cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322652 Numbers that are sums of consecutive heptagonal pyramidal numbers (A002413).

Original entry on oeis.org

0, 1, 8, 9, 26, 34, 35, 60, 86, 94, 95, 115, 175, 196, 201, 209, 210, 308, 311, 371, 397, 405, 406, 456, 504, 619, 645, 679, 705, 713, 714, 764, 880, 960, 1075, 1101, 1135, 1161, 1166, 1169, 1170, 1409, 1508, 1525, 1605, 1720, 1780, 1806, 1814, 1815, 1911, 1981, 2046, 2289, 2380
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 21 2018

Keywords

Links

Crossrefs

Cf. A002413, A002418, A321450, A322468, A322479, A322636, A322640, A322651, A322653.

Programs

  • Mathematica
    imax = 55;
A002413 = LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 60}, imax];
Join[{0}, Table[A002413[[i ;; j]] // Total, {i, 1, imax}, {j, i, imax}] // Flatten // Union][[;; imax]] (* Jean-François Alcover, Nov 11 2024 *)

A322855 Number of compositions (ordered partitions) of n into heptagonal pyramidal numbers (A002413).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 97, 121, 151, 188, 233, 287, 352, 431, 530, 654, 809, 1002, 1241, 1535, 1895, 2335, 2876, 3544, 4371, 5396, 6666, 8237, 10176, 12564, 15504, 19126, 23594, 29111, 35928
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 29 2018

Keywords

Links

Crossrefs

Cf. A002413, A279223, A282582, A322340, A322799, A322803, A322853, A322854, A322856.

Programs

  • Mathematica
    nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (5 k - 2)/6), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)*(5*k-2)/6)).

A000566 Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.

Original entry on oeis.org

0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Binomial transform of (0, 1, 5, 0, 0, 0, ...). Binomial transform is A084899. - Paul Barry, Jun 10 2003
Row sums of triangle A131413. - Gary W. Adamson, Jul 08 2007
Sequence starting (1, 7, 18, 34, ...) = binomial transform of (1, 6, 5, 0, 0, 0, ...). Also row sums of triangle A131896. - Gary W. Adamson, Jul 24 2007
Also the partial sums of A016861, a zero added in front; therefore a(n) = n (mod 5). - R. J. Mathar, Mar 19 2008
Also sequence found by reading the line from 0, in the direction 0, 7, ..., and the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. These parallel lines are the semi-axes perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011
Also sequence found by reading the line from 0, in the direction 0, 7, ... and the parallel line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
Partial sums give A002413. - Omar E. Pol, Jan 12 2013
The n-th heptagonal number equals the sum of the n consecutive integers starting at 2*n-1; for example, 1, 3+4, 5+6+7, 7+8+9+10, etc. In general, the n-th (2k+1)-gonal number is the sum of the n consecutive integers starting at (k-1)*n - (k-2). When k = 1 and 2, this result generates the triangular numbers, A000217, and the pentagonal numbers, A000326, respectively. - Charlie Marion, Mar 02 2022

Examples

			G.f. = x + 7*x^2 + 18*x^3 + 34*x^4 + 55*x^5 + 81*x^6 + 112*x^7 + ... - _Michael Somos_, Jan 25 2019

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
  • Leonard E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
  • 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).
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Links

Crossrefs

Cf. A014637, A014640, A014773, A014792, A069099, A131413, A131896, A134483, A000384.
a(n)= A093562(n+1, 2), (5, 1)-Pascal column.
Cf. A006564, A147875, A244639.
Cf. sequences listed in A254963.

