 n^2)/6, {n, 50}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 11, 54, 170, 415}, 40] (* Vincenzo Librandi, Aug 01 2015 *)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1 + 3*x)^2/(1 - x)^5.
From Mircea Dan Rus, Aug 26 2020: (Start)
a(n) = A000332(n+3) + 6*A000332(n+2) + 9*A000332(n+1).
a(n) = A002417(n) + 3*A002417(n-1). (End)

Edited (with simpler definition) by N. J. A. Sloane, Sep 19 2017

A261721 Fourth-dimensional figurate numbers.

Original entry on oeis.org

1, 1, 5, 1, 6, 15, 1, 7, 20, 35, 1, 8, 25, 50, 70, 1, 9, 30, 65, 105, 126, 1, 10, 35, 80, 140, 196, 210, 1, 11, 40, 95, 175, 266, 336, 330, 1, 12, 45, 110, 210, 336, 462, 540, 495, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 1, 14, 55, 140, 280, 476, 714, 960, 1155, 1210, 1001, 1

Offset: 1

Gary W. Adamson, Aug 30 2015

Generating polygons for the sequences are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ... .
n-th row sequence is the binomial transform of the fourth row of Pascal's triangle (1,n) followed by zeros; and the fourth partial sum of (1, n, n, n, ...).
n-th row sequence is the binomial transform of:
((n-1) * (0, 1, 3, 3, 1, 0, 0, 0) + (1, 4, 6, 4, 1, 0, 0, 0)).
Given the n-th row of the array (1, b, c, d, ...), the next row of the array is (1, b, c, d, ...) + (0, 1, 5, 15, 35, ...)

			The array as shown in A257200:
  1,  5, 15,  35,  70, 126, 210,  330, ... A000332
  1,  6, 20,  50, 105, 196, 336,  540, ... A002415
  1,  7, 25,  65, 140, 266, 462,  750, ... A001296
  1,  8, 30,  80, 175, 336, 588,  960, ... A002417
  1,  9, 35,  95, 210, 406, 714, 1170, ... A002418
  1, 10, 40, 110, 245, 476, 840, 1380, ... A002419
  ...
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of the fourth row of Pascal's triangle (1,3) followed by zeros: (1, 6, 12, 10, 3, 0, 0, 0, ...); and the fourth partial sum of (1, 3, 3, 3, 0, 0, 0).
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of: (2 * (0, 1, 3, 3, 1, 0, 0, 0, ...) + (1, 4, 6, 4, 1, 0, 0, 0, ...)); that is, the binomial transform of (1, 6, 12, 10, 3, 0, 0, 0, ...).
Row 2 of the array is (1, 5, 15, 35, 70, ...) + (0, 1, 5, 15, 35, ...), = (1, 6, 20, 50, 105, ...).

Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 195 (Table 80).

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index to sequences related to pyramidal numbers

Cf. A257200, A261720 (pyramidal numbers), A000332, A002415, A001296, A002417, A002418, A002419.
Similar to A080852 but without row n=0.
Main diagonal gives A256859.

A:= (n, k)-> binomial(k+3, 3) + n*binomial(k+3, 4):
seq(seq(A(d-k, k), k=0..d-1), d=1..13);  # Alois P. Heinz, Aug 31 2015

row[1] = LinearRecurrence[{5, -10, 10, -5, 1}, {1, 5, 15, 35, 70}, m = 10];
row1 = Join[{0}, row[1] // Most]; row[n_] := row[n] = row[n-1] + row1;
Table[row[n-k+1][[k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

A(n, k) = binomial(k+3, 3) + n*binomial(k+3, 4)
table(n, k) = for(x=1, n, for(y=0, k-1, print1(A(x, y), ", ")); print(""))
/* Print initial 6 rows and 8 columns as follows: */
table(6, 8) \\ Felix Fröhlich, Jul 28 2016

G.f. for row n: (1 + (n-1)*x)/(1 - x)^5.
A(n,k) = C(k+3,3) + n * C(k+3,4) = A080852(n,k).
E.g.f. as array: exp(y)*(exp(x)*(24 + 24*(3 + x)*y + 36*(1 + x)*y^2 + 4*(1 + 3*x)*y^3 + x*y^4) - 4*(6 + 18*y + 9*y^2 + y^3))/24. - Stefano Spezia, Aug 15 2024

A264850 a(n) = n(n + 1)(n + 2)(7n - 5)/12.

