A129863 -id:A129863 - cp's OEIS frontend

A130423 Main diagonal of array A[k,n] = n-th sum of 3 consecutive k-gonal numbers, k>2.

4, 14, 39, 88, 170, 294, 469, 704, 1008, 1390, 1859, 2424, 3094, 3878, 4785, 5824, 7004, 8334, 9823, 11480, 13314, 15334, 17549, 19968, 22600, 25454, 28539, 31864, 35438, 39270, 43369, 47744, 52404, 57358, 62615, 68184, 74074, 80294, 86853

Offset: 1

Jonathan Vos Post, May 26 2007

The first row of the array is the sum of 3 consecutive triangular numbers = A000217(n) + A000217(n+1) + A000217(n+2) = Centered triangular numbers: 3*n*(n-1)/2 + 1, for n>1. The second row of the array is the sum of 3 consecutive squares = Number of points on surface of square pyramid: 3*n^2 + 2 (n>1). The first column of the array is k+1 = 4, 5, 6, 7, 8, 9, ... The second column of the array is A016825 = 4*n + 2 (for n>2). The third column of the array is A017377 = 10*n + 9 (for n>0).

			The array begins:
k / A[k,n]
3.|.4.10.19.31..46..64..85.109.136.166....=A005448(n+1).
4.|.5.14.29..50..77.110.149.194.245.302...=A005918(n).
5.|.6.18.39..69.108.156.213.279.354.438...=A129863(n).
6.|.7.22.49..88.139.202.277.364.463.574...
7.|.8.26.59.107.170.248.341.449.572.710...
8.|.9.30.69.126.201.294.405.534.681.846...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A005448, A005918, A016825, A017377, A129803, A129863.

I:=[4, 14, 39, 88]; [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..40]]; // Vincenzo Librandi, Jun 28 2012

P := proc(k,n) n*((k-2)*n-k+4)/2 ; end: A := proc(k,n) add( P(k,i),i=n..n+2) ; end: A130423 := proc(n) A(n+3,n) ; end: seq(A130423(n),n=0..40) ; # R. J. Mathar, Jun 14 2007

CoefficientList[Series[(4-2*x+7*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 28 2012 *)
Table[n (3n^2-3n+8)/2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{4,14,39,88},40] (* Harvey P. Dale, Aug 15 2012 *)

a(n) = A[n+2,n] = P(k+2,n) + P(k+2,n+1) + P(k+2,n+2) where P(k,n) = k*((n-2)*k - (n-4))/2.
a(n) = n*(3*n^2-3*n+8)/2. G.f.: x*(4-2*x+7*x^2)/(1-x)^4. [Colin Barker, Apr 30 2012]
a(1)=4, a(2)=14, a(3)=39, a(4)=88, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Aug 15 2012

More terms from R. J. Mathar, Jun 14 2007

A129109 Sums of three consecutive hexagonal numbers.

Original entry on oeis.org

7, 22, 49, 88, 139, 202, 277, 364, 463, 574, 697, 832, 979, 1138, 1309, 1492, 1687, 1894, 2113, 2344, 2587, 2842, 3109, 3388, 3679, 3982, 4297, 4624, 4963, 5314, 5677, 6052, 6439, 6838, 7249, 7672, 8107, 8554, 9013, 9484, 9967, 10462, 10969, 11488, 12019, 12562

Offset: 0

Jonathan Vos Post, May 24 2007

Arises in hexagonal number analog to A129803 Triangular numbers which are the sum of three consecutive triangular numbers. What are the hexagonal numbers which are the sum of three consecutive hexagonal numbers? Prime for a(0) = 7, a(4) = 139, a(6) = 277, a(8) = 463, a(18) = 2113, a(22) = 3109, a(26) = 4297, a(38) = 9013, a(40) = 9967.

			a(0) = H(0) + H(1) + H(2) = 0 + 1 + 6 = 7 = 6*0^2 + 9*0 + 7.
a(1) = H(1) + H(2) + H(3) = 1 + 6 + 15 = 22 = 6*1^2 + 9*1 + 7.
a(2) = H(2) + H(3) + H(4) = 6 + 15 + 28 = 49 = 6*2^2 + 9*2 + 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A007667, A034961, A129803, A129863.

I:=[7,22,49]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

LinearRecurrence[{3,-3,1},{7,22,49},50] (* Vincenzo Librandi, Feb 20 2012 *)
Total/@Partition[PolygonalNumber[6,Range[0,50]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 14 2020 *)

a(n)=6*n^2+9*n+7 \\ Charles R Greathouse IV, Feb 20 2012

a(n) = H(n) + H(n+1) + H(n+2) where H(n) = A000384(n) = n*(2*n-1).
a(n) = 6*n^2 + 9*n + 7.
From Colin Barker, Feb 20 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (7 + x + 4*x^2)/(1-x)^3. (End)
E.g.f.: (7 + 15*x + 6*x^2)*exp(x). - Elmo R. Oliveira, Nov 16 2024

A129111 Sums of three consecutive heptagonal numbers.