Programs

  • Haskell
    a000566 n = n * (5 * (n - 1) + 2) `div` 2
a000566_list = scanl (+) 0 a016861_list  -- Reinhard Zumkeller, Jun 16 2013
  • Magma
    a000566:=func< n | n*(5*n-3) div 2 >; [ a000566(n): n in [0..50] ];
  • Maple
    A000566 := proc(n)
        n*(5*n-3)/2 ;
end proc:
seq(A000566(n),n=0..30); # R. J. Mathar, Oct 02 2020
  • Mathematica
    Table[n (5n - 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 7}, 50] (* Harvey P. Dale, Oct 13 2011 *)
Join[{0},Accumulate[Range[1,315,5]]] (* Harvey P. Dale, Mar 26 2016 *)
(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[7], n], {n, 0, 48}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[7,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 23 2021 *)
  • Maxima
    makelist(n*(5*n-3)/2, n, 0, 20); /* Martin Ettl, Dec 11 2012 */
  • PARI
    a(n) = n * (5*n - 3) / 2
  • Python
    # Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 5, y + 5
A000566 = aList()
print([next(A000566) for i in range(49)]) # Peter Luschny, Aug 04 2019
  • Python
    [n*(5*n-3)//2 for n in range(50)] # Gennady Eremin, Mar 24 2022

Formula

G.f.: x*(1 + 4*x)/(1 - x)^3. Simon Plouffe in his 1992 dissertation.
a(n) = C(n, 1) + 5*C(n, 2). - Paul Barry, Jun 10 2003
a(n) = Sum_{k = 1..n} (4*n - 3*k). - Paul Barry, Sep 06 2005
a(n) = n + 5*A000217(n-1) - Floor van Lamoen, Oct 14 2005
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for a(0) = 0, a(1) = 1, a(2) = 7. - Jaume Oliver Lafont, Dec 02 2008
a(n+1) = A153126(n) + n mod 2; a(2*n+1) = A033571(n); a(2*(n+1)) = A153127(n) + 1. - Reinhard Zumkeller, Dec 20 2008
40*a(n)+ 9 = A017354(n-1). - Ken Rosenbaum, Dec 02 2009.
a(n) = 2*a(n-1) - a(n-2) + 5, with a(0) = 0 and a(1) = 1. - Mohamed Bouhamida, May 05 2010
a(n) = A000217(n) + 4*A000217(n-1). - Vincenzo Librandi, Nov 20 2010
a(n) = a(n-1) + 5*n - 4, with a(0) = 0. - Vincenzo Librandi, Nov 20 2010
a(n) = A130520(5*n). - Philippe Deléham, Mar 26 2013
a(5*a(n) + 11*n + 1) = a(5*a(n) + 11*n) + a(5*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n>=1} 1/a(n) = sqrt(1 - 2/sqrt(5))*Pi/3 + 5*log(5)/6 - sqrt(5)*log((1 + sqrt(5))/2)/3 = 1.32277925312238885674944226131... . See A244639. - Vaclav Kotesovec, Apr 27 2016
E.g.f.: x*(2 + 5*x)*exp(x)/2. - Ilya Gutkovskiy, Aug 27 2016
From Charlie Marion, May 02 2017: (Start)
a(n+m) = a(n) + 5*n*m + a(m);
a(n-m) = a(n) - 5*n*m + a(m) + 3*m;
a(n) - a(m) = (5*(n + m) - 3)*(n - m)/2.
In general, let P(k,n) be the n-th k-gonal number. Then
P(k,n+m) = P(k,n) + (k - 2)*n*m + P(k,m);
P(k,n-m) = P(k,n) - (k - 2)*n*m + P(k,m) + (k - 4)*m;
P(k,n) - P(k,m) = ((k-2)*(n + m) + 4 - k)*(n - m)/2.
(End)
a(n) = A147875(-n) for all n in Z. - Michael Somos, Jan 25 2019
a(n) = A000217(n-1) + A000217(2*n-1). - Charlie Marion, Dec 19 2019
Product_{n>=2} (1 - 1/a(n)) = 5/7. - Amiram Eldar, Jan 21 2021
a(n) + a(n+1) = (2*n+1)^2 + n^2 - 2*n. In general, if we let P(k,n) = the n-th k-gonal number, then P(k^2-k+1,n)+ P(k^2-k+1,n+1) = ((k-1)*n+1)^2 + (k-2)*(n^2-2*n) = ((k-1)*n+1)^2 + (k-2)*A005563(n-2). When k = 2, this formula reduces to the well-known triangular number formula: T(n) + T(n+1) = (n+1)^2. - Charlie Marion, Jul 01 2021

Extensions

Partially edited by Joerg Arndt, Mar 11 2010

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

Original entry on oeis.org

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413
Offset: 0

Views

Author

Clark Kimberling, Apr 27 2012

Keywords

Comments

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}. The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent. In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or x
A212750 ... w=R or x=R or y
A212751 ... w=R or x=R or y
A212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

Examples

			a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).