Original entry on oeis.org

0, 1, 18, 80, 230, 525, 1036, 1848, 3060, 4785, 7150, 10296, 14378, 19565, 26040, 34000, 43656, 55233, 68970, 85120, 103950, 125741, 150788, 179400, 211900, 248625, 289926, 336168, 387730, 445005, 508400, 578336, 655248, 739585, 831810, 932400, 1041846

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of 16-gonal (or hexadecagonal) pyramidal numbers. Therefore, this is the case k=7 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A172076.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12: A000292 (k=0), A002415 (which arises from k=1), A002417 (k=2), A002419 (k=3), A051797 (k=4), A051799 (k=5), A220212 (k=6), this sequence (k=7), A264851 (k=8), A264852 (k=9).

[n*(n+1)*(n+2)*(7*n-5)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (n + 2) (7 n - 5)/12, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,18,80,230},40] (* Harvey P. Dale, Sep 27 2018 *)

a(n)=n*(n+1)*(n+2)*(7*n-5)/12 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 13*x)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A172076(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

A290939 Number of 5-cycles in the n-triangular graph.

Original entry on oeis.org

0, 0, 24, 312, 1584, 5376, 14448, 33264, 68544, 129888, 230472, 387816, 624624, 969696, 1458912, 2136288, 3055104, 4279104, 5883768, 7957656, 10603824, 13941312, 18106704, 23255760, 29565120, 37234080, 46486440, 57572424, 70770672, 86390304, 104773056, 126295488

Offset: 2

Eric W. Weisstein, Aug 14 2017

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

Cf. A002417 (number of 3-cycles in the triangular graph), A151974 (4-cycles), A290940 (6-cycles).

Table[12/5 Binomial[n, 4] (n^2 + 7 n - 34), {n, 2, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 24, 312, 1584, 5376, 14448}, 20]
CoefficientList[Series[(24 (-x^2 - 6 x^3 + 4 x^4))/(-1 + x)^7, {x, 0, 20}], x]

a(n)=12*binomial(n, 4)*(n^2+7*n-34)/5 \\ Charles R Greathouse IV, Aug 14 2017

a(n) = 12/5 * binomial(n, 4) * (n^2 + 7*n - 34).
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).
G.f.: (24 x^2 (-x^2 - 6 x^3 + 4 x^4))/(-1 + x)^7.

A290940 Number of 6-cycles in the n-triangular graph.

Original entry on oeis.org

0, 0, 16, 920, 7800, 36260, 122080, 334656, 794640, 1696200, 3334320, 6137560, 10706696, 17859660, 28683200, 44591680, 67393440, 99365136, 143334480, 202771800, 281890840, 385759220, 520418976, 693017600, 911950000, 1187011800, 1529564400, 1952712216, 2471492520

Offset: 2

Eric W. Weisstein, Aug 14 2017

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Triangular Graph
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A002417 (3-cycles), A151974 (4-cycles), A290939 (5-cycles).

Table[2 Binomial[n, 4] (n^3 + 27 n^2 - 220 n + 392), {n, 2, 20}]
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 16, 920, 7800, 36260, 122080, 334656}, 20]
CoefficientList[Series[-((4 (-4 x^2 - 198 x^3 - 222 x^4 + 319 x^5))/(-1 + x)^8), {x, 0, 20}], x]

a(n) = 2*binomial(n, 4) (n^3 + 27*n^2 - 220*n + 392).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: -((4*x^2 (-4*x^2 - 198*x^3 - 222*x^4 + 319*x^5))/(-1 + x)^8).

A317019 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*binomial(k+2,3)).

Original entry on oeis.org

1, 1, 9, 39, 155, 570, 2131, 7599, 26667, 90996, 305144, 1004173, 3254123, 10385884, 32704819, 101678860, 312435675, 949498206, 2855953018, 8507079361, 25108844890, 73468004480, 213201630328, 613871526178, 1754365814430, 4978113020152, 14029639217532, 39281646364737

Offset: 0

Ilya Gutkovskiy, Jul 19 2018

nonn

Euler transform of A002417.

