A002412 -id:A002412 - cp's OEIS frontend

A143690 a(n) = A007318 * [1, 6, 14, 9, 0, 0, 0, ...].

1, 7, 27, 70, 145, 261, 427, 652, 945, 1315, 1771, 2322, 2977, 3745, 4635, 5656, 6817, 8127, 9595, 11230, 13041, 15037, 17227, 19620, 22225, 25051, 28107, 31402, 34945, 38745, 42811, 47152, 51777, 56695, 61915, 67446, 73297, 79477, 85995, 92860, 100081, 107667

Offset: 0

Gary W. Adamson, Aug 29 2008

Binomial transform of [1, 6, 14, 9, 0, 0, 0,...].

Row sums of triangle A033292.

			a(3) = 70 = (1, 3, 3, 1) dot (1, 6, 14, 9) = (1 + 18 + 42 + 9). a(3) = 70 = sum of row 3 terms of triangle A033292: (13 + 16 + 19, + 22).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000326, A002412, A033292.

Cf. A226449. - Bruno Berselli, Jun 09 2013

Table[(n+1)*(3*n^2+2*n+2)/2, {n,0,50}] (* G. C. Greubel, May 30 2021 *)

[(n+1)*(3*n^2+2*n+2)/2 for n in (0..50)] # G. C. Greubel, May 30 2021

From R. J. Mathar, Aug 29 2008: (Start)
G.f.: (1 +3*x +5*x^2)/(1-x)^4.
a(n) = A002412(n+1) + 5*A000292(n-1). (End)
a(n) = A000326(n+1) + (n+1)*A000326(n). - Bruno Berselli, Jun 07 2013
From G. C. Greubel, May 30 2021: (Start)
a(n) = (n+1)*(3*n^2 +2*n +2)/2.
E.g.f.: (1/2)*(2 +12*x +14*x^2 +3*x^3)*exp(x). (End)

Extended beyond a(14) by R. J. Mathar, Aug 29 2008

A266677 Alternating sum of hexagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 6, -16, 34, -61, 100, -152, 220, -305, 410, -536, 686, -861, 1064, -1296, 1560, -1857, 2190, -2560, 2970, -3421, 3916, -4456, 5044, -5681, 6370, -7112, 7910, -8765, 9680, -10656, 11696, -12801, 13974, -15216, 16530, -17917, 19380, -20920, 22540

Offset: 0

Ilya Gutkovskiy, Feb 02 2016

More generally, the ordinary generating function for the alternating sum of k-gonal pyramidal numbers is x*(1 + (3 - k)*x)/((x - 1)*(x + 1)^4).

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

Cf. A000292, A002412, A002717, A173196.

Table[((-1)^n (2 n (n + 2) (4 n + 1) - 3) + 3)/24, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 6, -16, 34}, 40]

concat(0, Vec(x*(1 - 3*x)/((x - 1)*(x + 1)^4) + O(x^50))) \\ Michel Marcus, Feb 02 2016

G.f.: x*(1 - 3*x)/((x - 1)*(x + 1)^4).
a(n) = ((-1)^n*(2*n*(n + 2)*(4*n + 1) - 3) + 3)/24.
a(n) = Sum_{k = 0..n} (-1)^k*A002412(k).

A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4x + 3)(x + 1)^(2n) + (x^2 - 1)(x^2 + 3*x + 3).

Original entry on oeis.org

0, 1, 3, 3, 1, 0, 7, 14, 9, 2, 0, 13, 37, 43, 26, 8, 1, 0, 19, 72, 129, 141, 98, 42, 10, 1, 0, 25, 119, 291, 463, 504, 378, 192, 63, 12, 1, 0, 31, 178, 553, 1156, 1716, 1848, 1452, 825, 330, 88, 14, 1, 0, 37, 249, 939, 2432, 4576, 6435, 6864, 5577, 3432, 1573

Offset: 0

Franck Maminirina Ramaharo, Jul 18 2018

The triangle is related to the Kauffman bracket polynomial evaluated at the shadow diagram of the two-bridge knot with Conway's notation C(2n,3).

			The triangle T(n,k) begins:
n\k| 0   1    2    3     4     5     9     7     8     9    10   11   12  13 14
-------------------------------------------------------------------------------
0  | 0   1    3    3     1
1  | 0   7   14    9     2
2  | 0  13   37   43    26     8     1
3  | 0  19   72  129   141    98    42    10     1
4  | 0  25  119  291   463   504   378   192    63    12     1
5  | 0  31  178  553  1156  1716  1848  1452   825   330    88   14    1
6  | 0  37  249  939  2432  4576  6435  6864  5577  3432  1573  520  117  16  1
...