References

  • A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
  • P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Links

Crossrefs

Cf. A000583 (n^4), A210000, A211809, A212959.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Formula

a(n) = n^4 - A211809(n).

A002411 Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.

Original entry on oeis.org

0, 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126, 10206, 11368, 12615, 13950, 15376, 16896, 18513, 20230, 22050, 23976, 26011, 28158, 30420, 32800, 35301, 37926, 40678
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = n^2(n+1)/2 is half the number of colorings of three points on a line with n+1 colors. - R. H. Hardin, Feb 23 2002
Sum of n smallest multiples of n. - Amarnath Murthy, Sep 20 2002
a(n) = number of (n+6)-bit binary sequences with exactly 7 1's none of which is isolated. A 1 is isolated if its immediate neighbor(s) are 0. - David Callan, Jul 15 2004
Also as a(n) = (1/6)*(3*n^3+3*n^2), n > 0: structured trigonal prism numbers (cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
If Y is a 3-subset of an n-set X then, for n >= 5, a(n-4) is the number of 5-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
a(n-1), n >= 2, is the number of ways to have n identical objects in m=2 of altogether n distinguishable boxes (n-2 boxes stay empty). - Wolfdieter Lang, Nov 13 2007
a(n+1) is the convolution of (n+1) and (3n+1). - Paul Barry, Sep 18 2008
The number of 3-character strings from an alphabet of n symbols, if a string and its reversal are considered to be the same.
Partial sums give A001296. - Jonathan Vos Post, Mar 26 2011
a(n-1):=N_1(n), n >= 1, is the number of edges of n planes in generic position in three-dimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p.506. - Wolfdieter Lang, May 27 2011
Partial sums of pentagonal numbers A000326. - Reinhard Zumkeller, Jul 07 2012
From Ant King, Oct 23 2012: (Start)
For n > 0, the digital roots of this sequence A010888(A002411(n)) form the purely periodic 9-cycle {1,6,9,4,3,9,7,9,9}.
For n > 0, the units' digits of this sequence A010879(A002411(n)) form the purely periodic 20-cycle {1,6,8,0,5,6,6,8,5,0,6,6,3,0,0,6,1,8,0,0}.
(End)
a(n) is the number of inequivalent ways to color a path graph having 3 nodes using at most n colors. Note, here there is no restriction on the color of adjacent nodes as in the above comment by R. H. Hardin (Feb 23 2002). Also, here the structures are counted up to graph isomorphism, where as in the above comment the "three points on a line" are considered to be embedded in the plane. - Geoffrey Critzer, Mar 20 2013
After 0, row sums of the triangle in A101468. - Bruno Berselli, Feb 10 2014
Latin Square Towers: Take a Latin square of order n, with symbols from 1 to n, and replace each symbol x with a tower of height x. Then the total number of unit cubes used is a(n). - Arun Giridhar, Mar 29 2015
This is the case k = n+4 of b(n,k) = n*((k-2)*n-(k-4))/2, which is the n-th k-gonal number. Therefore, this is the 3rd upper diagonal of the array in A139600. - Luciano Ancora, Apr 11 2015
For n > 0, a(n) is the number of compositions of n+7 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
Also the Wiener index of the n-antiprism graph. - Eric W. Weisstein, Sep 07 2017
For n > 0, a(2n+1) is the number of non-isomorphic 5C_m-snakes, where m = 2n+1 or m = 2n (for n >= 2). A kC_n-snake is a connected graph in which the k >= 2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 15 2019
For n >= 1, a(n-1) is the number of 0°- and 45°-tilted squares that can be drawn by joining points in an n X n lattice. - Paolo Xausa, Apr 13 2021
a(n) is the number of all possible products of n rolls of a six-sided die. This can be easily seen by the recursive formula a(n) = a(n - 1) + 2 * binomial(n, 2) + binomial(n + 1, 2). - Rafal Walczak, Jun 15 2024
a(n) is the number of all triples consisting of nonnegative integers smaller than n such that the sum of the first two integers is less than n. - Ruediger Jehn, Aug 17 2025

Examples

			a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable boxes in binomial(4,2)*(2!/(1!*1!) + 2!/2!) = 6*(2+1) = 18 ways. The m=2 part partitions of 4, namely (1,3) and (2,2), specify the filling of each of the 6 possible two-box choices. - _Wolfdieter Lang_, Nov 13 2007

References

  • V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_1.
  • Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., Vol. 60 (2001), pp. 85-96.
  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
  • S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/5).
  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see Vol. 2, p. 2.
  • 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).

