A002413 -id:A002413 - cp's OEIS frontend

A269428 Alternating sum of heptagonal pyramidal numbers.

0, -1, 7, -19, 41, -74, 122, -186, 270, -375, 505, -661, 847, -1064, 1316, -1604, 1932, -2301, 2715, -3175, 3685, -4246, 4862, -5534, 6266, -7059, 7917, -8841, 9835, -10900, 12040, -13256, 14552, -15929, 17391, -18939, 20577, -22306, 24130, -26050, 28070

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A000292, A002413, A002717, A173196, A266677.

[((20*n^3+42*n^2+4*n-9)*(-1)^n+9)/48: n in [0..50]]; // Vincenzo Librandi, Feb 26 2016

Table[((20 n^3 + 42 n^2 + 4 n - 9) (-1)^n + 9)/48, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 7, -19, 41}, 41]

a(n)=((20*n^3 + 42*n^2 + 4*n - 9)*(-1)^n + 9)/48 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 - 4*x)/((x - 1)*(x + 1)^4).
a(n) =  ((20*n^3 + 42*n^2 + 4*n - 9)*(-1)^n + 9)/48.
a(n) = Sum_{k = 0..n} (-1)^k*A002413(k).
Sum_{n>=1} 1/a(n) = -0.8939139178060972723185724267951741... . - Vaclav Kotesovec, Feb 26 2016
E.g.f.: (9*sinh(x) - (33*x - 51*x^2 + 10*x^3)*exp(-x))/24. - Franck Maminirina Ramaharo, Nov 11 2018

A279663 a(n) = (5/6)^nGamma(n+3/5)Gamma(n+1)*Gamma(n+2)/Gamma(3/5).

Original entry on oeis.org

1, 1, 8, 208, 12480, 1435200, 281299200, 86640153600, 39507910041600, 25482601976832000, 22424689739612160000, 26147188236387778560000, 39429959860472770068480000, 75350653293363463600865280000, 179334554838205043370059366400000, 523656900127558726640573349888000000

Offset: 0

Ilya Gutkovskiy, Dec 16 2016

nonn

Heptagonal pyramidal factorial numbers.

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to factorial numbers

Cf. A002413.
Cf. A084940 (heptagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279662 (hexagonal pyramidal factorial numbers).

[Round((5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5)): n in [0..20]]; // Vincenzo Librandi Dec 17 2016

FullSimplify[Table[(5/6)^n Gamma[n + 3/5] Gamma[n + 1] Gamma[n + 2]/Gamma[3/5], {n, 0, 15}]]

a(n) = Product_{k=1..n} k*(k + 1)*(5*k - 2)/6, a(0)=1.
a(n) = Product_{k=1..n} A002413(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(5/6)^n*n^(3*n+21/10)/(Gamma(3/5)*exp(3*n)).

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479

Offset: 0

Ilya Gutkovskiy, Aug 15 2017

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A085787.
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017

A338458 Least number of heptagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 7, 8, 6, 7, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 1, 2, 3, 4, 5, 6, 6, 7, 2, 3, 4, 5, 6, 7, 7, 8, 3, 4, 3, 4, 5, 6, 7, 8, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 7, 8, 6, 7

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002413, A104246, A338480, A338492, A338493, A338494, A338495, A338496.

A104587 Triangle read by rows, given by the matrix product A * B where A (A094727) = [1; 2, 3; 3, 4, 5; 4, 5, 6, 7; ...] and B = [1; 1, 1; 1, 1, 1; ...] (both are infinite lower triangular matrices with the other terms zero).

Original entry on oeis.org

1, 5, 3, 12, 9, 5, 22, 18, 13, 7, 35, 30, 24, 17, 9, 51, 45, 38, 30, 21, 11, 70, 63, 55, 46, 36, 25, 13, 92, 84, 75, 65, 54, 42, 29, 15, 117, 108, 98, 87, 75, 62, 48, 33, 17, 145, 135, 124, 112, 99, 85, 70, 54, 37, 19, 176, 165, 153, 140, 126, 111, 95, 78, 60, 41, 21

Offset: 0

Gary W. Adamson, Mar 17 2005

Left column of the triangle = pentagonal numbers, A000326 (starting with 1).