Ji-Young Ham and Joongul Lee, An explicit formula for the A-polynomial of the knot with Conway’s notation C(2n,3), Journal of Knot Theory and Its Ramifications 25 (2016), 1-9.
Ryo Hanaki, On scannable properties of the original knot from a knot shadow, Topology and its Applications 194 (2015), 296-305.
Bin Lu and Jianyuan K. Zhong, The Kauffman Polynomials of 2-bridge Knots, arXiv:math/0606114 [math.GT], 2006.

Row sums: 4*A004171.

Cf. A093560, A137396, A299989, A300184, A300192, A300453, A300454, A316659.

T := proc (n, k) if k = 1 then 6*n + 1 else binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) end if end proc:
for n from 0 to 12 do seq(T(n, k), k = 0 .. max(4, 2*(n + 1))) od;

row[n_] := CoefficientList[(x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3), x];
Array[row, 12, 0] // Flatten

T(n, k) := binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) - kron_delta(1, k)$
for n:0 thru 12 do print(makelist(T(n, k), k, 0, max(4, 2*(n + 1))));

T(n,1) = A016921(n) and T(n,k) = C(2*n+3,k+1) + (C(2*n+1,k)*(2*k - 2*n) + C(4,k)*(2*k - 3))/(k + 1) for k > 1.
T(n,2) = A173247(2*n+1) = A300401(2*n,3).
T(n,3) = 2*A099721(n) + 3.
T(n,4) = A244730(n) - A002412(n) + 1.
T(n,k) = A093560(2*n,k) for n > 2 and k > 4.
G.f.: (x^2 + 4*x + 3)/(1 - y*(x + 1)^2) + (x^4 + 3*x^3 + 2*x^2 - 3*x - 3)/(1 - y).

A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

Original entry on oeis.org

0, 1, 6, 7, 15, 21, 22, 28, 43, 45, 49, 50, 66, 73, 88, 91, 94, 95, 111, 120, 139, 153, 154, 157, 160, 161, 190, 202, 211, 230, 231, 245, 251, 252, 273, 276, 277, 322, 325, 343, 350, 364, 365, 371, 372, 378, 421, 430, 435, 463, 475, 496, 503, 507, 518, 524, 525, 554, 561

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A002412, A034705, A034706, A319184, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[anmax/2]] + 1; Select[Union[Flatten[Table[Sum[k*(2*k-1), {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A338495 Least number of hexagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002412, A104246, A234533, A338479, A338493, A338494.

N:= 200: # for a(1)..a(N)
V:= Vector(N):
S:= {seq(n*(n+1)*(4*n-1)/6,n=1..floor((N*3/2)^(1/3)))}:
V[convert(S,list)]:= 1:
T:= S:
for m from 2 do
  Tn:= select(`<=`,map(t -> op(t +~ S),T),N) minus T;
  if nops(Tn) = 0 then break fi;
  T:= T union Tn;
  V[convert(Tn,list)]:= m
od:
convert(V,list); # Robert Israel, Nov 02 2020

A366015 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 7, 76, 995, 14433, 223300, 3611016, 60305787, 1032115315, 18007816255, 319110233104, 5727667197044, 103913426353324, 1902498385538520, 35106179258551632, 652236828560562987, 12190651925663309175, 229059610932456616501, 4324334144117016053500, 81983637468108446363755

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for hexagonal pyramidal numbers (with signs).

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002293, A002412, A064087, A365754, A365816, A366014, A366016, A366017.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 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(4*n,n-k-1) * 3^k for n > 0.

A062368 Multiplicative with a(p^e) = (e+1)(e+2)(4*e+3)/6.

Original entry on oeis.org

1, 7, 7, 22, 7, 49, 7, 50, 22, 49, 7, 154, 7, 49, 49, 95, 7, 154, 7, 154, 49, 49, 7, 350, 22, 49, 50, 154, 7, 343, 7, 161, 49, 49, 49, 484, 7, 49, 49, 350, 7, 343, 7, 154, 154, 49, 7, 665, 22, 154, 49, 154, 7, 350, 49, 350, 49, 49, 7, 1078, 7, 49, 154, 252, 49, 343, 7, 154

Offset: 1

Vladeta Jovovic, Jul 07 2001

Conjecture: this is the third inverse Mobius transform of the sequence 4^A001221(n). - R. J. Mathar, Aug 09 2012

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001221, A002412, A007425, A060648, A062380.

f[p_, e_] := (e+1)*(e+2)*(4*e+3)/6; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(gcd(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Also a(n) = Sum_{i|n, j|n} tau(lcm(i, j)).
a(n) = Sum_{d|n} tau_3(d^2) = Sum_{d|n} A007425(d^2). - Enrique Pérez Herrero, Jan 17 2013

A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.