Links

Crossrefs

Cf. A001296, A011379, A015223, A015224, A014799, A014800, A127739, A132118, A139600.
A006002(n) = -a(-1-n).
a(n) = A093560(n+2, 3), (3, 1)-Pascal column.
A row or column of A132191.
Second column of triangle A103371.
Cf. similar sequences listed in A237616.

Programs

  • GAP
    List([0..45], n->n^2*(n+1)/2); # Muniru A Asiru, Feb 19 2018
  • Haskell
    a002411 n = n * a000217 n  -- Reinhard Zumkeller, Jul 07 2012
  • Magma
    [n^2*(n+1)/2: n in [0..40]]; // Wesley Ivan Hurt, May 25 2014
  • Maple
    seq(n^2*(n+1)/2, n=0..40);
  • Mathematica
    Table[n^2 (n + 1)/2, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 6, 18}, 50] (* Harvey P. Dale, Oct 20 2011 *)
Nest[Accumulate, Range[1, 140, 3], 2] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
CoefficientList[Series[x (1 + 2 x) / (1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jan 08 2016 *)
  • PARI
    a(n)=n^2*(n+1)/2
  • PARI
    concat(0, Vec(x*(1+2*x)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 07 2016

Formula

Average of n^2 and n^3.
G.f.: x*(1+2*x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
a(n) = n*Sum_{k=0..n} (n-k) = n*Sum_{k=0..n} k. - Paul Barry, Jul 21 2003
a(n) = n*A000217(n). - Xavier Acloque, Oct 27 2003
a(n) = (1/2)*Sum_{j=1..n} Sum_{i=1..n} (i+j) = (1/2)*(n^2+n^3) = (1/2)*A011379(n). - Alexander Adamchuk, Apr 13 2006
Row sums of triangle A127739, triangle A132118; and binomial transform of [1, 5, 7, 3, 0, 0, 0, ...] = (1, 6, 18, 40, 75, ...). - Gary W. Adamson, Aug 10 2007
G.f.: x*F(2,3;1;x). - Paul Barry, Sep 18 2008
Sum_{j>=1} 1/a(j) = hypergeom([1, 1, 1], [2, 3], 1) = -2 + 2*zeta(2) = A195055 - 2. - Stephen Crowley, Jun 28 2009
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Oct 20 2011
From Ant King, Oct 23 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3.
a(n) = (n+1)*(2*A000326(n)+n)/6 = A000292(n) + 2*A000292(n-1).
a(n) = A000330(n)+A000292(n-1) = A000217(n) + 3*A000292(n-1).
a(n) = binomial(n+2,3) + 2*binomial(n+1,3).
(End)
a(n) = (A000330(n) + A002412(n))/2 = (A000292(n) + A002413(n))/2. - Omar E. Pol, Jan 11 2013
a(n) = (24/(n+3)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+3). - Vladimir Kruchinin, Jun 04 2013
Sum_{n>=1} a(n)/n! = (7/2)*exp(1). - Richard R. Forberg, Jul 15 2013
E.g.f.: x*(2 + 4*x + x^2)*exp(x)/2. - Ilya Gutkovskiy, May 31 2016
From R. J. Mathar, Jul 28 2016: (Start)
a(n) = A057145(n+4,n).
a(n) = A080851(3,n-1). (End)
For n >= 1, a(n) = (Sum_{i=1..n} i^2) + Sum_{i=0..n-1} i^2*((i+n) mod 2). - Paolo Xausa, Apr 13 2021
a(n) = Sum_{k=1..n} GCD(k,n) * LCM(k,n). - Vaclav Kotesovec, May 22 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 + Pi^2/6 - 4*log(2). - Amiram Eldar, Jan 03 2022

A011379 a(n) = n^2*(n+1).