Row sums = heptagonal pyramidal numbers, A002413.

			Triangle begins:
   1;
   5,  3;
  12,  9,  5;
  22, 18, 13,  7;
  35, 30, 24, 17,  9;
  51, 45, 38, 30, 21, 11;
  70, 63, 55, 46, 36, 25, 13;
  92, 84, 75, 65, 54, 42, 29, 15;
  ...

Cf. A000326, A094727, A002413.

tabl(nn) = {ma = matrix(nn, nn, n, k, (n+k-1)*(k<=n)); mb = matrix(nn, nn, n, k, (k<=n)); mt = ma*mb; for (i=1, nn, for (j=1, i, print1(ma[i,j], ", ");); print(););} \\ Michel Marcus, Mar 03 2014

More terms from Michel Marcus, Mar 03 2014

Edited by Michel Marcus and N. J. A. Sloane, Mar 03 2014

A329530 a(n) = n * (7*binomial(n, 2) + 1).

Original entry on oeis.org

0, 1, 16, 66, 172, 355, 636, 1036, 1576, 2277, 3160, 4246, 5556, 7111, 8932, 11040, 13456, 16201, 19296, 22762, 26620, 30891, 35596, 40756, 46392, 52525, 59176, 66366, 74116, 82447, 91380, 100936, 111136, 122001, 133552, 145810, 158796, 172531, 187036, 202332, 218440

Offset: 0

Ilya Gutkovskiy, Nov 15 2019

Centered heptagonal prism numbers.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 144.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Centered m-gonal prism numbers: A100175 (m = 3), A059722 (m = 4), A006564 (m = 5), A005915 (m = 6), this sequence (m = 7), A139757 (m = 8), A006566 (m = 9).

Cf. A000217, A002413, A006597, A024966, A069099.

Table[n (7 Binomial[n, 2] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 + 12 x + 8 x^2)/(1 - x)^4, {x, 0, nmax}], x]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 16, 66}, 41]

G.f.: x * (1 + 12*x + 8*x^2) / (1 - x)^4.
E.g.f.: exp(x) * x * (2 + 14*x + 7*x^2) / 2.
a(n) = n * (7*n^2 - 7*n + 2) / 2.
a(n) = n * (7*A000217(n-1) + 1).
a(n) = n * A069099(n).

A353064 Numbers simultaneously square and heptagonal pyramidal.

Original entry on oeis.org

0, 1, 196, 99225

Offset: 1

Kelvin Voskuijl, Apr 21 2022

Is this sequence finite?

No other terms < 10^32. - Michael S. Branicky, Jul 12 2022

			196 is a term because 196 = 14^2 is a perfect square and 196 = 6*(6+1)*(5*6-2)/6 is the 6th heptagonal pyramidal number.

ProofWiki, Heptagonal Pyramidal Numbers which are Square

Intersection of A000290 and A002413.

Cf. A003556 (tetrahedral and square), 1 and 4900 are only squares that are square pyramidal, A277792 (pentagonal pyramidal and square).

select(issqr, [seq(n*(n+1)*(5*n-2)/6, n=0..50)])[];  # Alois P. Heinz, Apr 21 2022

Select[Table[n*(n + 1)*(5*n - 2)/6, {n, 0, 100}], IntegerQ @ Sqrt[#] &] (* Amiram Eldar, Apr 21 2022 *)

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A269428 Alternating sum of heptagonal pyramidal numbers.

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A279663 a(n) = (5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5).

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A287143 Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

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A338458 Least number of heptagonal pyramidal numbers needed to represent n.

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A104587 Triangle read by rows, given by the matrix product A * B where A (A094727) = [1; 2, 3; 3, 4, 5; 4, 5, 6, 7; ...] and B = [1; 1, 1; 1, 1, 1; ...] (both are infinite lower triangular matrices with the other terms zero).

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A329530 a(n) = n * (7*binomial(n, 2) + 1).

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A353064 Numbers simultaneously square and heptagonal pyramidal.

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A279663 a(n) = (5/6)^nGamma(n+3/5)Gamma(n+1)*Gamma(n+2)/Gamma(3/5).

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).