N. J. A. Sloane, Transforms

Cf. A000391, A002417, A278767, A279217, A305653, A317017, A317020, A317021.

a:=series(mul(1/(1-x^k)^(k*binomial(k+2,3)),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 3 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^3 (d + 1) (d + 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002417(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 3*x^k)/(k*(1 - x^k)^5)).
a(n) ~ 1/(2^(601/720) * 3^(359/480) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)) * exp(-Zeta(3)/(12 * Pi^2) + (491 * Zeta(5))/(400 * Pi^4) - (2250423 * Zeta(5)^3)/(10 * Pi^14) + (103355177121 * Zeta(5)^5)/(10 * Pi^24) + Zeta'(-3)/2 + ((-7 * 7^(1/6) * Pi)/(1200 * 2^(1/3) * sqrt(3)) + (27783 * sqrt(3) * 7^(1/6) * Zeta(5)^2)/(40 * 2^(1/3) * Pi^9) - (614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4)/(16 * 2^(1/3) * Pi^19)) * n^(1/6) + ((-63 * 7^(1/3) * Zeta(5))/(10 * 2^(2/3) * Pi^4) + (214326 * 14^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/3) * Pi)/30 - (1701 * sqrt(21) * Zeta(5)^2)/(2 * Pi^9)) * sqrt(n) + ((27 * 7^(2/3) * Zeta(5))/(2 * 2^(1/3) * Pi^4)) * n^(2/3) + ((2 * 2^(1/3) * sqrt(3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A322651 Numbers that are sums of consecutive hexagonal pyramidal numbers (A002412).

Original entry on oeis.org

0, 1, 7, 8, 22, 29, 30, 50, 72, 79, 80, 95, 145, 161, 167, 174, 175, 252, 256, 306, 328, 335, 336, 372, 413, 508, 525, 558, 580, 587, 588, 624, 715, 785, 880, 897, 930, 946, 952, 959, 960, 1149, 1222, 1240, 1310, 1405, 1455, 1477, 1484, 1485, 1547, 1612, 1661, 1864, 1925

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number

Cf. A002412, A002417, A152043, A319185, A321450, A322468, A322479, A322652, A322653.

A366035 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 8, 98, 1440, 23389, 404712, 7314724, 136476912, 2608808180, 50828498336, 1005682252458, 20152470321984, 408149824237302, 8341496306085040, 171812412714350280, 3562961488550366480, 74328284438252301996, 1558783863783469298016, 32844108784368485209320, 694957689921176181019520

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A002417 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002294, A002417, A064087, A365668, A365755, A365816, A366015, A366036, A366037.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 3^k for n > 0.

a(n) ~ 2^(4*n - 1) * 5^(5*n + 1/2) / (sqrt(Pi) * n^(3/2) * 3^(7*n + 5/2)). - Vaclav Kotesovec, Sep 27 2023

A068240 1/2 the number of colorings of a 4 X 4 square array with n colors.

Original entry on oeis.org

1, 3906, 3000366, 414425080, 19064362455, 428429377026, 5861180425996, 55823546748096, 403783634784285, 2353615149832210, 11531349080992026, 48981767072238936, 184656623163700051, 629125059062885490, 1964980839044519640, 5691311662142685376

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

a:= n-> n*(n-1)*(17493+(-94782+(251492+(-430857+(529770+(-492434 +(355622+(-202160+(90723+(-31939+(8675+(-1762+(253 +(-23+n)*n)*n) *n)*n)*n)*n) *n)*n) *n)*n) *n)*n)*n) /2:
seq(a(n), n=2..30); # Alois P. Heinz, Apr 27 2012

From Alois P. Heinz, Apr 27 2012 (Start)
G.f.: -(2507986*x^14 +349887529*x^13 +12282125725*x^12 +158263444274*x^11 +896159384816*x^10 +2455337616143*x^9 +3417678462327*x^8 +2453922059100*x^7 +895941969162*x^6 +158666067383*x^5 +12424532171*x^4 +363949394*x^3 +2934100*x^2 +3889*x+1)*x^2 / (x-1)^17.
a(n) = n*(n-1)*(n^14 -23*n^13 +253*n^12 -1762*n^11 +8675*n^10 -31939*n^9 +90723*n^8 -202160*n^7 +355622*n^6 -492434*n^5 +529770*n^4 -430857*n^3 +251492*n^2 -94782*n +17493)/2.
(End)

cp's OEIS Frontend

A076455 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly two ways.

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A213840 a(n) = n*(1 + n)*(3 - 4*n + 4*n^2)/6.

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A261721 Fourth-dimensional figurate numbers.

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A264850 a(n) = n*(n + 1)*(n + 2)*(7*n - 5)/12.

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A290939 Number of 5-cycles in the n-triangular graph.

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A290940 Number of 6-cycles in the n-triangular graph.

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A317019 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*binomial(k+2,3)).

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A076455 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly two ways.

A213840 a(n) = n(1 + n)(3 - 4n + 4n^2)/6.

A264850 a(n) = n(n + 1)(n + 2)(7n - 5)/12.