Original entry on oeis.org

0, 2, 12, 36, 80, 150, 252, 392, 576, 810, 1100, 1452, 1872, 2366, 2940, 3600, 4352, 5202, 6156, 7220, 8400, 9702, 11132, 12696, 14400, 16250, 18252, 20412, 22736, 25230, 27900, 30752, 33792, 37026, 40460, 44100, 47952, 52022, 56316, 60840
Offset: 0

Views

Author

Glen Burch (gburch(AT)erols.com), Felice Russo

Keywords

Comments

(1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (A000217) - the n-th partial sum of triangular numbers (A000217). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n). - Amarnath Murthy, Nov 05 2002
Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x) = (Product_{i=0..n-1} i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - Benoit Cloitre, Jan 15 2003
Y values of solutions of the equation: (X-Y)^3-X*Y=0. X values are a(n)=n*(n+1)^2 (see A045991) - Mohamed Bouhamida, May 09 2006
a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - Michael Somos, Sep 06 2006
a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. - Mohamed Bouhamida, Sep 05 2009
This sequence is related to A005449 by a(n) = n*A005449(n)-sum(A005449(i), i=0..n-1), and this is the case d=3 in the identity n^2*(d*n+d-2)/2 - Sum_{k=0..n-1} k*(d*k+d-2)/2 = n*(n+d)*(2*d*n+d-3)/6. - Bruno Berselli, Nov 18 2010
Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). - J. M. Bergot, Sep 22 2011
a(n+1) is the sum of n-th row of the triangle in A195437. - Reinhard Zumkeller, Nov 23 2011
Partial sums of A049450. - Omar E. Pol, Jan 12 2013
From Jon Perry, May 11 2013: (Start)
Define a 'stable brick triangle' as:
-----
| c |
---------
| a | | b |
----------
with a, b, c > 0 and c <= a + b. This can be visualized as two bricks with a third brick on top. The third brick can only be as strong as a+b, otherwise the wall collapses - for example, (1,2,4) is unstable.
a(n) gives the number of stable brick triangles that can be formed if the two supporting bricks are 1 <= a <= n and 1 <= b <= n: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_c 1 = n^3 + n^2 as given in the Adamchuk formula.
So for i=j=n=2 we have 4:
1 2 3 4
2 2 2 2 2 2 2 2
For example, n=2 gives 2 from [a=1,b=1], 3 from both [a=1,b=2] and [a=2,b=1] and 4 from [a=2,b=2] so a(2) = 2 + 3 + 3 + 4 = 12. (End)
Define the infinite square array m(n,k) by m(n,k) = (n-k)^2 if n >= k >= 0 and by m(n,k) = (k+n)*(k-n) if 0 <= n <= k. This contains A120070 below the diagonal. Then a(n) = Sum_{k=0..n} m(n,k) + Sum_{r=0..n} m(r,n), the "hook sum" of the terms to the left of m(n,n) and above m(n,n) with irrelevant (vanishing) terms on the diagonal. - J. M. Bergot, Aug 16 2013
a(n) is the sum of all pairs with repetition drawn from the set of odd numbers 2*n-3. This is similar to A027480 but using the odd integers instead. Example using n=3 gives the odd numbers 1,3,5: 1+1, 1+3, 1+5, 3+3, 3+5,5+5 having a total of 36=a(3). - J. M. Bergot, Apr 05 2016
a(n) is the first Zagreb index of the complete graph K[n+1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016
a(n-2) is the maximum sigma irregularity over all trees with n vertices. The extremal graphs are stars. (The sigma irregularity of a graph is the sum of squares of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 14 2023

Examples

			a(3) = 3^2+3^3 = 36.

References

  • L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.

Links

Crossrefs

Cf. A000217, A000290, A000292, A000330, A000578, A002378, A002411, A002412, A002413, A005449, A013661, A022549, A027480, A045991, A049450, A120070, A126890, A195437, A245334.
Cf. A011379, A181617, A270205 (sigma irregularities of maximal k-degenerate graphs).
Cf. A028724, A052289.