Original entry on oeis.org

1, 5, 7, 16, 24, 22, 38, 59, 65, 50, 75, 120, 141, 136, 95, 131, 215, 262, 274, 245, 161, 210, 352, 440, 480, 470, 400, 252, 316, 539, 687, 770, 790, 741, 609, 372, 453, 784, 1015, 1160, 1225, 1208, 1099, 880, 525, 625, 1095, 1436, 1666, 1795, 1825, 1750, 1556, 1221, 715, 836, 1480, 1962, 2304, 2520, 2616, 2590, 2432, 2124, 1640, 946, 1090, 1947, 2605, 3090, 3420, 3605, 3647, 3540

Offset: 1

Clark Kimberling, Feb 05 2011

See A144112 for the definition of accumulation array.

			Northwest corner:
   1,   5,  16,  38,  75
   7,  24,  59, 120, 215
  22,  54, 141, 262, 440
  50, 136, 174, 480, 770

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A185868.

Row 1: A174723; column 1: A002412.

f[n_,k_]:=2n-1+(2n+2k-4)(2n+2k-3)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185868 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]] (*formula for A185872 *)
g[n_]:=Sum[f[n+1-k,k],{k,1,n}];
Table[g[n],{n,50}] (* A185872 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]

T(n,k) = (k*n/6)*(4*n^2 + 6*n*k + 4*k^2 - 3*n - 9*k + 4), k>=1, n>=1.

A213836 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 2, 22, 13, 3, 50, 37, 19, 4, 95, 78, 52, 25, 5, 161, 140, 106, 67, 31, 6, 252, 227, 185, 134, 82, 37, 7, 372, 343, 293, 230, 162, 97, 43, 8, 525, 492, 434, 359, 275, 190, 112, 49, 9, 715, 678, 612, 525, 425, 320, 218, 127

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213837.
Antidiagonal sums: A071238.
Row 1, (1,5,9,13,...)**(1,2,3,4,...): A002412.
Row 2, (1,5,9,13,...)**(2,3,4,5,...): (4*k^3 + 15*k^2 - 7*k)/6.
Row 3, (1,5,9,13,...)**(3,4,5,6,...): (4*k^3 + 27*k^2 - 13*k)/6.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1...7....22...50....95
2...13...37...78....140
3...19...52...106...185
4...25...67...134...230
5...31...82...162...275
6...37...97...190...320

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500.

b[n_]:=4n-3;c[n_]:=n;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213836 *)
Table[t[n,n],{n,1,40}] (* A213837 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A071238 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(n + (2*n+1)*x - 3*(n-1)*x^2) and g(x) = (1-x)^4.

A213837 Principal diagonal of the convolution array A213836.

Original entry on oeis.org

1, 13, 52, 134, 275, 491, 798, 1212, 1749, 2425, 3256, 4258, 5447, 6839, 8450, 10296, 12393, 14757, 17404, 20350, 23611, 27203, 31142, 35444, 40125, 45201, 50688, 56602, 62959, 69775, 77066, 84848, 93137

Offset: 1

Clark Kimberling, Jul 04 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000384, A002412, A213836, A220084 (for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).

(See A213836.)
LinearRecurrence[{4,-6,4,-1},{1,13,52,134},40] (* Harvey P. Dale, Aug 07 2025 *)

a(n) = n*(5 - 15*n + 16*n^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: f(x)/g(x), where f(x) = x*(1 + 9*x + 6*x^2) and g(x) = (1-x)^4.
a(n) = n*A002412(n) - (n-1)*A002412(n-1). [Bruno Berselli, Dec 11 2012]
a(n) = n*A000384(n) + sum( A000384(i), i=0..n-1 ). [Bruno Berselli, Dec 18 2013]

cp's OEIS Frontend

A143690 a(n) = A007318 * [1, 6, 14, 9, 0, 0, 0, ...].

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A266677 Alternating sum of hexagonal pyramidal numbers.

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A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3).

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A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

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A338495 Least number of hexagonal pyramidal numbers needed to represent n.

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A366015 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 3 * A(x)).

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A062368 Multiplicative with a(p^e) = (e+1)*(e+2)*(4*e+3)/6.

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A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.

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A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4x + 3)(x + 1)^(2n) + (x^2 - 1)(x^2 + 3*x + 3).

A062368 Multiplicative with a(p^e) = (e+1)(e+2)(4*e+3)/6.