Original entry on oeis.org

8, 26, 59, 107, 170, 248, 341, 449, 572, 710, 863, 1031, 1214, 1412, 1625, 1853, 2096, 2354, 2627, 2915, 3218, 3536, 3869, 4217, 4580, 4958, 5351, 5759, 6182, 6620, 7073, 7541, 8024, 8522, 9035, 9563, 10106, 10664, 11237, 11825, 12428, 13046, 13679, 14327, 14990

Offset: 0

Jonathan Vos Post, May 24 2007

Arises in heptagonal number analog to A129803 (Triangular numbers which are the sum of three consecutive triangular numbers).
What are the heptagonal numbers which are the sum of three consecutive heptagonal numbers?
Prime for a(2) = 59, a(3) = 107, a(7) = 449, a(10) = 863, a(11) = 1031, a(23) = 4217, a(26) = 5351, a(31) = 7541, a(42) = 13679, a(43) = 14327, a(46) = 16361, a(51) = 20051.

			a(0) = Hep(0) + Hep(1) + Hep(2) = 0 + 1 + 7 = 8 = (15/2)*0^2 + (21/2)*0 + 8.
a(1) = Hep(1) + Hep(2) + Hep(3) = 1 + 7 + 18 = 26 = (15/2)*1^2 + (21/2)*1 + 8.
a(2) = Hep(2) + Hep(3) + Hep(4) = 7 + 18 + 34 = 59 = (15/2)*2^2 + (21/2)*2 + 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A007667, A034961, A129803, A129863.

I:=[8,26,59]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

LinearRecurrence[{3,-3,1},{8,26,59},50] (* Vincenzo Librandi, Feb 12 2012 *)

a(n)=3*n*(5*n+7)/2+8 \\ Charles R Greathouse IV, Jun 17 2017

def a(n): return 3*n*(5*n+7)//2 + 8
print([a(n) for n in range(44)]) # Michael S. Branicky, Aug 26 2021

a(n) = Hep(n) + Hep(n+1) + Hep(n+2) where Hep(n) = A000566(n) = n*(5*n-3)/2.
a(n) = (15/2)*n^2 + (21/2)*n + 8.
From Colin Barker, Feb 20 2012: (Start)
G.f.: (8 + 2*x + 5*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(16 + 36*x + 15*x^2)/2. - Elmo R. Oliveira, Nov 16 2024

A130424 Main diagonal of array A[k,n] = n-th sum of k consecutive k-gonal numbers, k>2.

Original entry on oeis.org

4, 30, 125, 365, 854, 1724, 3135, 5275, 8360, 12634, 18369, 25865, 35450, 47480, 62339, 80439, 102220, 128150, 158725, 194469, 235934, 283700, 338375, 400595, 471024, 550354, 639305, 738625, 849090, 971504, 1106699, 1255535, 1418900

Offset: 1

Jonathan Vos Post, May 26 2007

The first row of the array is the sum of 3 consecutive triangular numbers = A000217(n) + A000217(n+1) + A000217(n+2) = Centered triangular numbers: 3*n*(n-1)/2 + 1, for n>1. The second row of the array is the sum of 4 consecutive squares = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 = A027575(n). The third row of the array is the sum of 5 consecutive pentagonal numbers.

			The array begins:
k / A[k,n]
3.|...4..10..19...31...46...64...85..109.136.166...=A005448(n+1).
4.|..14..30..54...86..126..174..230..294.366.446...=A027575(n).
5.|..40..75.125..190..270..365..475..600.740...
6.|..95.161.251..365..503..665..851.1061.1295...
7.|.196.308.455..637..854.1106.1393.1715.2072...
8.|.364.540.764.1036.1356.1724.2140.2604.3116...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Polygonal Number.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A005448, A005918, A016825, A017377, A129803, A129863, A130423.

P := proc(k,n) n*((k-2)*n-k+4)/2 ; end: A := proc(k,n) add( P(k,i),i=n..n+k-1) ; end: A130424 := proc(n) A(n+3,n) ; end: seq(A130424(n),n=0..40) ; # R. J. Mathar, Oct 28 2007

LinearRecurrence[{5,-10,10,-5,1},{4,30,125,365,854},50] (* Harvey P. Dale, Jun 23 2020 *)

a(n) = A[n+2,n] = P(k+2,n) + P(k+2,n+1) + P(k+2,n+2) + ... P(k+2,n+k-1) where P(k,n) = k*((n-2)*k - (n-4))/2.
a(n) = (n+2)*(7*n^3-8*n^2+12*n-3)/6. [R. J. Mathar, Oct 30 2008]
G.f.: x*(4+10*x+15*x^2-x^4)/(1-x)^5. [Colin Barker, Sep 08 2012]

More terms from R. J. Mathar, Oct 28 2007

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A130423 Main diagonal of array A[k,n] = n-th sum of 3 consecutive k-gonal numbers, k>2.

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A129109 Sums of three consecutive hexagonal numbers.

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A129111 Sums of three consecutive heptagonal numbers.

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A130424 Main diagonal of array A[k,n] = n-th sum of k consecutive k-gonal numbers, k>2.

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