Programs

Formula

a(n) = 2*A002411(n).
a(n) = Sum_{j=1..n} (Sum_{i=1..n} (i+j)), row sums of A126890 skipping numbers in the first column. - Alexander Adamchuk, Oct 12 2004
Sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340... [Jolley eq 272] - Gary W. Adamson, Dec 22 2006
a(n) = 2*n*binomial(n+1,2) = 2*n*A000217(n). - Arkadiusz Wesolowski, Feb 10 2012
G.f.: 2*x*(1 + 2*x)/(1 - x)^4. - Arkadiusz Wesolowski, Feb 11 2012
a(n) = A000330(n) + A002412(n) = A000292(n) + A002413(n). - Omar E. Pol, Jan 11 2013
a(n) = A245334(n+1,2), n > 0. - Reinhard Zumkeller, Aug 31 2014
Sum_{n>=1} 1/a(n) = A013661-1. - R. J. Mathar, Oct 18 2019 [corrected by Jason Yuen, Aug 04 2024]
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 - 2*log(2). - Amiram Eldar, Jul 04 2020
E.g.f.: exp(x)*x*(2 + 4*x + x^2). - Stefano Spezia, May 20 2021
a(n) = n*A002378(n) = A000578(n) + A000290(n). - J.S. Seneschal, Jun 18 2024

A002414 Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2.

Original entry on oeis.org

1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030, 8190, 9471, 10879, 12420, 14100, 15925, 17901, 20034, 22330, 24795, 27435, 30256, 33264, 36465, 39865, 43470, 47286, 51319, 55575, 60060, 64780
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes. - Yong Kong (ykong@curagen.com), May 06 2000
Equals binomial transform of [0, 1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jun 14 2008, corrected Oct 25 2012
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF3 denominators of A156927. See A157704 for background information. - Johannes W. Meijer, Mar 07 2009
This sequence is related to A000326 by a(n) = n*A000326(n) - Sum_{i=0..n-1} A000326(i) and this is the case d=3 in the identity n*(n*(d*n-d+2)/2)-Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Apr 21 2010
2*a(n) gives the principal diagonal of the convolution array A213819. - Clark Kimberling, Jul 04 2012
Partial sums of the figurate octagonal numbers A000567. For each sequence with a linear recurrence with constant coefficients, the values reduced modulo some constant m generate a periodic sequence. For this sequence, these Pisano periods have length 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, ... for m >= 1. - Ant King, Oct 26 2012
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
On a square grid of side length n+1, the number of embedded rectangles (where each side is greater than 1). For example, in a 2 X 2 square there is one rectangle, in a 3 X 3 square there are nine rectangles, etc. - Peter Woodward, Nov 26 2017
a(n) is the sum of the numbers in the n X n square array A204154(n). - Ali Sada, Jun 21 2019
Sum of all multiples of n and n+1 that are <= n^2. - Wesley Ivan Hurt, May 25 2023

Examples

			a(2) = 9 since there are 9 ways to cover a 4 X 4 lattice with 8 dominoes, 2 of which is horizontal and the other 6 are vertical. - Yong Kong (ykong@curagen.com), May 06 2000
G.f. = x + 9*x^2 + 30*x^3 + 70*x^4 + 135*x^5 + 231*x^6 + 364*x^7 + 540*x^8 + 765*x^9 + ...

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
  • 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).

Links

Crossrefs

Cf. A000578, A004003, A160378.
Cf. A093563 (( 6, 1) Pascal, column m=3). A000567 (differences).
Cf. A156927, A157704. - Johannes W. Meijer, Mar 07 2009
Cf. A000326.
Cf. similar sequences listed in A237616.
Cf. A260234 (largest prime factor of a(n+1)).

Programs

  • Magma
    [n*(n+1)*(2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, May 24 2016
  • Maple
    A002414 := n-> 1/2*n*(n+1)*(2*n-1): seq(A002414(n), n=1..100);
  • Mathematica
    LinearRecurrence[{4,-6,4,-1},{1,9,30,70},40] (* Harvey P. Dale, Apr 12 2013 *)
  • PARI
    {a(n) = (2*n - 1) * n * (n + 1) / 2} \\ Michael Somos, Mar 17 2011

Formula

a(n) = odd numbers * triangular numbers = (2*n-1)* binomial(n+1,2). - Xavier Acloque, Oct 27 2003
G.f.: x*(1+5*x)/(1-x)^4. [Conjectured by Simon Plouffe in his 1992 dissertation.]
a(n) = A000578(n) + A000217(n-1). - Kieren MacMillan, Mar 19 2007
a(-n) = -A160378(n). - Michael Somos, Mar 17 2011
From Ant King, Oct 26 2012: (Start)
a(n) = a(n-1) + n*(3*n-2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = n*A000326(n) - A002411(n-1), see Berselli's comment.
a(n) = (n+1)*(2*A000567(n)+n)/6.
a(n) = A000292(n) + 5*A000292(n-1) = binomial(n+2,3)+5*binomial(n+1,3).
a(n) = A002413(n) + A000292(n-1).
a(n) = A000217(n) + 6*A000292(n-1).
Sum_{n>=1} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...
(End)
a(n) = Sum_{i=0..n-1} (n-i)*(6*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(2 + 7*x + 2*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 23 2016
a(n) = A080851(6,n-1). - R. J. Mathar, Jul 28 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi + 1 - 4*log(2))/3. - Amiram Eldar, Jul 02 2020

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
Incorrect formula deleted by Ant King, Oct 04 2012

A080851 Square array of pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78
Offset: 0

Views

Author

Paul Barry, Feb 21 2003

Keywords

Comments

The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

Examples

			Array begins (n>=0, k>=0):
1,  3,  6, 10,  15,  21,  28,  36,  45,   55, ... A000217
1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584

Links

Crossrefs

Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145, A027660 (antidiagonal sums).
See A257199 for another version of this array.

Programs

  • Derive
    vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4,x,11),x,n),n,0,11),k,-1,10) VECTOR(VECTOR(comb(k+2,2)+comb(k+2,3)n, k, 0, 11), n, 0, 11)
  • Maple
    A080851 := proc(n,k)
    binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
end proc:
seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021
  • Mathematica
    pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)

Formula

T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016

A237616 a(n) = n*(n + 1)*(5*n - 4)/2.

Original entry on oeis.org

0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, 39325, 44226, 49518, 55216, 61335, 67890, 74896, 82368, 90321, 98770, 107730, 117216, 127243, 137826, 148980, 160720
Offset: 0

Views

Author

Bruno Berselli, Feb 10 2014

Keywords

Comments

Also 17-gonal (or heptadecagonal) pyramidal numbers.
This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).

Examples

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  16;
   3,  32,  31;
   4,  48,  62,  46;
   5,  64,  93,  92,  61;
   6,  80, 124, 138, 122,  76;
   7,  96, 155, 184, 183, 152,  91;
   8, 112, 186, 230, 244, 228, 182, 106;
   9, 128, 217, 276, 305, 304, 273, 212, 121;
  10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 15*r-14 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.

References

  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).

Links

Crossrefs

Cf. A051869, A104728.
Cf. sequences with formula n*(n+1)*(k*n-k+3)/6: A000217 (k=0), A000292 (k=1), A000330 (k=2), A002411 (k=3), A002412 (k=4), A002413 (k=5), A002414 (k=6), A007584 (k=7), A007585 (k=8), A007586 (k=9), A007587 (k=10), A050441 (k=11), A172073 (k=12), A177890 (k=13), A172076 (k=14), this sequence (k=15), A172078(k=16), A237617 (k=17), A172082 (k=18), A237618 (k=19), A172117(k=20), A256718 (k=21), A256716 (k=22), A256645 (k=23), A256646(k=24), A256647 (k=25), A256648 (k=26), A256649 (k=27), A256650(k=28).

Programs

  • GAP
    List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019
  • Magma
    [n*(n+1)*(5*n-4)/2: n in [0..40]];
  • Magma
    I:=[0,1,18,66]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 12 2014
  • Maple
    seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019
  • Mathematica
    Table[n(n+1)(5n-4)/2, {n, 0, 40}]
CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* Harvey P. Dale, Jan 11 2015 *)
  • PARI
    a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015
  • Sage
    [n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

G.f.: x*(1 + 14*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(15*i+1). More generally, the sequence with the closed form n*(n+1)*(k*n-k+3)/6 is also given by Sum_{i=0..n-1} (n-i)*(k*i+1) for n>0.
a(n) = A104728(A001844(n-1)) for n>0.
Sum_{n>=1} 1/a(n) = (2*sqrt(5*(5 + 2*sqrt(5)))*Pi + 10*sqrt(5)*arccoth(sqrt(5)) + 25*log(5) - 16)/72 = 1.086617842136293176... . - Vaclav Kotesovec, Dec 07 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

A002418 4-dimensional figurate numbers: a(n) = (5*n-1)*binomial(n+2,3)/4.

Original entry on oeis.org

0, 1, 9, 35, 95, 210, 406, 714, 1170, 1815, 2695, 3861, 5369, 7280, 9660, 12580, 16116, 20349, 25365, 31255, 38115, 46046, 55154, 65550, 77350, 90675, 105651, 122409, 141085, 161820, 184760, 210056, 237864, 268345, 301665, 337995
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Partial sums of A002413.
Principal diagonal of the convolution array A213550, for n>0. - Clark Kimberling, Jun 17 2012
Convolution of A000027 with A000566. - Bruno Berselli, Dec 06 2012
Coefficients in the hypergeometric series identity 1 - 9*(x - 1)/(4*x + 1) + 35*(x - 1)*(x - 2)/((4*x + 1)*(4*x + 2)) - 95*(x - 1)*(x - 2)*(x - 3)/((4*x + 1)*(4*x + 2)*(4*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002412. Column 4 of A103450. - Peter Bala, Mar 14 2019

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
  • 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).

Links

Crossrefs

Cf. A093562 ((5, 1) Pascal, column m=4).
Cf. A000332, A000566, A001622, A080852, A128064.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

Programs

  • GAP
    List([0..40],n->(5*n-1)*Binomial(n+2,3)/4); # Muniru A Asiru, Mar 18 2019
  • Magma
    [(5*n - 1)*Binomial(n + 2, 3)/4: n in [0..40]]; // Vincenzo Librandi, Oct 17 2012
  • Magma
    /* A000027 convolved with A000566: */ A000566:=func; [&+[(n-i+1)*A000566(i): i in [0..n]]: n in [0..35]]; // Bruno Berselli, Dec 06 2012
  • Mathematica
    Table[(5n-1) Binomial[n+2,3]/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,1,9,35,95},40] (* Harvey P. Dale, Oct 16 2012 *)
CoefficientList[Series[x*(1 + 4*x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 17 2012 *)
  • PARI
    a(n)=(5*n-1)*binomial(n+2,3)/4 \\ Charles R Greathouse IV, Sep 24 2015
  • Sage
    [(5*n-1)*binomial(n+2,3)/4 for n in (0..40)] # G. C. Greubel, Jul 03 2019

Formula

G.f.: x*(1+4*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
Starting (1, 9, 35, 95, ...), = A128064 * A000332, (A000332 starting 1, 5, 15, 35, 70, ...), such that a(n) = n*C(n+3,4) - (n-1)*C(n+2,4). E.g., a(5) = 210 = 5*C(8,4) - 4*C(7,4) = 5*70 - 4*35. - Gary W. Adamson, Dec 28 2007
Unit digit, A010879(a(n)), is one of {0,1,9,5,6,4} [Eric Desbiaux] because a(n) mod 5 = 0,1,4,0,0, periodic with period 5. [Proof: A002413(n) mod 5 = 1,3,1,0,0 with period 5 and a(n) are the partial sums of A002413.] - R. J. Mathar, Mar 19 2008
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Oct 16 2012
a(n) = A080852(5,n-1). - R. J. Mathar, Jul 28 2016
a(n) = Sum_{i=0..n} (n-i) * Sum_{j=i..n} j. - J. M. Bergot, May 30 2017
E.g.f.: x*(24 + 84*x + 44*x^2 + 5*x^3)*exp(x)/4!. - G. C. Greubel, Jul 03 2019
Sum_{n>=1} 1/a(n) = (50*sqrt(5)*log(phi) + 125*log(5) - 50*sqrt(1+2/sqrt(5))*Pi - 26)/11, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 11 2022
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